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## Progressive Hybrid and Adaptive Censoring and Related Inference

### The Art of Progressive Censoring (2014-01-01): 327-340 , January 01, 2014

Inferential results for progressive hybrid and adaptive progressive Type-II censored data are shown. A special focus is given to one- and two-parameter exponential distributions.

## Locally minimax test of independence in elliptically symmetrical distributions with additional observations

### Metrika (1992-12-01) 39: 75-84 , December 01, 1992

### Summary

Let*X*=(*X*_{ij})=(*X*_{1}, ...,*X*_{n})’,*X*’_{i}=(*X*_{i1}, ...,*X*_{ip})’,*i*=1,2, ...,*n* be a matrix having a multivariate elliptical distribution depending on a convex function*q* with parameters, 0,*σ*. Let ϱ^{2}=ϱ
_{2}^{-2}
be the squared multiple correlation coefficient between the first and the remaining*p*_{2}+*p*_{3}=*p*−1 components of each*X*_{i}. We have considered here the problem of testing*H*_{0}:ϱ^{2}=0 against the alternatives*H*_{1}:ϱ
_{1}^{-2}
=0, ϱ
_{2}^{-2}
>0 on the basis of*X* and*n*_{1} additional observations*Y*_{1} (*n*_{1}×1) on the first component,*n*_{2} observations*Y*_{2}(*n*_{2}×*p*_{2}) on the following*p*_{2} components and*n*_{3} additional observations*Y*_{3}(*n*_{3}×*p*_{3}) on the last*p*_{3} components and we have derived here the locally minimax test of*H*_{0} against*H*_{1} when ϱ
_{2}^{-2}
→0 for a given*q*. This test, in general, depends on the choice of*q* of the family*Q* of elliptically symmetrical distributions and it is not optimality robust for*Q*.

## Progressive Type-II Censoring: Distribution Theory

### The Art of Progressive Censoring (2014-01-01): 21-66 , January 01, 2014

The distribution theory of progressively Type-II censored order statistics is presented with a focus on particular baseline distributions like exponential and generalized Pareto distributions. The discussion includes joint, marginal, and conditional distributions as well as the fundamental quantile representation. The connection to generalized order statistics and sequential order statistics is highlighted. Further topics discussed are shapes of density functions, recurrence relations, exceedances, and discrete progressively Type-II censored order statistics.

## Exact two-sample nonparametric confidence, prediction, and tolerance intervals based on ordinary and progressively type-II right censored data

### TEST (2010-05-01) 19: 68-91 , May 01, 2010

It is shown how various exact nonparametric inferences based on an ordinary right or progressively Type-II right censored sample can be generalized to situations where two independent samples are combined. We derive the relevant formulas for the combined ordered samples to construct confidence intervals for a given quantile, prediction intervals, and tolerance intervals. The results are valid for every continuous distribution function. The key results are the derivations of the marginal distribution functions in the combined ordered samples. In the case of ordinary Type-II right censored order statistics, it is shown that the combined ordered sample is no longer distributed as order statistics. Instead, the distribution in the combined ordered sample is closely related to progressively Type-II censored order statistics.

## Selecting the Best Population Using a Test for Equality Based on Minimal Wilcoxon Rank-sum Precedence Statistic

### Methodology and Computing in Applied Probability (2007-06-01) 9: 263-305 , June 01, 2007

In this paper, we first give an overview of the precedence-type test procedures. Then we propose a nonparametric test based on early failures for the equality of two life-time distributions against two alternatives concerning the best population. This procedure utilizes the minimal Wilcoxon rank-sum precedence statistic (Ng and Balakrishnan, 2002, 2004) which can determine the difference between populations based on early (100*q*%) failures. Hence, this procedure can be useful in life-testing experiments in biological as well as industrial settings. After proposing the test procedure, we derive the exact null distribution of the test statistic in the two-sample case with equal or unequal sample sizes. We also present the exact probability of correct selection under the Lehmann alternative. Then, we generalize the test procedure to the *k*-sample situation. Critical values for some sample sizes are presented. Next, we examine the performance of this test procedure under a location-shift alternative through Monte Carlo simulations. Two examples are presented to illustrate our test procedure with selecting the best population as an objective.

## Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues

### Methodology and Computing in Applied Probability (2016-12-01) 18: 937-966 , December 01, 2016

Cure rate models offer a convenient way to model time-to-event data by allowing a proportion of individuals in the population to be completely cured so that they never face the event of interest (say, death). The most studied cure rate models can be defined through a competing cause scenario in which the random variables corresponding to the time-to-event for each competing causes are conditionally independent and identically distributed while the actual number of competing causes is a latent discrete random variable. The main interest is then in the estimation of the cured proportion as well as in developing inference about failure times of the susceptibles. The existing literature consists of parametric and non/semi-parametric approaches, while the expectation maximization (EM) algorithm offers an efficient tool for the estimation of the model parameters due to the presence of right censoring in the data. In this paper, we study the cases wherein the number of competing causes is either a binary or Poisson random variable and a piecewise linear function is used for modeling the hazard function of the time-to-event. Exact likelihood inference is then developed based on the EM algorithm and the inverse of the observed information matrix is used for developing asymptotic confidence intervals. The Monte Carlo simulation study demonstrates the accuracy of the proposed non-parametric approach compared to the results attained from the true correct parametric model. The proposed model and the inferential method is finally illustrated with a data set on cutaneous melanoma.

## Concepts of Stochastic Dependence

### Continuous Bivariate Distributions (2009-01-01): 105-140 , January 01, 2009

Dependence relations between two variables are studied extensively in probability and statistics. No meaningful statistical models can be constructed without some assumptions regarding dependence although in many cases one may simply assume the variables are not dependent, i.e., they are independent.

## Quantile-Quantile Plots and Goodness-of-Fit Test

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 39-40 , January 01, 1999

In any statistical study based on the assumption of particular distribution for the data at hand, one will naturally be interested in assessing the validity of that assumption; more specifically, one will be interested in testing for the hypothesis that the data has come from that specific distribution wherein only the functional form of the distribution is assumed to be known while it may involve some unknown parameters. For example, we may be interested in testing whether the data at hand has possibly arisen from the three-parameter lognormal distribution in (4.1), wherein we may assume that all three parameters μ, σ and *k* are unknown.

## Pooled parametric inference for minimal repair systems

### Computational Statistics (2015-06-01) 30: 605-623 , June 01, 2015

Consider two independent and identically structured systems, each with a certain number of observed repair times. The repair process is assumed to be performed according to a minimal-repair strategy. In this strategy, the state of the system after the repair is the same as it was immediately before the failure of the system. The resulting pooled sample is then used to obtain best linear unbiased estimators (BLUEs) as well as best linear invariant estimators of the location and scale parameters of the presumed parametric families of life distributions. It is observed that the BLUEs based on the pooled sample are overall more efficient than those based on one sample of the same size and also than those based on independent samples. Furthermore, the best linear unbiased predictor and the best linear invariant predictor of a future repair time from an independent system are also obtained. A real data set of Boeing air conditioners, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes, is used to illustrate the inferential results developed here.

## Precedence-type tests based on record values

### Metrika (2008-09-01) 68: 233-255 , September 01, 2008

Precedence-type tests based on order statistics are simple and efficient nonparametric tests that are very useful in the context of life-testing, and they have been studied quite extensively in the literature; see Balakrishnan and Ng (Precedence-type tests and applications. Wiley, Hoboken, 2006). In this paper, we consider precedence-type tests based on record values and develop specifically record precedence test, record maximal precedence test and record-rank-sum test. We derive their exact null distributions and tabulate some critical values. Then, under the general Lehmann alternative, we derive the exact power functions of these tests and discuss their power under the location-shift alternative. We also establish that the record precedence test is the uniformly most powerful test for testing against the one-parameter family of Lehmann alternatives. Finally, we discuss the situation when we have insufficient number of records to apply the record precedence test and then make some concluding remarks.

## Success runs of lengthk in Markov dependent trials

### Annals of the Institute of Statistical Mathematics (1994-12-01) 46: 777-796 , December 01, 1994

The geometric type and inverse Polýa-Eggenberger type distributions of waiting time for success runs of length*k* in two-state Markov dependent trials are derived by using the probability generating function method and the combinatorial method. The second is related to the minimal sufficient partition of the sample space. The first two moments of the geometric type distribution are obtained. Generalizations to ballot type probabilities of which negative binomial probabilities are special cases are considered. Since the probabilities do not form a proper distribution, a modification is introduced and new distributions of order*k* for Markov dependent trials are developed.

## Relationships between moments of two related sets of order statistics and some extensions

### Annals of the Institute of Statistical Mathematics (1993-06-01) 45: 243-247 , June 01, 1993

Govindarajulu expressed the moments of order statistics from a symmetric distribution in terms of those from its folded form. He derived these relations analytically by dividing the range of integration suitably into parts. In this paper, we establish these relations through probabilistic arguments which readily extend to the independent and non-identically distributed case. Results for random variables having arbitrary multivariate distributions are also derived.

## Front Matter - The Art of Progressive Censoring

### The Art of Progressive Censoring (2014-01-01) , January 01, 2014

## Robust Clustering in Regression Analysis via the Contaminated Gaussian Cluster-Weighted Model

### Journal of Classification (2017-07-01) 34: 249-293 , July 01, 2017

The Gaussian cluster-weighted model (CWM) is a mixture of regression models with random covariates that allows for flexible clustering of a random vector composed of response variables and covariates. In each mixture component, a Gaussian distribution is adopted for both the covariates and the responses given the covariates. To make the approach robust with respect to the presence of mildly atypical observations, the contaminated Gaussian CWM is introduced. In addition to the parameters of the Gaussian CWM, each mixture component has a parameter controlling the proportion of outliers, one controlling the proportion of leverage points, one specifying the degree of contamination with respect to the response variables, and another specifying the degree of contamination with respect to the covariates. Crucially, these parameters do not have to be specified *a priori*, adding flexibility to the approach. Furthermore, once the model is estimated and the observations are assigned to the components, a finer intra-group classification in typical points, (mild) outliers, good leverage points, and bad leverage points—concepts of primary importance in robust regression analysis—can be directly obtained. Relations with other mixture-based contaminated models are analyzed, identifiability conditions are provided, an expectation-conditional maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments and compared with other procedures. A sensitivity study is also conducted based on a real data set.

## General relations and identities for order statistics from non-independent non-identical variables

### Annals of the Institute of Statistical Mathematics (1992-03-01) 44: 177-183 , March 01, 1992

Some recurrence relations and identities for order statistics are extended to the most general case where the random variables are assumed to be non-independent non-identically distributed. In addition, some new identities are given. The results can be used to reduce the computations considerably and also to establish some interesting combinatorial identities.

## On the identifiability of start-up demonstration mixture models

### Annals of the Institute of Statistical Mathematics (2017-08-01) 69: 717-735 , August 01, 2017

In start-up demonstration testing, the performance of the unit on successive start-ups is taken into account and several different types of decision criteria (most of them are inspired by the theory of runs and scans) for accepting or rejecting the unit have been introduced. Although the use of a start-up demonstration test assumes the existence of units of lower quality, when the estimation of the respective probability comes up, there is still much work to be done. Therefore, in this paper, we study binary start-up demonstration tests, assuming that we have at hand two different types of units with potentially different probabilities of successful start-up. In this case, the waiting time distributions are expressed as two-component mixture models and their identifiability is discussed. Finally, an estimation method based on the EM algorithm for the model parameters is described and some numerical examples are presented to illustrate the methods developed here.

## Progressive Hybrid Censoring: Distributions and Properties

### The Art of Progressive Censoring (2014-01-01): 125-142 , January 01, 2014

The distribution theory of progressive hybrid censored order statistics is developed. A special emphasis is given to the exponential distribution where also the distributions of spacings and the total time on test statistic are discussed.

## Accelerated Life Testing

### The Art of Progressive Censoring (2014-01-01): 481-505 , January 01, 2014

Methods of accelerated life testing are applied to several kinds of progressively censored data. This includes step-stress testing as well as progressive stress models.

## Front Matter - The Art of Progressive Censoring

### The Art of Progressive Censoring (2014-01-01) , January 01, 2014

## Review of Designs for Accommodating Patients’ or Physicians’ Preferences in Randomized Controlled Trials

### Developments in Statistical Evaluation of Clinical Trials (2014-01-01): 305-333 , January 01, 2014

The randomized controlled trial (RCT) is regarded as the principal way to collect scientific data on the efficacy of health interventions. Despite the advantages of RCT design in reducing extraneous variation that may confound interpretation of intervention results, the design may not be suitable for interventions in which patients are likely to have a strong preference for a particular treatment. Some designs incorporating patients or physician preferences by allowing at least a subgroup of them to choose their treatment have been proposed. In this chapter, we review various randomized control trials designs for accommodating participants’ and professionals’ preferences. Specifically, we discuss the advantages, limitations, applicability, ethical issues and statistical issues of each design. We also discuss the estimation of treatment effect (a measure of the extent to which treatment difference is attributable to treatments); selection effect (a measure of the extent to which treatment response is influenced by self-selection of treatment by patients); and preference effect (a measure of the extent to which treatment difference is caused by an interaction between the patient’s choice of treatment and the treatment actually received).

## Recurrence relations for order statistics from n independent and non-identically distributed random variables

### Annals of the Institute of Statistical Mathematics (1988-06-01) 40: 273-277 , June 01, 1988

Some well-known reeurrence relations for order statistics in the i.i.d. case are generalized to the case when the variables are independent and non-identically distributed. These results could be employed in order to reduce the amount of direct computations involved in evaluating the moments of order statistics from an outlier model.

## Short-tailed distributions and inliers

### TEST (2008-08-01) 17: 282-296 , August 01, 2008

We consider two families of short-tailed distributions (kurtosis less than 3) and discuss their usefulness in modeling numerous real life data sets. We develop estimation and hypothesis testing procedures which are efficient and robust to short-tailed distributions and inliers.

## Elliptically Symmetric Bivariate Distributions and Other Symmetric Distributions

### Continuous Bivariate Distributions (2009-01-01): 591-622 , January 01, 2009

This chapter is devoted to describing a class of bivariate distributions whose contours of probability densities are ellipses; in particular, those ellipses with constant eccentricity. These distributions are generally known as elliptically contoured or elliptically symmetric distributions. A subclass of distributions with contours that are circles are known as spherically symmetric (or simply spherical) distributions. The chapter also includes other symmetric bivariate distributions.

## Simulation of Progressively Censored Order Statistics

### The Art of Progressive Censoring (2014-01-01): 193-200 , January 01, 2014

Several accounts to the simulation of progressively censored data are presented. This includes procedures mimicking the generation process ofprogressively censored order statistics as well as methods based on the quantile representation.

## Bayesian Inference for Progressively Type-II Censored Data

### The Art of Progressive Censoring (2014-01-01): 341-353 , January 01, 2014

Bayesian approaches for progressively Type-II censored data are reviewed. The presentation includes, e.g., exponential, Weibull, Pareto, and Burr distributions.

## On properties of dependent progressively Type-II censored order statistics

### Metrika (2013-10-01) 76: 909-917 , October 01, 2013

In the context of life-testing, progressive censoring has been studied extensively. But, all the results have been developed under the key assumption that the units under test are independently distributed. In this paper, we consider progressively Type-II censored order statistics (PCOS-II) arising from dependent units that are jointly distributed according to an Archimedean copula. Density and distribution functions of dependent general PCOS-II (GPCOS-II) are derived under this set-up. These results include those in Kamps and Cramer (Statistics 35:269–280, 2001) as special cases. Some bounds for the mean of PCOS-II from dependent data are then established. Finally, through an example, a special case of PCOS-II from $$N$$ dependent components is illustrated.

## Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under Type-I and Type-II censoring

### Metrika (2010-07-01) 72: 89-109 , July 01, 2010

In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum likelihood estimators
$${\hat{\theta}_{1}}$$
and
$${\hat{\theta}_{2}}$$
of the parameters *θ*_{1} and *θ*_{2} have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively. Here, we prove that these estimators are stochastically monotone with respect to *θ*_{1} and *θ*_{2}, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters *θ*_{1} and *θ*_{2}. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions under truncation, which is also of independent interest.

## Precedence-Type Tests for the Comparison of Treatments with a Control

### Scan Statistics (2009-01-01): 27-54 , January 01, 2009

Precedence-type tests are proposed for comparing several treatments with a control. The null distributions of these test statistics are derived, and critical values for some combination of sample sizes are then presented. Next, the exact power function of these tests under the Lehmann alternative is derived and used to compare the power properties of the proposed test procedures. Finally, an example is presented to illustrate all the test procedures discussed here.

## Progressive Type-II Censoring Under Nonstandard Conditions

### The Art of Progressive Censoring (2014-01-01): 229-244 , January 01, 2014

The mixture representation for progressively Type-II censored order statistics from arbitrary baseline distributions is proven. Furthermore, results for INID progressively Type-II censored order statistics and their connection to permanents are shown. Finally, results on progressively censored samples from dependent samples are presented.

## Compound weighted Poisson distributions

### Metrika (2013-05-01) 76: 543-558 , May 01, 2013

In this paper, we discuss discrete compound distributions, in which the counting distribution is a weighted Poisson distribution. The over- and under-dispersion of these distributions are then discussed by analyzing the Fisher index of dispersion as well as a newly introduced factorial moment to mean measure. Several cases of compounding distributions and weight functions are subsequently examined in detail.

## Families of Parsimonious Finite Mixtures of Regression Models

### Advances in Statistical Models for Data Analysis (2015-01-01): 73-84 , January 01, 2015

Finite mixtures of regression (FMR) models offer a flexible framework for investigating heterogeneity in data with functional dependencies. These models can be conveniently used for unsupervised learning on data with clear regression relationships. We extend such models by imposing an eigen-decomposition on the multivariate error covariance matrix. By constraining parts of this decomposition, we obtain families of parsimonious mixtures of regressions and mixtures of regressions with concomitant variables. These families of models account for correlations between multiple responses. An expectation-maximization algorithm is presented for parameter estimation and performance is illustrated on simulated and real data.

## A Survey of Modeling and Application of Non-destructive and Destructive Degradation Tests

### Statistical Modeling for Degradation Data (2017-01-01): 105-124 , January 01, 2017

These days, most products are highly reliable which makes it very difficult or even impossible to obtain failure data on such products within a reasonable period of time prior to product release. Degradation tests are one way to overcome this obstacle by collecting degradation data (measurement of degradation) on such products. Based on different measurement processes, degradation tests can be divided into non-destructive and destructive degradation tests. In this chapter, we discuss a number of these two types of degradation models that have been developed in the literature to describe the degradation paths of products. In addition, some applications of degradation models of these two classes are also discussed.

## Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables

### Annals of the Institute of Statistical Mathematics (1989-06-01) 41: 323-329 , June 01, 1989

Some recurrence relations among moments of order statistics from two related sets of variables are quite well-known in the i.i.d. case and are due to Govindarajulu (1963*a*, *Technometrics*, *5*, 514–518 and 1966, *J. Amer. Statist. Assoc.*, *61*, 248–258). In this paper, we generalize these results to the case when the order statistics arise from two related sets of independent and non-identically distributed random variables. These relations can be employed to simplify the evaluation of the moments of order statistics in an outlier model for symmetrically distributed random variables.

## Forms of four-word indicator functions with implications to two-level factorial designs

### Annals of the Institute of Statistical Mathematics (2011-04-01) 63: 375-386 , April 01, 2011

Indicator functions are new tools to study fractional factorial designs. In this paper, we study indicator functions with four words and provide possible forms of the indicator functions and explain their implications to two-level factorial designs.

## Back Matter - Handbook of Tables for Order Statistics from Lognormal Distributions with Applications

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01) , January 01, 1999

## Measures of Dependence

### Continuous Bivariate Distributions (2009-01-01): 141-177 , January 01, 2009

A measure of dependence indicates in some particular manner how closely the variables *X* and *Y* are related; one extreme will include a case of complete linear dependence, and the other extreme will be complete mutual independence. Although it is customary in bivariate data analysis to compute a correlation measure of some sort, one number (or index) alone can never fully reveal the nature of dependence; hence a variety of measures are needed.

## Characterization of Bivariate Generalized Logistic Family of Distributions Through Conditional Specification

### Sankhya B (2017-05-01) 79: 170-186 , May 01, 2017

The univariate logistic distribution and its properties and applications have been studied quite extensively in the literature. Some generalizations as well as multivariate extensions of it have also been proposed for greater flexibility in modeling univariate and multivariate data. In this paper, we construct three different types of generalized bivariate logistic type distributions through conditional specification, and discuss some of their properties. Finally, we use a data set to fit the proposed models for the purpose of illustration.

## Likelihood ratio order of parallel systems with heterogeneous Weibull components

### Metrika (2016-08-01) 79: 693-703 , August 01, 2016

In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let
$$X_{1},\ldots ,X_{n}$$
be independent Weibull random variables with
$$X_{i}$$
having shape parameter
$$0<\alpha \le 1$$
and scale parameter
$$\lambda _{i}$$
,
$$i=1,\ldots ,n$$
, and
$$Y_{1},\ldots ,Y_{n}$$
be a random sample of size *n* from a Weibull distribution with shape parameter
$$0<\alpha \le 1$$
and a common scale parameter
$$\overline{\lambda }=\frac{1}{n}\sum \nolimits _{i=1}^{n}\lambda _{i}$$
, the arithmetic mean of
$$\lambda _{i}^{'}s$$
. Let
$$X_{n:n}$$
and
$$Y_{n:n}$$
denote the corresponding largest order statistics, respectively. We then prove that
$$X_{n:n}$$
is stochastically larger than
$$Y_{n:n}$$
in terms of the likelihood ratio order, and provide numerical examples to illustrate the results established here.

## Efficient iterative computation of mixture weights for pooled order statistics for meta-analysis of multiple type-II right censored data

### Computational Statistics (2013-10-01) 28: 2231-2239 , October 01, 2013

This paper considers computation of mixture weights of the marginal distribution of pooled order statistics that arise from combining and ordering multiple independent Type-II right censored samples. The proposed method is an iterative procedure which is computationally efficient and produces the same mixture representations as direct methods. It is shown that the resultant mixtures are independent of the order in which the samples are entered into the algorithm. Some comparative computational results are finally presented.

## Some binary start-up demonstration tests and associated inferential methods

### Annals of the Institute of Statistical Mathematics (2014-08-01) 66: 759-787 , August 01, 2014

During the past few decades, substantial research has been carried out on start-up demonstration tests. In this paper, we study the class of binary start-up demonstration tests under a general framework. Assuming that the outcomes of the start-up tests are described by a sequence of exchangeable random variables, we develop a general form for the exact waiting time distribution associated with the length of the test (i.e., number of start-ups required to decide on the acceptance or rejection of the equipment/unit under inspection). Approximations for the tail probabilities of this distribution are also proposed. Moreover, assuming that the probability of a successful start-up follows a beta distribution, we discuss several estimation methods for the parameters of the beta distribution, when several types of observed data have been collected from a series of start-up tests. Finally, the performance of these estimation methods and the accuracy of the suggested approximations for the tail probabilities are illustrated through numerical experimentation.

## Robust classification procedures based on dichotomous and continuous variables

### Journal of Classification (1988-03-01) 5: 53-80 , March 01, 1988

For classifying a univariate or a multivariate observation in one of the two populations, Tiku and Balakrishnan (1984) and Balakrishnan, Tiku and Shaarawi (1985) developed robust (to departures from normality) procedures. These procedures are extended here to situations where the classification has to be based on the observed value of a pair of variables, one being a dichotomous random variable and the other a univariate or a multivariate continuous random variable.

## On a Multiparameter Version of Tukey's Linear Sensitivity Measure and its Properties

### Annals of the Institute of Statistical Mathematics (2002-12-01) 54: 796-805 , December 01, 2002

A multiparameter version of Tukey's (1965, *Proc. Nat. Acad. Sci. U.S.A.*, *53*, 127–134) linear sensitivity measure, as a measure of informativeness in the joint distribution of a given set of random variables, is proposed. The proposed sensitivity measure, under some conditions, is a matrix which is non-negative definite, weakly additive, monotone and convex. Its relation to Fisher information matrix and the best linear unbiased estimator (BLUE) are investigated. The results are applied to the location-scale model and it is observed that the dispersion matrix of the BLUE of the vector location-scale parameter is the inverse of the sensitivity measure. A similar property was established by Nagaraja (1994, *Ann. Inst. Statist. Math.*, *46*, 757–768) for the single parameter case when applied to the location and scale models. Two illustrative examples are included.

## Dispersive ordering of fail-safe systems with heterogeneous exponential components

### Metrika (2011-09-01) 74: 203-210 , September 01, 2011

Let *X*_{1}, . . . , *X*_{n} be independent exponential random variables with respective hazard rates λ_{1}, . . . , λ_{n}, and *Y*_{1}, . . . , *Y*_{n} be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that *X*_{2:n}, the second order statistic from *X*_{1}, . . . , *X*_{n}, is larger than *Y*_{2:n}, the second order statistic from *Y*_{1}, . . . , *Y*_{n}, in terms of the dispersive order if and only if
$$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$$
We also show that *X*_{2:n} is smaller than *Y*_{2:n} in terms of the dispersive order if and only if
$$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $$
Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.

## Stress–Strength Models with Progressively Censored Data

### The Art of Progressive Censoring (2014-01-01): 507-513 , January 01, 2014

Step-stress models based on two progressively Type-II censored data sets are reviewed. The discussion includes likelihood inference as well minimum variance unbiased estimation. A special emphasis is put on exponentially distributed stress and strength.

## A procedure for outlier identification in data sets from continuous distributions

### Test (2004-06-01) 13: 247-262 , June 01, 2004

We propose a procedure, based on sums of reciprocals of*p*-values, for the identification of outliers in univariate or multivariate data sets coming from continuous distributions. Using results of Csörgő (1990), we find the limiting distribution of the relevant statistic for completely specified models. By simulations, we obtain approximate quantiles for the asymptotic distribution, (which does not depend on the specific model or the dimension where the data live) and for the finite sample distribution in different dimensions of our statistic when parameters are estimated, for the multivariate Gaussian model and a multivariate double exponential model with independent coordinates. Monte Carlo evaluation shows that the procedure proposed is effective in the identification of outliers, and that it is sensitive to sample size, a feature seldom found in outlier identification methods.

## Representations of the inactivity time for coherent systems with heterogeneous components and some ordered properties

### Metrika (2016-01-01) 79: 113-126 , January 01, 2016

In this paper, we present several useful mixture representations for the reliability function of the inactivity time of systems with heterogeneous components based on order statistics, signatures and mean reliability functions. Some stochastic comparisons of inactivity times between two systems are discussed. These results form nice extensions of some existing results for the case when the components are independent and identically distributed.

## Likelihood inference for the destructive exponentially weighted Poisson cure rate model with Weibull lifetime and an application to melanoma data

### Computational Statistics (2017-06-01) 32: 429-449 , June 01, 2017

In this paper, we develop the steps of the expectation maximization algorithm (EM algorithm) for the determination of the maximum likelihood estimates (MLEs) of the parameters of the destructive exponentially weighted Poisson cure rate model in which the lifetimes are assumed to be Weibull. This model is more flexible than the promotion time cure rate model as it provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest by including a destructive process of the initial number of causes in a competitive scenario. The standard errors of the MLEs are obtained by inverting the observed information matrix. An extensive Monte Carlo simulation study is carried out to evaluate the performance of the developed method of estimation. Finally, a known melanoma data are analyzed to illustrate the method of inference developed here. With these data, a comparison is also made with the scenario when the destructive mechanism is not included in the analysis.

## Stochastic Order and MLE of the Mean of the Exponential Distribution

### Methodology And Computing In Applied Probability (2002-03-01) 4: 83-93 , March 01, 2002

The maximum likelihood estimator of the mean of the exponential distribution, based on various data structures has been studied extensively. However, the order preserving property of these estimators is not found in the literature. This article discusses this property. Suppose that two samples of the same size are drawn from two independent exponential populations that have different means. It is shown in this article that the regular stochastic ordering holds between the two MLEs corresponding to the two exponential means, based on various censored data. In particular, conditions are given on inspection times so that the result is also true for grouped data.

## FIFO Versus LIFO Issuing Policies for Stochastic Perishable Inventory Systems

### Methodology and Computing in Applied Probability (2011-06-01) 13: 405-417 , June 01, 2011

We consider an inventory system for perishable items in which the arrival times of the items to be stored and the ones of the demands for those items form independent Poisson processes. The shelf lifetime of every item is finite and deterministic. Every demand is for a single item and is satisfied by one of the items on the shelf, if available. A demand remains unsatisfied if it arrives at an empty shelf. The aim of this paper is to compare two issuing policies: under FIFO (‘first in, first out’) any demand is satisfied by the item with the currently longest shelf life, while under LIFO (‘last in, first out’) always the youngest item on the shelf is assigned first. We determine the long-run net average profit as a function of the system parameters under each of the two policies, taking into account the revenue earned from satisfied demands, the cost of shelf space, penalties for unsatisfied demands, and the purchase cost of incoming items. The analytical results are used in several numerical examples in which the optimal input rate and the maximum expected long-run average profit under FIFO and under LIFO are determined and compared. We also provide a sensitivity analysis of the optimal solution for varying parameter values.

## Order Statistics and Moments

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 7-12 , January 01, 1999

Let *Z*_{1}, Z_{1},…, *Z*_{n} be a random sample of size *n* from the standard lognormal distribution in (2.3). Let *Z*_{1:n} ≤*Z*_{2:n} ≤ … ≤ *Z*_{n:n} be the order statistics obtained by arranging this sample in increasing order of magnitude.

## Generating Functions of Waiting Times and Numbers of Visits for Random Walks on Graphs

### Methodology and Computing in Applied Probability (2013-06-01) 15: 349-362 , June 01, 2013

In this paper, we consider some cover time problems for random walks on graphs in a wide class of waiting time problems. By using generating functions, we present a unified approach for the study of distributions associated with waiting times. In addition, the distributions of the numbers of visits for the random walks on the graphs are also studied. We present the relationship between the distributions of the waiting times and the numbers of visits. We also show that these theoretical results can be easily carried out through some computer algebra systems and present some numerical results for cover times in order to demonstrate the usefulness of the results developed. Finally, the study of cover time problems through generating functions leads to more extensive development.

## Bivariate Normal Distribution

### Continuous Bivariate Distributions (2009-01-01): 477-561 , January 01, 2009

In introductory statistics courses, one has to know why the (univariate) normal distribution is important—especially that the random variables that occur in many situations are approximately normally distributed and that it arises in theoretical work as an approximation to the distribution of many statistics, such as averages of independent random variables. More or less, the same reasons apply to the bivariate normal distribution. “But the prime stimulus has undoubtedly arisen from the strange tractability of the normal model: a facility of manipulation which is absent when we consider almost any other multivariate data-generating mechanism.”—Barnett (1979).

## Dual connections in nonparametric classical information geometry

### Annals of the Institute of Statistical Mathematics (2010-10-01) 62: 873-896 , October 01, 2010

We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixture can be extended to an open segment containing the extreme points. For this class, we define an infinite-dimensional analogue of the mixture parallel transport and prove that it is dual to the exponential parallel transport with respect to the Fisher information. We also define α-derivatives and prove that they are convex mixtures of the extremal (±1)-derivatives.

## Computational aspects of statistical intervals based on two Type-II censored samples

### Computational Statistics (2013-06-01) 28: 893-917 , June 01, 2013

In this paper, we propose an efficient branch and bound procedure to compute exact nonparametric statistical intervals based on two Type-II right censored data sets. The procedure is based on some recurrence relations for the distribution and density functions of progressively Type-II censored order statistics which can be applied to compute the coverage probabilities. We illustrate the method for both confidence and prediction intervals of a given level.

## Point Prediction from Progressively Type-II Censored Samples

### The Art of Progressive Censoring (2014-01-01): 355-377 , January 01, 2014

Several prediction problems for progressively Type-II censored data are considered. This includes prediction of progressively censored lifetime as well as prediction of future observations. After introducing several concepts of point prediction, applications to exponential, extreme value, normal, and Pareto distributions are presented.

## Prediction of censored exponential lifetimes in a simple step-stress model under progressive Type II censoring

### Computational Statistics (2017-12-01) 32: 1665-1687 , December 01, 2017

In this article, we consider the problem of predicting survival times of units from the exponential distribution which are censored under a simple step-stress testing experiment. Progressive Type-II censoring are considered for the form of censoring. Two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP)—are derived. Some numerical examples are presented to illustrate the prediction methods developed here. Using simulation studies, prediction intervals are generated for these examples. We then compare the MLP and the CMP with respect to mean squared prediction error and the prediction interval.

## Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

### Annals of the Institute of Statistical Mathematics (1996-12-01) 48: 773-787 , December 01, 1996

The probability generating functions of the waiting times for the first success run of length *k* and for the sooner run and the later run between a success run of length *k* and a failure run of length *r* in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^{.m}conditional probability generating functions of the waiting times in the *m*-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If *m* is large, some computer algebra systems are available to solve the linear systems of equations.

## Progressive censoring methodology: an appraisal

### TEST (2007-06-28) 16: 211-259 , June 28, 2007

Properties of progressively censored order statistics and inferential procedures based on progressively censored samples have recently attracted considerable attention in the literature. In this paper, I provide an overview of various developments that have taken place in this direction and also suggest some potential problems of interest for further research.

## Characterizations of Proportional Hazard and Reversed Hazard Rate Models Based on Symmetric and Asymmetric Kullback-Leibler Divergences

### Sankhya B (2017-11-06): 1-13 , November 06, 2017

Kullback-Leibler divergence
$(\mathcal {K}\mathcal {L})$
is widely used for selecting the best model from a given set of candidate parametrized probabilistic models as an approximation to the true density function *h*(·). In this paper, we obtain a necessary and sufficient condition to determine proportional hazard and reversed hazard rate models based on symmetric and asymmetric Kullback-Leibler divergences. Obtained results show that if *h* belongs to proportional hazard rate (reversed hazard) model, then the Kullback-Leibler divergence between *h* and baseline density function, *f*_{0}, is independent of the choice of *ξ*, the cut point of left (right) truncated distribution.

## Generalized mixtures of Weibull components

### TEST (2014-09-01) 23: 515-535 , September 01, 2014

Weibull mixtures have been used extensively in reliability and survival analysis, and they have also been generalized by allowing negative mixing weights, which arise naturally under the formation of some structures of reliability systems. These models provide flexible distributions for modeling dependent lifetimes from heterogeneous populations. In this paper, we study conditions on the mixing weights and the parameters of the Weibull components under which the considered generalized mixture is a well-defined distribution. Specially, we characterize the generalized mixture of two Weibull components. In addition, some reliability properties are established for these generalized two-component Weibull mixture models. One real data set is also analyzed for illustrating the usefulness of the studied model.

## Visualizing hypothesis tests in multivariate linear models: the heplots package for R

### Computational Statistics (2009-05-01) 24: 233-246 , May 01, 2009

Hypothesis-error (or “HE”) plots, introduced by Friendly (J Stat Softw 17(6):1–42, 2006a; J Comput Graph Stat 16:421–444, 2006b), permit the visualization of hypothesis tests in multivariate linear models by representing hypothesis and error matrices of sums of squares and cross-products as ellipses. This paper describes the implementation of these methods in the *heplots* package for R, as well as their extension, for example from two to three dimensions and by scaling hypothesis ellipses and ellipsoids in a natural manner relative to error.

## On inaccuracy generating functions of probability distributions

### Metrika (1972-12-01) 19: 185-192 , December 01, 1972

## Progressive Censoring: Data and Models

### The Art of Progressive Censoring (2014-01-01): 3-20 , January 01, 2014

The notion of progressive censoring is explained by introducing progressive Type-I and Type-II censoring in detail. The presentation includes detailed descriptions of the procedures as well as graphical illustrations and data. Additionally, progressive hybrid censoring is discussed. Finally, the chapter is supplemented by introducing particular probability models assumed in progressive censoring.

## Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution

### Annals of the Institute of Statistical Mathematics (2003-06-01) 55: 319-330 , June 01, 2003

Chen and Bhattacharyya (1988,*Comm. Statist. Theory Methods*,*17*, 1857–1870) derived the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution and an exact lower confidence bound for the mean based on a hybrid censored sample. In this paper, an alternative simple form for the distribution is obtained and is shown to be equivalent to that of Chen and Bhattacharyya (1988). Noting that this scheme, which would guarantee the experiment to terminate by a fixed time*T*, may result in few failures, we propose a new hybrid censoring scheme which guarantees at least a fixed number of failures in a life testing experiment. The exact distribution of the MLE as well as an exact lower confidence bound for the mean is also obtained for this case. Finally, three examples are presented to illustrate all the results developed here.

## Mixture model averaging for clustering

### Advances in Data Analysis and Classification (2015-06-01) 9: 197-217 , June 01, 2015

In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the ‘best’ one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw away all but the best model, we average multiple models that are in some sense close to the best one, thereby producing a weighted average of clustering results. Two (weighted) averaging approaches are considered: averaging component membership probabilities and averaging models. In both cases, Occam’s window is used to determine closeness to the best model and weights are computed within a Bayesian model averaging paradigm. In some cases, we need to merge components before averaging; we introduce a method for merging mixture components based on the adjusted Rand index. The effectiveness of our model-based clustering averaging approaches is illustrated using a family of Gaussian mixture models on real and simulated data.

## Nonparametric confidence intervals for ranked set samples

### Computational Statistics (2017-12-01) 32: 1689-1725 , December 01, 2017

In this work, we propose several different confidence interval methods based on ranked-set samples. First, we develop bootstrap bias-corrected and accelerated method for constructing confidence intervals based on ranked-set samples. Usually, for this method, the accelerated constant is computed by employing jackknife method. Here, we derive an analytical expression for the accelerated constant, which results in reducing the computational burden of this bias-corrected and accelerated bootstrap method. The other proposed confidence interval approaches are based on a monotone transformation along with normal approximation. We also study the asymptotic properties of the proposed methods. The performances of these methods are then compared with those of the conventional methods. Through this empirical study, it is shown that the proposed confidence intervals can be successfully applied in practice. The usefulness of the proposed methods is further illustrated by analyzing a real-life data on shrubs.

## Ordering properties of the smallest order statistics from generalized Birnbaum–Saunders models with associated random shocks

### Metrika (2017-10-20): 1-17 , October 20, 2017

Let $$X_{1},\ldots , X_{n}$$ be lifetimes of components with independent non-negative generalized Birnbaum–Saunders random variables with shape parameters $$\alpha _{i}$$ and scale parameters $$\beta _{i},~ i=1,\ldots ,n$$ , and $$I_{p_{1}},\ldots , I_{p_{n}}$$ be independent Bernoulli random variables, independent of $$X_{i}$$ ’s, with $$E(I_{p_{i}})=p_{i},~i=1,\ldots ,n$$ . These are associated with random shocks on $$X_{i}$$ ’s. Then, $$Y_{i}=I_{p_{i}}X_{i}, ~i=1,\ldots ,n,$$ correspond to the lifetimes when the random shock does not impact the components and zero when it does. In this paper, we discuss stochastic comparisons of the smallest order statistic arising from such random variables $$Y_{i},~i=1,\ldots ,n$$ . When the matrix of parameters $$(h({\varvec{p}}), {\varvec{\beta }}^{\frac{1}{\nu }})$$ or $$(h({\varvec{p}}), {\varvec{\frac{1}{\alpha }}})$$ changes to another matrix of parameters in a certain mathematical sense, we study the usual stochastic order of the smallest order statistic in such a setup. Finally, we apply the established results to two special cases: classical Birnbaum–Saunders and logistic Birnbaum–Saunders distributions.

## Joint Distributions of Numbers of Success-Runs and Failures Until the First Consecutive k Successes in a Binary Sequence

### Annals of the Institute of Statistical Mathematics (1997-09-01) 49: 519-529 , September 01, 1997

Joint distributions of the numbers of failures, successes andsuccess-runs of length less than k until the first consecutive k successesin a binary sequence were derived recently by Aki and Hirano (1995, Ann.Inst. Statist. Math., 47, 225-235). In this paper, we present an alternatederivation of these results and also use this approach to establish someadditional results. Extensions of these results to binary sequences of orderh are also presented.

## Best Linear Unbiased Prediction

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 31-38 , January 01, 1999

Prediction problems arise naturally in life-testing experiments. For example, let us consider the life-test experiment described in Example 3 of Chapter 5. In this case, twenty three ball bearings were placed on a life-test and the data on the number of million revolutions before failure of each of these ball bearings were observed. The experiment itself was terminated as soon as the twentieth ball bearing failed with three ball bearings still surviving at the time of termination of the experiment. It is, therefore, natural for the experimenter to be interested in predicting the number of million revolutions before failure for the remaining three surviving bearings. In particular, the experimenter may be interested in predicting the very next failure or the very last failure.

## Two-Stage Start-Up Demonstration Testing

### Statistical and Probabilistic Models in Reliability (1999-01-01): 251-263 , January 01, 1999

Start-up demonstration tests and various extensions and generalizations of them (in order to accommodate dependence between the trials, to allow for corrective action to be taken once the equipment fails for the first time, etc.) have been discussed quite extensively in the literature. In this paper, we propose a start-up demonstration test to be performed in two stages which would facilitate an early rejection of a potentially bad equipment and would also enable the experimenter to place a more stringent requirement for acceptance upon observing a certain number of failures. Specifically, the decision procedure proposed is as follows. Perform start-up demonstration tests on the equipment under study consecutively and decide to: 1.

Accept the equipment (in the first stage) if a run of *c*_{1} successes occurs before *d*_{1} failures.

Accept the equipment if no run of *c*_{1} successes occurs before *d*_{1} failures, but a run of *c*_{2} successes is observed before the next *d*_{2} failures.

Reject the equipment if no run of *c*_{1} successes occurs before *d*_{1} failures and also no run of *c*_{1} successes occurs before the next *d*_{2} failures.

We then derive the probability generating function of the waiting time for the termination of the start-up demonstration testing, and the mean of this waiting time. We also establish some recurrence relations satisfied by the probability mass function which will facilitate easy recursive computation of probabilities. We also discuss the distributions of some related random variables such as the numbers of successes and failures.

## Production/Clearing Models Under Continuous and Sporadic Reviews

### Methodology and Computing in Applied Probability (2005-06-01) 7: 203-224 , June 01, 2005

We consider production/clearing models where random demand for a product is generated by customers (e.g., retailers) who arrive according to a compound Poisson process. The product is produced uniformly and continuously and added to the buffer to meet future demands. Allowing to operate the system without a clearing policy may result in high inventory holding costs. Thus, in order to minimize the average cost for the system we introduce two different clearing policies (continuous and sporadic review) and consider two different issuing policies (“all-or-some” and “all-or-none”) giving rise to four distinct production/clearing models. We use tools from level crossing theory and establish integral equations representing the stationary distribution of the buffer’s content level. We solve the integral equations to obtain the stationary distributions and develop the average cost objective functions involving holding, shortage and clearing costs for each model. We then compute the optimal value of the decision variables that minimize the objective functions. We present numerical examples for each of the four models and compare the behaviour of different solutions.

## Percentile estimators in location-scale parameter families under absolute loss

### Metrika (2010-11-01) 72: 351-367 , November 01, 2010

Estimators of percentiles of location-scale parameter families are optimized based on median unbiasedness and absolute risk. Median unbiased estimators and minimum absolute risk estimators are shown to exist within a class of equivariant estimators and depend upon medians of two completely specified distributions. This work extends earlier findings to a larger class of equivariant estimators. These estimators are illustrated in the normal and exponential distributions.

## Constrained test in linear models with multivariate power exponential distribution

### Computational Statistics (2016-12-01) 31: 1569-1592 , December 01, 2016

We investigate the problem of testing equality and inequality constraints on regression coefficients in linear models with multivariate power exponential (MPE) distribution. This distribution has received considerable attention in recent years and provides a useful generalization of the multivariate normal distribution. We examine the performance of the power of the likelihood ratio, Wald and Score tests for grouped data and in the presence of regressors, in small and moderate sample sizes, using Monte Carlo simulations. Additionally, we present a real example to illustrate the performance of the proposed tests under the MPE model.

## Evaluating expectations of L-statistics by the Steffensen inequality

### Metrika (2006-01-10) 63: 371-384 , January 10, 2006

By combining the Moriguti and Steffensen inequalities, we obtain sharp upper bounds for the expectations of arbitrary linear combinations of order statistics from iid samples. The bounds are expressed in terms of expectations of the left truncated parent distribution and constants that depend only on the coefficients of the linear combination. We also present analogous results for dependent id samples. The bounds are especially useful for *L*-estimates of the scale parameter of the distribution.

## Multi-sample Models

### The Art of Progressive Censoring (2014-01-01): 515-530 , January 01, 2014

Several models involving multiple samples based on progressively Type-II censored data are discussed. The presentation includes competing risk models, joint progressive censoring, concomitants, and progressively censored systems data.

## Progressive Type-I Censoring: Basic Properties

### The Art of Progressive Censoring (2014-01-01): 115-124 , January 01, 2014

The distribution theory of progressive Type-I censored order statistics is presented. Furthermore, the dependence structure and the distribution of the number of observations up to some given threshold are addressed.

## Parameter Estimation and the CRLB with Uncertain Origin Measurements

### Methodology And Computing In Applied Probability (2001-12-01) 3: 387-410 , December 01, 2001

Parameter estimation in the presence of false measurements due to false alarms and missed true detections, i.e., in the presence of measurement origin uncertainty, is a difficult problem because of the need for data association, the process of deciding which, if any, is the true measurement and which are false. An additional aspect of estimation is performance evaluation via, for example, the Cramer-Rao Lower Bound (CRLB), which quantifies the achievable performance. With measurement origin uncertainty and the ensuing data association, the CRLB has to be modified to account for the loss of information due to false alarms and missed true detections. This is the focus of our paper—we show that the loss of information can be accounted for by a single scalar, known as the information reduction factor, under certain conditions. We illustrate the evaluation of the generalized CRLB on parameter estimation from direction-of-arrival measurements with applications to target tracking, communications and signal processing. Simulation results on a realistic scenario show that the lower bounds quantified via the information reduction factor are statistically compatible with the observed errors.

## Proportional hazards regression under progressive Type-II censoring

### Annals of the Institute of Statistical Mathematics (2008-04-03) 61: 887-903 , April 03, 2008

This paper proposes an inferential method for the semiparametric proportional hazards model for progressively Type-II censored data. We establish martingale properties of counting processes based on progressively Type-II censored data that allow to derive the asymptotic behavior of estimators of the regression parameter, the conditional cumulative hazard rate functions, and the conditional reliability functions. A Monte Carlo study and an example are provided to illustrate the behavior of our estimators and to compare progressive Type-II censoring sampling plans with classical Type-II right censoring sampling plan.

## Construction of Bivariate Distributions

### Continuous Bivariate Distributions (2009-01-01): 179-228 , January 01, 2009

In this chapter, we review methods of constructing bivariate distributions. There is no satisfactory mathematical scheme for classifying the methods. Instead, we offer a classification that is based on loosely connected common structures, with the hope that a new bivariate distribution can be fitted into one of these schemes. We focus especially on application-oriented methods as well as those with mathematical nicety.

## Parametric inference from system lifetime data under a proportional hazard rate model

### Metrika (2012-04-01) 75: 367-388 , April 01, 2012

In this paper, we discuss the statistical inference of the lifetime distribution of components based on observing the system lifetimes when the system structure is known. A general proportional hazard rate model for the lifetime of the components is considered, which includes some commonly used lifetime distributions. Different estimation methods—method of moments, maximum likelihood method and least squares method—for the proportionality parameter are discussed. The conditions for existence and uniqueness of method of moments and maximum likelihood estimators are presented. Then, we focus on a special case when the lifetime distributions of the components are exponential. Computational formulas for point and interval estimations of the unknown mean lifetime of the components are provided. A Monte Carlo simulation study is used to compare the performance of these estimation methods and recommendations are made based on these results. Finally, an example is provided to illustrate the methods proposed in this paper.

## Testing Outliers in Multivariate Data

### Statistical Distributions in Scientific Work (1981-01-01) 79: 203-218 , January 01, 1981

### Summary

Given n random observations on a p-dimensional random vector x, the problem is to test whether a specified number (usually^{~}small) of suspected observations are outliers (too discordant as compared to the bulk of observations). As a generalization of Tiku’s (1975, 1977) univariate statistic, we propose a statistic g for testing a specified number of outliers in multivariate data; g is the ratio of the product of robust estimators (Tiku, 1980) to the product of ordinary estimators of the scale parameters. For the multivariate normal, g is shown to be considerably more powerful than the prominent statistic R (restricted to the multivariate normal) due to Wilks (1963) under location shifts (model A; Barnett and Lewis, 1978) although slightly less powerful under scale changes (model B; Barnett and Lewis). Like R, g is not sensitive to changes in correlations (orientation). The statistic g can be used (under models A or B) for testing outliers in samples from any multivariate distribution whose marginal distributions are of the type (l/σ)f((x-μ)/σ).

## Counting and Quantile Processes and Progressive Censoring

### The Art of Progressive Censoring (2014-01-01): 439-450 , January 01, 2014

Results for counting and quantile processes based on progressively Type-II censored data are presented.

## Front Matter - Continuous Bivariate Distributions

### Continuous Bivariate Distributions (2009-01-01) , January 01, 2009

## Factor probabilistic distance clustering (FPDC): a new clustering method

### Advances in Data Analysis and Classification (2016-12-01) 10: 441-464 , December 01, 2016

Factor clustering methods have been developed in recent years thanks to improvements in computational power. These methods perform a linear transformation of data and a clustering of the transformed data, optimizing a common criterion. Probabilistic distance (PD)-clustering is an iterative, distribution free, probabilistic clustering method. Factor PD-clustering (FPDC) is based on PD-clustering and involves a linear transformation of the original variables into a reduced number of orthogonal ones using a common criterion with PD-clustering. This paper demonstrates that Tucker3 decomposition can be used to accomplish this transformation. Factor PD-clustering alternatingly exploits Tucker3 decomposition and PD-clustering on transformed data until convergence is achieved. This method can significantly improve the PD-clustering algorithm performance; large data sets can thus be partitioned into clusters with increasing stability and robustness of the results. Real and simulated data sets are used to compare FPDC with its main competitors, where it performs equally well when clusters are elliptically shaped but outperforms its competitors with non-Gaussian shaped clusters or noisy data.

## Simple Forms of the Bivariate Density Function

### Continuous Bivariate Distributions (2009-01-01): 351-400 , January 01, 2009

When one considers a bivariate distribution, it is perhaps common to think of a joint density function rather than a joint distribution function, and it is also conceivable that such a density may be simple in expression, while the corresponding distribution function may involve special functions, can be expressed only as an infinite series, and sometimes may even be more complicated. Such distributions form the subject matter of this chapter. Although the standard form of these densities is simple, their generalizations are often not so simple. To include these generalizations would undoubtedly place the title of this chapter under question, but the alternative of leaving them out would be remiss. Therefore, for the sake of completeness, generalized forms of these simple densities will also be included in this discussion.

## Statistical Intervals for Progressively Type-II Censored Data

### The Art of Progressive Censoring (2014-01-01): 379-419 , January 01, 2014

Results on interval prediction based on progressively Type-II censored order statistics are reviewed. The discussion includes exact, conditional, and asymptotic confidence intervals as well as prediction and tolerance intervals.

## Univariate Distributions

### Continuous Bivariate Distributions (2009-01-01): 1-32 , January 01, 2009

A study of bivariate distributions cannot be complete without a sound background knowledge of the univariate distributions, which would naturally form the marginal or conditional distributions. The two encyclopedic volumes by Johnson et al. (1994, 1995) are the most comprehensive texts to date on continuous univariate distributions. Monographs by Ord (1972) and Hastings and Peacock (1975) are worth mentioning, with the latter being a convenient handbook presenting graphs of densities and various relationships between distributions. Another useful compendium is by Patel et al. (1976); Chapters 3 and 4 of Manoukian (1986) present many distributions and relations between them. Extensive collections of illustrations of probability density functions (denoted by p.d.f. hereafter) may be found in Hirano et al. (1983) (105 graphs, each with typically about five curves shown, grouped in 25 families of distributions) and in Patil et al. (1984).

## Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

### Annals of the Institute of Statistical Mathematics (1980-12-01) 32: 223-240 , December 01, 1980

Let*F* and*G* denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functions*f* and*g*, respectively. A measure of the closeness between*F* and*G* is defined by:
$$\lambda = \lambda (F,G) = 2\int {f(x)g(x)dx} /\left[ {\int {f^2 (x)dx + \int {g^2 (x)dx} } } \right]$$
. Based on two independent samples it is proposed to estimate λ by
$$\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]$$
, where*F*_{n}*(x)* and*G*_{n}*(x)* are the empirical distribution functions of*F(x)* and*G(x)* respectively and
$$\hat f(x)$$
and
$$\hat g(x)$$
are taken to be the so-called kernel estimates of*f(x)* and*g(x)* respectively, as defined by Parzen [16]. Large sample theory of
$$\hat \lambda $$
is presented and a two sample goodness-of-fit test is presented based on
$$\hat \lambda $$
. Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., when*g(x)=f*_{0}*(x)*, with*f*_{0}*(x)* completely known and for testing symmetry, i.e., testing*H*_{0}:*f(x)=f(−x)*.

## Moments of Progressively Type-II Censored Order Statistics

### The Art of Progressive Censoring (2014-01-01): 155-191 , January 01, 2014

Results on moments of progressively Type-II censored order statistics are reviewed. After presenting general expressions and existence results, explicit expressions for particular population distributions are given. Further, results for symmetric population distributions are developed. The presentation is completed by a survey on reccurence relations, bounds, and first-order approximations.

## Start-Up Demonstration Tests with Rejection of Units upon Observing d Failures

### Annals of the Institute of Statistical Mathematics (2000-03-01) 52: 184-196 , March 01, 2000

The probability generating function of number of trials for the start-up demonstration test with rejection upon *d* failures is derived. The exact distribution of the number of trials is obtained. Some recurrence relations for the probabilities are also established. The average length of the test is derived. Some illustrative examples are finally presented.

## On Generalized Wishart Distributions - I: Likelihood Ratio Test for Homogeneity of Covariance Matrices

### Sankhya A (2014-08-01) 76: 179-194 , August 01, 2014

In this paper, we define and discuss a class of generalized Wishart distributions under elliptical models. We derive the non-central moments of the likelihood ratio statistic for testing the equality of two covariance matrices under elliptical models for the corresponding matrices. Known classical expressions for the Gaussian model are then deduced from these general results. Finally, the exact distribution of the Wilks’ statistic under a specific distribution, including the Gaussian distribution as a particular member, is derived.

## Back Matter - Continuous Bivariate Distributions

### Continuous Bivariate Distributions (2009-01-01) , January 01, 2009

## A simple method for combining estimates to improve the overall error rates in classification

### Computational Statistics (2015-12-01) 30: 1033-1049 , December 01, 2015

We present a new and easy-to-implement procedure for combining $$J\ge 2$$ different classifiers in order to develop more effective classification rules. The method works by finding nonparametric estimates of the class conditional expectation of a new observation (that has to be classified), conditional on the vector of $$J$$ predicted values corresponding to the $$J$$ individual classifiers. Here, we propose a data-splitting method to carry out the estimation of various class conditional expectations. It turns out that, under rather minimal assumptions, the proposed combined classifier is optimal in the sense that its overall misclassification error rate is asymptotically less than (or equal to) that of any one of the individual classifiers. Simulation studies are also carried out to evaluate the proposed method. Furthermore, to make the numerical results more challenging, we also consider stable distributions (Cauchy) with rather high dimensions.

## Fisher information ink-records

### Annals of the Institute of Statistical Mathematics (2004-06-01) 56: 383-396 , June 01, 2004

We derive some general results on the Fisher information (FI) contained in the upper (or lower)*k*-record values and associated*k*-record times generated from an i.i.d. sample of fixed size from a continuous distribution. We apply the results to obtain the FI in both upper and lower*k*-record data from an exponential distribution. We propose two estimators of the exponential mean, based on the upper and lower*k*-record data, and discuss their small sample properties. We also consider*k*-record data from an inverse sampling plan, and present general formulas for the FI contained in it.

## Vector Extensions of the Dirichlet HC and HD Functions, with Applications to the Sharing Problem

### Methodology and Computing in Applied Probability (2010-03-01) 12: 91-109 , March 01, 2010

In this paper we apply the Dirichlet HC and HD functions to a generalization of the sharing problem in which the population is finite, and sampling is without replacement. In doing so we extend the Dirichlet HC and HD functions, and associated waiting time results, from Sobel and Frankowski (Congressus Numerantium 106:171–191, 1995) to handle vector arguments. We also provide Maple procedures for their computation. Our results for the sharing problem generalize the results for with replacement sampling given in Sobel and Frankowski (Am Math Mon 101:833–847, 1994a).

## Classification of Three-word Indicator Functions of Two-level Factorial Designs

### Annals of the Institute of Statistical Mathematics (2006-09-01) 58: 595-608 , September 01, 2006

Indicator functions have been in the literature for several years, and yet only a few of their properties have been examined. In this paper, we study some properties of indicator functions of two-level fractional factorial designs. For example, we show that there is no indicator function with only two words, and also classify all indicator functions with only three words. The results imply that there is no valuable non-regular design with only three or less words in its indicator function.

## Whitworth runs on a circle

### Annals of the Institute of Statistical Mathematics (1977-12-01) 29: 287-293 , December 01, 1977

### Summary

Suppose different classes of items, for example, beads of different colours, are placed in a circle. Two probability models have been proposed, which lead to different distributions of runs, i.e. sequences of one colour. Barton and David [3] have called these Whitworth runs and Jablonski runs, and have tabulated the distributions for small samples. Asano [1] has extended the tabulations for Jablonski runs. In this paper, Whitworth runs are examined, particularly some approximations to the distributions which avoid extensive tabulations. Some potential uses of Whitworth runs are also pointed out.

## Maximum likelihood estimation of Laplace parameters based on general type-II censored examples

### Statistical Papers (1997-09-01) 38: 343-349 , September 01, 1997

In this paper, we derive the maximum likelihood estimators of the parameters of a Laplace distribution based on general Type-II censored samples. The resulting explicit MLE's turn out to be simple linear functions of the order statistics. We then examine the asymptotic variance of the estimates by calculating the elements of the Fisher information matrix.

## Gamma Degradation Models: Inference and Optimal Design

### Statistical Modeling for Degradation Data (2017-01-01): 171-191 , January 01, 2017

During the past two decades, degradation analysis has been widely used to assess the lifetime information of highly reliable products. Usually, random effect models and/or Wiener processes are well suited for modelling stochastic degradation. But in many situations, such as materials that lead to fatigue, it is more appropriate to model the degradation data by a gamma degradation process which exhibits a monotone increasing pattern. This article surveys the theoretical aspects as well as the application of gamma processes in degradation analysis. Some statistical properties of degradation models based on gamma processes under different tests are also given. Furthermore, the corresponding optimal designs for conducting the degradation experiments efficiently are reviewed. Finally, some extensions and their applications of gamma-process degradation model are presented.

## An information theoretical algorithm for analyzing supersaturated designs for a binary response

### Metrika (2013-01-01) 76: 1-18 , January 01, 2013

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the number of runs. It is used in many experiments, for screening purpose, i.e., for studying a large number of factors and identifying the active ones. In this paper, we propose a method for screening out the important factors from a large set of potentially active variables through the symmetrical uncertainty measure combined with the information gain measure. We develop an information theoretical analysis method by using Shannon and some other entropy measures such as Rényi entropy, Havrda–Charvát entropy, and Tsallis entropy, on data and assuming generalized linear models for a Bernoulli response. This method is quite advantageous as it enables us to use supersaturated designs for analyzing data on generalized linear models. Empirical study demonstrates that this method performs well giving low Type I and Type II error rates for any entropy measure we use. Moreover, the proposed method is more efficient when compared to the existing ROC methodology of identifying the significant factors for a dichotomous response in terms of error rates.