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## 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.

## 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.

## Multiple Comparisons between Several Treatments and a Specified Treatment

### Linear Statistical Inference (1985-01-01) 35: 39-47 , January 01, 1985

Suppose that there are several treatments to be compared in an experiment. The experimental design may be any one to which an analysis of variance model is applied; here, for simplicity, we assume the design is a one-way classification (i.e., the only factor in the design is the treatment factor). However, usually the analysis of variance F-tests do not satisfy the needs of the experimenter, who often wishes to make specific comparisons between certain treatment means. This leads to multiple tests if there are more than two treatments and the problem is how to make valid inferences concerning the treatment comparisons of interest in the experiment when there is more than one to be considered. The early work by Duncan (1951), Scheffé (1953) and Tukey (1953) laid the foundations for the subject of multiple comparisons or, as it is sometimes called, simultaneous inference. For a review, see Miller (1966 or 1981).

## 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.

## Bivariate Distributions Constructed by the Conditional Approach

### Continuous Bivariate Distributions (2009-01-01): 229-278 , January 01, 2009

In Section 5.6, we outlined the construction of a bivariate p.d.f. as the product of a marginal p.d.f. and a conditional p.d.f., *h*(*x,y*)=*f*(*x*)*g*(*y*|*x*). This construction is easily understood, and has been a popular choice in the literature, especially when *Y* can be thought of as being caused by, or predicted from, *X*. Arnold et al. (1999, p. 1) contend that it is often easier to visualize conditional densities or features of conditional densities than marginal or joint densities. They cite, for example, that it is not unreasonable to visualize that, in the human population, the distribution of heights for a given weight will be unimodal, with the mode of the conditional distribution varying monotonically with weight. Similarly, we may visualize a unimodal distribution of weights for a given height, this time with the mode varying monotonically with the height. Thus, construction of a bivariate distribution using two conditional distributions may be practically useful.

## Front Matter - Advances in Mathematical and Statistical Modeling

### Advances in Mathematical and Statistical Modeling (2008-01-01) , January 01, 2008

## Higher Order Moments of Order Statistics From the Power Function Distribution and Edgeworth Approximate Inference

### Advances in Stochastic Simulation Methods (2000-01-01): 245-282 , January 01, 2000

In this paper, we first derive exact explicit expressions for the triple and quadruple moments of order statistics from the power function distribution. Also, we present recurrence relations for single, double, triple and quadruple moments of order statistics from the power function distribution. These relations will enable one to find all moments (of order up to four) of order statistics for all sample sizes in a simple recursive manner. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. We then derive approximate confidence intervals for the parameters of the power function distribution using the Edgeworth approximation. Finally, we extend the recurrence relations to the case of the doubly truncated power function distribution.

## 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).

## Acceptance Sampling Plans

### Progressive Censoring (2000-01-01): 215-222 , January 01, 2000

In this Chapter, we discuss the construction of acceptance sampling plans based on progressively Type-II right censored samples. For this purpose, we may either use exact (ML/BLU) estimators of the parameters and their exact distributional properties or use first-order approximate estimators of the parameters and their distributional properties. We consider the exponential and log-normal distributions for illustration with the exponential being an example of the first kind and the log-normal being an example of the second kind.

## An Assessment of Up-and-Down Designs and Associated Estimators in Phase I Trials

### Advances in Statistical Methods for the Health Sciences (2007-01-01): 361-386 , January 01, 2007

In this article, we consider some up-and-down designs that are discussed in Ivanova *et al.* (2003) for estimating the maximum tolerated dose (MTD) in phase I trials: the biased coin design, *k*-in-a-row rule, Narayana rule, and continual reassessment method (CRM). A large-scale Monte Carlo simulation study, which is substantially more extensive than Ivanova *et al.* (2003), is conducted to examine the performance of these five designs for different sample sizes and underlying dose-response curves. For the estimation of MTD, we propose a modified maximum likelihood estimator (MMLE) in addition to those in Ivanova *et al.* (2003). The selection of different dose-response curves and their parameters allows us to evaluate the robustness features of the designs as well as the performance of the estimators. The results obtained, in addition to revealing that the new estimator performs better than others in many situations, enable us to make recommendations on designs.

## 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.

## Simulation of Bivariate Observations

### Continuous Bivariate Distributions (2009-01-01): 623-653 , January 01, 2009

Devroye (1986) has provided an exhaustive treatment on the generation of random variates. Gentle (2003) has also recently provided a state-of-the-art treatise on random number generation and Monte Carlo methods. For this reason, we provide here a brief review of this subject and refer readers to these two references for a comprehensive treatment. In view of the importance of simulation as a tool while analyzing practical data using different parametric statistical models as well as while examining the properties and performance of estimators and hypothesis tests, we feel that it is very important for a reader of this book to know at least some essential details about the simulation of observations from a specified bivariate probability function.

## Front Matter - The Art of Progressive Censoring

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

## 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.

## Linear Estimation in Progressive Type-II Censoring

### The Art of Progressive Censoring (2014-01-01): 247-266 , January 01, 2014

Linear inference for progressively Type-II censored order statistics is discussed for location, scale, and location-scale families of population distributions. After a general introduction, results for exponential, generalized Pareto, extreme value, Weibull, Laplace, and logistic distributions are presented in detail.

## A Hybrid Logistic Model for Case-Control Studies

### Methodology And Computing In Applied Probability (2003-12-01) 5: 419-426 , December 01, 2003

For logistic regression in case-control studies, when risk factors associated with the outcome are exceedingly rare in the control group, the estimation of parameters in the model becomes difficult. In this paper, we propose a two-stage hybrid method to achieve this. In the first stage, we model the risk due to the rare factor, and in the second stage we model the residual risk due to the other factors using standard logistic model.

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

### Metrika (2015-12-11): 1-11 , December 11, 2015

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.

## 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.

## 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.

## 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.

## Front Matter - Advances in Mathematical and Statistical Modeling

### Advances in Mathematical and Statistical Modeling (2008-01-01) , January 01, 2008

## 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.

## Front Matter - The Art of Progressive Censoring

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

## A note on Bernstein and Müntz-Szasz theorems with applications to the order statistics

### Annals of the Institute of Statistical Mathematics (1978-12-01) 30: 167-176 , December 01, 1978

## 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.

## Order statistics from non-identical right-truncated Lomax random variables with applications

### Statistical Papers (2001-04-01) 42: 187-206 , April 01, 2001

In this paper, we derive some recurrence relations for the single and the product moments of order statistics from n independent and non-identically distributed Lomax and right-truncated Lomax random variables. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and product moments of all order statistics in a simple recursive manner. The results for order statistics from the multiple-outlier model (with a slippage of p observations) are deduced as special cases. We then apply these results by examining the robustness of censored BLUE’s to the presence of multiple outliers.

## Linear Prediction

### Progressive Censoring (2000-01-01): 139-165 , January 01, 2000

Until now, we have discussed many properties of progressively Type-II right censored order statistics and also the estimation of location and scale parameters of different distributions based on progressively censored samples.

## 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.

## Illustrative Examples

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 17-30 , January 01, 1999

In this chapter, we shall present four examples in order to illustrate the use of Tables 4–6 for the computation of the best linear unbiased estimates of the mean and standard deviation of the assumed lognormal distribution for the observed data.

## Maximum Likelihood Estimation in Progressive Type-II Censoring

### The Art of Progressive Censoring (2014-01-01): 267-312 , January 01, 2014

Likelihood inference in progressive Type-II censoring is presented for a wide range of distributions including exponential, Weibull, extreme value, generalized Pareto, Laplace, and normal distributions. The chapter is supplemented by a review for other distributions as well as by a discussion of related methods like modified and approximate likelihood estimation. Finally, results for $M$-estimation and order restricted inference are presented.

## 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.

## 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.

## 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.

## Conditional Inference for the Parameters of Pareto Distributions when Observed Samples are Progressively Censored

### Advances in Stochastic Simulation Methods (2000-01-01): 293-302 , January 01, 2000

In this paper we develop procedures for obtaining confidence intervals for the location and scale parameters of a Pareto distribution as well as upper and lower *γ* probability tolerance intervals for a proportion *β* when the observed samples are progressively censored. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since the procedures assume that the shape parameter *ν* is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in *ν*.

## Alternative Computational Methods

### Progressive Censoring (2000-01-01): 67-83 , January 01, 2000

Clever transformations and efficient recursive algorithms are often useful in obtaining moments and for establishing mathematical properties of progressively Type-II right censored order statistics from a number of distributions, as we have already seen. However, such elegant methods are not possible for all distributions that may be of interest to a practitioner. For this reason, alternative methods for computing moments of progressively censored order statistics must be sought. In this chapter, we present two such methods for the computation of moments of progressively Type-II right censored order statistics. The first applies to an arbitrary continuous distributions for which the moments of *usual* order statistics are known, and the second applies specifically to symmetric distributions for which the moments of progressively Type-II right censored order statistics from the corresponding folded distribution are known. These two methods will further enhance our repertoire of distributions that we can consider as models for lifetime data, and hence will make the use of progressive censoring in real-life situations much more attractive. Finally, we present some first-order approximations to the means, variances and covariances of progressively Type-II right censored order statistics based on Taylor series expansions. These expressions will be used later in Chapter 6 in order to develop and illustrate first-order approximations to the best linear unbiased estimators of location and scale parameters of any distribution of interest.

## State Estimation in Wastewater Engineering: Application to an Anaerobic Process

### Statistical Methods for the Assessment of Point Source Pollution (1989-01-01): 171-182 , January 01, 1989

The use of an extended Kaiman filter for state estimation in biological wastewater treatment processes is discussed. The application of the technique requires an adequate mechanistic dynamic model, and the identification and modelling of the major sources of stochastic disturbances in the process. The filter allows the on-line tracking of process variables which are not directly measurable. The use of an extended Kaiman filter is illustrated through a simulated application to a high rate anaerobic wastewater treatment process.

## 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.

## Lognormal Distributions and Properties

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 5-6 , January 01, 1999

If *Y* is normally distributed with mean μ and variance σ^{2}, then the random variable *X* defined by the relationship *Y* =*log*(*X* - γ) is distributed as lognormal, and is denoted as lognormal(γ,μ,σ^{2}).

## Point Estimation in Progressive Type-I Censoring

### The Art of Progressive Censoring (2014-01-01): 313-325 , January 01, 2014

Results on likelihood inference in progressive Type-I censoring are reviewed for a plenty of distributions including one- and two-parameter exponential, extreme value, Weibull, normal, and Burr distributions.

## Nonparametric Inferential Issues in Progressive Type-II Censoring

### The Art of Progressive Censoring (2014-01-01): 451-464 , January 01, 2014

Nonparametric statistical tests are reviewed. This includes precedence-type tests as well as test for hazard rate ordering.

## 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.

## Optimal Experimental Designs

### The Art of Progressive Censoring (2014-01-01): 531-570 , January 01, 2014

The problem of an optimal censoring plan in progressive Type-II censoring is discussed for several criteria including minimum experimental time, maximum Fisher information, minimum variance of estimates, as well as further criteria proposed in the literature.

## 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.

## Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

### Annals of the Institute of Statistical Mathematics (1996-12-01) 48: 757-771 , December 01, 1996

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes *n* and all censoring schemes (*R*_{1}, *R*_{2}, ..., *R*_{m}), *m≤n*. The results presented in the paper generalize the results given by Joshi (1978, *Sankhyā Ser. B*, *39*, 362–371; 1982, *J. Statist. Plann. Inference*, *6*, 13–16) for the single moments and product moments of order statistics from the exponential distribution.

To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.

## 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.

## 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.

## 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

## Mathematical Properties of Progressively Type-II Right Censored Order Statistics

### Progressive Censoring (2000-01-01): 11-29 , January 01, 2000

We mentioned earlier in Chapter 1 that marginal distributions of progressively Type-II right censored order statistics (except for the first one, X_{1:m:n}), unlike usual order statistics, are quite complicated and cumbersome and are not easily obtained (Try it — you won’t like it!). However, certain joint marginal distributions are readily obtained and they do aid in studying mathematical properties of progressively Type-II censored order statistics and also in developing inferential procedures based on progressively censored samples. Furthermore, they aid in providing some efficient algorithms for simulating progressively Type-II censored samples.

## Order Statistics from a Sample Containing a Single Outlier

### Relations, Bounds and Approximations for Order Statistics (1989-01-01) 53: 108-140 , January 01, 1989

Density functions and joint density functions of order statistics arising from a sample containing a single outlier have been given by Shu (1978) and David and Shu (1978), and have been made use of by David et al. (1977) in tabulating means, variances and covariances of order statistics from a normal sample comprising one outlier. One may also refer to Vaughan and Venables (1972) for more general expressions of distributions of order statistics when the sample, in fact, includes k outliers. The importance of a systematic study of the order statistics from an outlier model and the usefulness of the tables of means, variances and covariances of these order statistics in the context of robustness has been well demonstrated by several authors including Andrews et al. (1972), David and Shu (1978), David (1981), and Tiku et al. (1986).

## 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.

## Exact Likelihood Inference for an Exponential Parameter Under Progressive Hybrid Censoring Schemes

### Statistical Models and Methods for Biomedical and Technical Systems (2008-01-01): 319-330 , January 01, 2008

The purpose of this chapter is to propose two types of progressive hybrid censoring schemes in life-testing experiments and develop exact inference for the mean of the exponential distribution. The exact distribution of the maximum likelihood estimator and an exact lower confidence bound for the mean lifetime are obtained under both types of progressive hybrid censoring schemes. Illustrative examples are finally presented.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## Recurrence Relations and Identities for Order Statistics

### Relations, Bounds and Approximations for Order Statistics (1989-01-01) 53: 5-37 , January 01, 1989

Order statistics and their moments have been of great interest from the turn of this century since Sir Francis Galton (1902) and Karl Pearson (1902) studied the distribution of the difference of two successive order statistics. The moments of order statistics did, subsequently, assume considerable importance in the statistical literature and have been numerically tabulated extensively for several distributions. For example, one can refer to Harter (1970a,b) and David (1981) for a detailed list of these tables. Meanwhile, with the primary intention of reducing the amount of direct computation of these moments, many authors including Jones (1948), Godwin (1949), Cole (1951) and Sillitto (1951, 1964) carried out independent investigations and derived several recurrence relations and identities satisfied by these moments of order statistics. Many of these relations and identities are quite useful as they express the higher order moments in terms of the lower order moments thus making the evaluation of higher order moments easy and, in addition, provide some simple checks to test the accuracy of the computation of moments of order statistics. It was only 25 years ago, however, that Govindarajulu (1963a) nicely summarized all these results and established some more recurrence relations and identities satisfied by the single and the product moments of order statistics. He then systematically applied these results in order to determine the maximum number of single and double integrals to be evaluated for the calculation of means, variances and covariances of order statistics in a sample of size n, assuming these quantities for all sample sizes less than n to be known. By a simple generalization of one of the results of Govindarajulu (1963a), Joshi (1971) determined that for distributions symmetric about zero the number of double integrals to be evaluated for even values of n is in fact zero. Recently, Joshi and Balakrishnan (1982) established similar results for any arbitrary continuous distribution and applied them to improve over the bounds of Govindarajulu. Yet another interesting application of these recurrence relations and identities among order statistics is in establishing some combinatorial identities and this has been demonstrated by Joshi (1973) and Joshi and Balakrishnan (1981a).

## 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.

## 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.

## 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.

## Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

### Annals of the Institute of Statistical Mathematics (1996-09-01) 48: 519-534 , September 01, 1996

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter *P*. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, *J. Statist. Comput. Simulation*, *52*, 385–398).

## 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.

## 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.

## 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.

## Survival Models for Step-Stress Experiments With Lagged Effects

### Advances in Degradation Modeling (2010-01-01): 355-369 , January 01, 2010

In this chapter, we consider models for experiments in which the stress levels are altered at intermediate stages during the exposure. These experiments, referred to as step-stress tests, belong to the class of accelerated models that are extensively used in reliability and life-testing applications. Models for step-stress tests have largely relied on the cumulative exposure model (CEM) discussed by Nelson. Unfortunately, the assumptions of the model are fairly restrictive and quite unreasonable for applications in survival analysis. In particular, under the CEM the hazard function has discontinuities at the points at which the stress levels are changed.

We introduce a new step-stress model where the hazard function is continuous. We consider a simple experiment with only two stress levels. The hazard function is assumed to be constant at the two stress levels and linear in the intermediate period. This model allows for a lag period before the effects of the change in stress are observed. Using this formulation in terms of the hazard function, we obtain the maximum likelihood estimators of the unknown parameters. A simple least squares-type procedure is also proposed that yields closed-form solutions for the underlying parameters. A Monte Carlo simulation study is performed to study the behavior of the estimators obtained by the two methods for different choices of sample sizes and parameter values. We analyze a real data set and show that the model provides an excellent fit.

## 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.

## 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).

## 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.

## 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.

## 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.

## Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

### Annals of the Institute of Statistical Mathematics (1979-12-01) 31: 279-288 , December 01, 1979

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(*f*)=∫*f*^{2}*(x)dx* were strong consistency is established. One of the suggested two estimates of Δ(*f*) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate.

## 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.

## 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.

## Front Matter - Advances in Mathematical and Statistical Modeling

### Advances in Mathematical and Statistical Modeling (2008-01-01) , January 01, 2008

## 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.

## Estimation for one- and two-parameter exponential distributions under multiple type-II censoring

### Statistical Papers (1992-12-01) 33: 203-216 , December 01, 1992

In this paper, we derive explicit best linear unbiased estimators for one- and two-parameter exponential distributions when the available sample is multiply Type-II censored. Further, after noting that the maximum likelihood estimators do not exist explicitly, we propose some linear estimators by approximating the likelihood equations appropriately. Some illustrative examples from life-testing experiments are also presented.

## 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.

## 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.

## 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.

## Front Matter - Continuous Bivariate Distributions

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

## Clinical Epidemiology

### Handbook of Epidemiology (2005-01-01): 1169-1223 , January 01, 2005

This chapter will begin with providing a brief overview of the history of clinical epidemiology and describe its relation with evidence basedmedicine. Clinical epidemiology differs from classical epidemiology in that clinical epidemiology supports other basic medical sciences such as biochemistry, anatomy and physiology because it facilitates their application in research through formulation of sound clinical research methods and, thus, puts these disciplines into clinical context. Therefore, clinical epidemiology goes beyond clinical trials. We will describe this concept in the following paragraphs (see Sect. 8.1.1 through 8.1.3). The following sections include case scenarios that facilitate the introduction of the key concepts about developing clinical questions, using diagnostic tests, evaluating therapy, appraising systematic reviews, developing guidelines and making clinical decisions.

## 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.

## 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.

## Robust Location-Tests and Classification Procedures

### Robustness of Statistical Methods and Nonparametric Statistics (1984-01-01) 1: 152-155 , January 01, 1984

Two sample problems and classifcation procedures are discussed. It is shown that adaptation of the MML estimators leads to tests which have Type I robustness and have remarkably high power.

## Front Matter - Advances in Mathematical and Statistical Modeling

### Advances in Mathematical and Statistical Modeling (2008-01-01) , January 01, 2008

## Skewed multivariate models related to hidden truncation and/or selective reporting

### Test (2002-06-01) 11: 7-54 , June 01, 2002

The univariate skew-normal distribution was introduced by Azzalini in 1985 as a natural extension of the classical normal density to accommodate asymmetry. He extensively studied the properties of this distribution and in conjunction with coauthors, extended this class to include the multivariate analog of the skew-normal. Arnold et al. (1993) introduced a more general skew-normal distribution as the marginal distribution of a truncated bivariate normal distribution in which*X* was retained only if*Y* satisfied certain constraints. Using this approach more general univariate and multivariate skewed distributions have been developed. A survey of such models is provided together with discussion of related inference questions.

## 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.

## Likelihood Inference: Type-I and Type-II Censoring

### Progressive Censoring (2000-01-01): 117-138 , January 01, 2000

In Chapter 6, we considered linear inference based on progressively Type-II censored order statistics. Linear inference is popular because, in addition to having many desirable properties which we associate with good estimators, linear estimators have a very simple form, viz, the estimators are linear combinations of observed data values. As a result, these estimators are often quite simple to interpret from a practitioner’s point of view. Furthermore, we are able to very easily calculate the variances and covariance of linear estimators. This is not always true for other types of estimators, such as maximum likelihood and moment estimators. However, the linear inference that we have discussed applies only to scale- or location-scale families of distributions. If a new parameter, such as a shape parameter or a threshold parameter that can not be written as a simple shift from the standard distribution, is introduced, other methods of estimating these parameters need to be considered.

## Best Linear Unbiased Estimation of Location and Scale Parameters

### Handbook of Tables for Order Statistics from Lognormal Distributions with Applications (1999-01-01): 13-15 , January 01, 1999

Let us now assume that we have a random sample of size *n*, *X*_{1}, *X*_{2},…, *X*_{n}, from a three-parameter lognormal distribution [obtained by introducing location and scale parameters in (2.3)] with probability density function
4.1
$$
\begin{gathered}
f(x|\mu ,\sigma ,k) \hfill \\
\,\,\,\,\,\, = \frac{1}{{\left( {{{(k - 1)}^{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/
{\vphantom {{ - 1} 2}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$2$}}}} + \frac{{x - \mu }}{\sigma }} \right)\sqrt {2\pi \log (k)\sigma } }}{e^{ - {{[\log \{ \sqrt k \left( {1 + {{(k -
1)}^{^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 2}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$2$}}\frac{{x - \mu }}{\sigma }}}}} \right)\} ]}^2}/(2\log (k)),}} \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mu - \frac{\sigma }{{\sqrt {k - 1} }} < x < \infty . \hfill \\
\end{gathered}
$$

## Robust estimation and hypothesis testing under short-tailedness and inliers

### Test (2005-06-01) 14: 129-150 , June 01, 2005

Estimation and hypothesis testing based on normal samples censored in the middle are developed and shown to be remarkably efficient and robust to symmetric shorttailed distributions and to inliers in a sample. This negates the perception that sample mean and variance are the best robust estimators in such situations (Tiku, 1980; Dunnett, 1982).

## A New Statistic for Testing an Assumed Distribution

### A Modern Course on Statistical Distributions in Scientific Work (1975-01-01) 17: 113-124 , January 01, 1975

### Summary

A new statistic, T, for testing an assumed distribution of the form
$$\frac{1}{\sigma }f\left( {\frac{{x - \mu }}{\sigma }} \right)$$
is proposed. This statistic is the ratio of V. the maximum likelihood estimator or modified maximum likelihood estimator (Tiku, 1967, 1973) of O calculated from a censored sample, to the maximum likelihood estimator calculated from the whole sample. T is both location and scale invariant. The asymptotic distribution of T is normal and for some populations the distribution of T is exactly beta. T is, in general, easy to compute and has good power properties. The statistic T can also be generalized to multi-sample situations in a straightforward fashion. However, in the use of T one has to have *a priori* knowledge whether the alternative distribution is *skewed* or *symmetric*.

## 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.