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.

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.

Many of the bivariate gamma distributions considered in this chapter may be derived from the bivariate normal in some fashion, such as by marginal transformation. It is well known that a univariate chi-squared distribution can be obtained from one or more independent and identically distributed normal variables and that a chi-squared random variable is a special case of gamma; hence, it is not surprising that a bivariate gamma model is related to the bivariate normal one.

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.

A feature common to all the distributions in this chapter is that H(x,y) is a simple function of the uniform marginals F(x) and G(y). These types of joint distributions are known as copulas, as mentioned in the last chapter, and will be denoted by C(u,v); the corresponding random variables will be denoted by U and V, respectively.

In this paper, we present a general method which can be used in order to show that the maximum likelihood estimator (MLE) of an exponential mean θ is stochastically increasing with respect to θ under different censored sampling schemes. This propery is essential for the construction of exact confidence intervals for θ via “pivoting the cdf” as well as for the tests of hypotheses about θ. The method is shown for Type-I censoring, hybrid censoring and generalized hybrid censoring schemes. We also establish the result for the exponential competing risks model with censoring.

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.

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.

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.

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distribution more quickly than under normal operating conditions. In this article, we consider a simple step-stress model under the exponential distribution when the available data are progressively Type-II censored. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We then construct confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Next, we investigate optimal progressive censoring schemes as well as optimal time for change of stress level based on the simple step-stress model. Finally, we present two examples to illustrate all the methods of inference discussed here.

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.

The EM algorithm is used to find the maximum likelihood estimates (MLEs) of the parameters of a bivariate normal distribution based on progressively Type-II right censored samples. The asymptotic variances and covariances of the MLEs are derived, using the missing information principle, from the Fisher information matrix as well as from the partially observed information matrix. Optimal censoring schemes are then investigated with respect to minimum trace of the variance-covariance matrix of the MLEs and also with respect to the maximum information about ρ.

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).

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.

A reliability experimenter is often interested in studying the effects of extreme or varying stress factors such as load, pressure, temperature and voltage on the lifetimes of experimental units. Accelerated life-tests allow the experimenter to vary the levels of these stress factors in order to obtain information on the parameters of the lifetime distributions more rapidly than under normal operating conditions. Step-stress tests are a particular class of accelerated life-tests which allow the experimenter to change the stress levels at pre-fixed times during the life-testing experiment. One of the prominent models assumed in step-stress tests is the cumulative exposure model which connects the lifetime distribution of units at one stress level to the lifetime distributions at preceding stress levels. Under such a cumulative exposure model and the assumption that the lifetimes at different stress levels are exponentially distributed, we review in this article various developments on exact inferential methods for the model parameters based on different forms of censored data. We also describe the approximate confidence intervals based on the asymptotic properties of maximum likelihood estimators as well as the bootstrap confidence intervals, and provide some comparisons between these methods. Finally, we present some examples to illustrate all the inferential methods discussed here.

Here, we study runs associated with record values. After we first discuss the distributions and moments of runs, we develop the asymptotic theory of these runs. Finally, we apply the concept of runs obtained from record values to a hypothesis testing problem.

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.

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.

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 (R₁, R₂, ..., 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.

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.

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.

The study of copulas is a growing field. The construction and properties of copulas have been studied rather extensively during the last 15 years or so. Hutchinson and Lai (1990) were among the early authors who popularized the study of copulas. Nelsen (1999) presented a comprehensive treatment of bivariate copulas, while Joe (1997) devoted a chapter of his book to multivariate copulas. Further authoritative updates on copulas are given in Nelsen (2006). Copula methods have many important applications in insurance and finance [Cherubini et al. (2004) and Embrechts et al. (2003)].

Let X₁, . . . , X_n be independent exponential random variables with respective hazard rates λ₁, . . . , λ_n, and Y₁, . . . , 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 X₁, . . . , X_n, is larger than Y_2:n, the second order statistic from Y₁, . . . , 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.

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.

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.

The geometric type and inverse Polýa-Eggenberger type distributions of waiting time for success runs of lengthk 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 orderk for Markov dependent trials are developed.

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.

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.

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 θ₁ and θ₂ 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 θ₁ and θ₂, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters θ₁ and θ₂. 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.

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).

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).

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.

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.

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)=∫f²(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.

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 (1963a, 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.

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.

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.

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.

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).

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.

Indicator functions are new tools for studying two-level fractional factorial designs. This article discusses some properties of indicator functions. Using indicator functions, we study the connection between general two-level factorial designs of generalized resolutions.

Normal distribution based discriminant methods have been used for the classification of new entities into different groups based on a discriminant rule constructed from the learning set. In practice if the groups are not homogeneous, then mixture discriminant analysis of Hastie and Tibshirani (J R Stat Soc Ser B 58(1):155–176, 1996) is a useful approach, assuming that the distribution of the feature vectors is a mixture of multivariate normals. In this paper a new logistic regression model for heterogenous group structure of the learning set is proposed based on penalized multinomial mixture logit models. This approach is shown through simulation studies to be more effective. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using this penalized multinomial mixture logit model as compared to the classical discriminant rules. It also showed better results when applied to practical life data problems producing smaller errors.

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.

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.

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 whichX was retained only ifY 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.

In this paper, the problem of reconstructing past records from the known values of future records is investigated. Different methods are applied when the underlying distributions are Exponential and Pareto, and several reconstructors are obtained and then compared. A data set representing the record values of average July temperatures in Neuenburg, Switzerland, is used to illustrate the proposed procedure in the Pareto case. The results may be used for studying the past epoch times of non-homogeneous Poisson process or the past failure times in a reliability problem when the repair policy is minimal repair.

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.

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 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.

A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.

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.

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.

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.

This paper considers the largest and smallest observations at the times when a new record of either kind (upper or lower) occurs. These are called the upper and lower current records and are denoted by $${R^l_m}$$ and $${R^s_m}$$ , respectively. The interval $${(R^s_m,R^l_m)}$$ is then referred to as the record coverage. The prediction problem in the two-sample case is then discussed and, specifically, the exact outer and inner prediction intervals are derived for order statistics intervals from an independent future Y-sample based on the m-th record coverage from the X-sequence when the underlying distribution of the two samples are the same. The coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Distribution-free prediction intervals as well as upper and lower prediction limits for spacings from a future Y-sample are obtained in terms of the record range from the X-sequence.

Panel count data usually refer to data arising from studies on recurrent events in which the subjects under study are followed or observed only periodically rather than continuously. In such situations, an objective of interest is about the occurrence of some events that can occur multiple times or repeatedly and the studies resulting in this type of information are often referred to as event history studies. There are many fields such as medical studies, reliability experiments and social sciences wherein panel count data are encountered commonly. This article reviews basic concepts about panel count data, some common issues and questions of interest regarding them as well as the corresponding statistical procedures that are suitable for their analysis. In particular, we will discuss an estimation of the mean function of the underlying counting process characterizing the occurrence of the events, comparison of several processes and analysis of multiple state panel count data. Some discussion is also presented of situations involving dependent or informative observation processes.

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 (100q%) 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.

Hoadley (Ann Math Stat 42:1977–1991, 1971) studied the weak law of large numbers for independent and non-identically distributed random variables. Using that result along with the missing information principle, we establish the consistency and asymptotic normality of maximum likelihood estimators based on progressively Type-II censored samples.

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.

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.

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.

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.

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.

In this paper we present analogues of Balakrishnan's (1989) relations that relate the triple and quadruple moments of order statistics from independent and nonidentically distributed (I.NI.D.) random variables from a symmetric distribution to those of the folded distribution. We then apply these results, along with the corresponding recurrence relations for the exponential distribution derived recently by Childs (2003), to study the robustness of the Winsorized variance.

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG 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 $$\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]$$ , whereF_n(x) andG_n(x) are the empirical distribution functions ofF(x) andG(x) respectively and $$\hat f(x)$$ and $$\hat g(x)$$ are taken to be the so-called kernel estimates off(x) andg(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., wheng(x)=f₀(x), withf₀(x) completely known and for testing symmetry, i.e., testingH₀:f(x)=f(−x).

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.

Summary

LetX=(X_ij)=(X₁, ...,X_n)’,X’_i=(X_i1, ...,X_ip)’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂^-2 be the squared multiple correlation coefficient between the first and the remainingp₂+p₃=p−1 components of eachX_i. We have considered here the problem of testingH₀:ϱ²=0 against the alternativesH₁:ϱ ₁^-2 =0, ϱ ₂^-2 >0 on the basis ofX andn₁ additional observationsY₁ (n₁×1) on the first component,n₂ observationsY₂(n₂×p₂) on the followingp₂ components andn₃ additional observationsY₃(n₃×p₃) on the lastp₃ components and we have derived here the locally minimax test ofH₀ againstH₁ when ϱ ₂^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.

(1995) derived the Fisher information and discussed the maximum likelihood estimation (MLE) of the parameters of a location-scale family $$ F\left( {\tfrac{{x - \mu }} {\sigma }} \right) $$ based on the ranked set sample (RSS). She found that a RSS provided more information about both μ and σ than a simple random sample (SRS) of the same size. We also focus here on the location-scale family. We use the idea of order statistics from independent and nonidentical random variables (INID) to propose an ordered ranked set sample (ORSS) and develop the Fisher information and the maximum likelihood estimation based on such an ORSS. We use logistic, normal, and one-parameter exponential distributions as examples and conclude that in all these three cases, the ORSS does not provide as much Fisher information as the RSS, and consequently the MLEs based on the ORSS (MLE-ORSS) are not as efficient as the MLEs based on the RSS (MLE-RSS). In addition to the MLEs, we are also interested in best linear unbiased estimators (BLUE). For this purpose, we apply another measure of information, viz., Tukey’s linear sensitivity. Tukey (1965) proposed linear sensitivity to measure information contained in an ordered sample. We use logistic, normal, one- and two-parameter exponential, two-parameter uniform, and right triangular distributions as examples and show that in all these cases except the one-parameter the RSS, and consequently the BLUEs based on the ORSS (BLUE-ORSS) are more efficient than the BLUEs based on the RSS (BLUE-RSS). In the case of one-parameter exponential, the ORSS has only slightly less information than the RSS with the relative efficiency being very close to 1.

Creation of a ranked set sample, by its nature, involves judgment ranking error within set units. This ranking error usually distorts statistical inference of the population characteristics. Tests may have inflated sizes, confidence intervals may have incorrect coverage probabilities, and the estimators may become biased. In this paper, we develop an exact two-sample nonparametric test for quantile shift between two populations based on ranked set samples. This test is based on two independent exact confidence intervals for the quantile of interest corresponding to the two populations and rejects the null hypothesis of equal quantiles if these intervals are disjoint. It is shown that a pair of 83 and 93% confidence intervals provide a 5 and 1% test for the equality of quantiles. The proposed test is calibrated for the effect of judgment ranking error so that the test has the correct size even under a wide range of judgment ranking errors. A small scale simulation study suggests that the test performs quite well for cycle sizes as small as 2.

The elliptical laws are a class of symmetrical probability models that include both lighter and heavier tailed distributions. These models may adapt well to the data, even when outliers exist and have other good theoretical properties and application perspectives. In this article, we present a new class of models, which is generated from symmetrical distributions in $${\mathbb{R}}$$ and generalize the well known inverse Gaussian distribution. Specifically, the density, distribution function, properties, transformations and moments of this new model are obtained. Also, a graphical analysis of the density is provided. Furthermore, we estimate parameters, propose asymptotic inference and discuss influence diagnostics by using likelihood methods for the new distribution. In particular, we show that the maximum likelihood estimates parameters of the new model under the t kernel are down-weighted for the outliers. Thus, smaller weights are attributed to outlying observations, which produce robust parameter estimates. Finally, an illustrative example with real data shows that the new distribution fits better to the data than some other well known probabilistic models.

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.

In this paper we develop recurrence relations for the third and fourth order moments of order statistics from I.NI.D exponential random variables. Recurrence relations for the p-outlier model (with a slippage of p observations) are derived as a special case. Applications of these results will also be described.

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 timeT, 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.

Progressively censored order statistics from heterogeneous distributions are introduced and their properties are investigated. After deriving the joint density function, some properties are established. In particular, the case of proportional hazards leads to an interesting connection to the model of generalized order statistics. Finally, the special case of exponential distribution is considered and some known results are generalized to this heterogeneous case, and their implications to robustness are highlighted.

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.

In the two-sample prediction problem, record values from the present sample may be used as predictors of order statistics from a future sample. In this paper, we investigate the nearness of record statistics (upper and lower) to order statistics from a location-scale family of distributions in the sense of Pitman closeness and discuss the corresponding monotonicity properties. We then determine the closest record value to a specific order statistic from a future sample. Even though in general it depends on the parent distribution, exact and explicit expressions are derived for the required probabilities in the case of exponential and uniform distributions, and some computational results are presented as well. Finally, we consider the mean squared error criterion and examine the corresponding results in the exponential case.

The vast majority of the bivariate exponential distributions arise in the reliability context one way or another. When we talk of reliability, we have in mind the failure of an item or death of a living organism. We especially think of time elapsing between the equipment being put into service and its failure. In the bivariate or multivariate context, we are concerned with dependencies between two failure times, such as those of two components of an electrical, mechanical, or biological system.

For reasons of time constraint and cost reduction, censoring is commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I right censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing for the removal of test units at non-terminal time points. In this article, we consider a progressively Type-I censored life-test under the assumption that the lifetime of each test unit is exponentially distributed. For small to moderate sample sizes, a practical modification is proposed to the censoring scheme in order to guarantee a feasible life-test under progressive Type-I censoring. Under this setup, we obtain the maximum likelihood estimator (MLE) of the unknown mean parameter and derive the exact sampling distribution of the MLE under the condition that its existence is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we then discuss the construction of confidence intervals for the mean parameter and their performance is assessed through Monte Carlo simulations. Finally, an example is presented in order to illustrate all the methods of inference discussed here.

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^.mconditional 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.

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider the simple step-stress model from the exponential distribution when there is time constraint on the duration of the experiment. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.

In this paper, we develop a flexible cure rate survival model by assuming the number of competing causes of the event of interest to follow a compound weighted Poisson distribution. This model is more flexible in terms of dispersion than the promotion time cure model. Moreover, it gives an interesting and realistic interpretation of the biological mechanism of the occurrence of event of interest as it includes a destructive process of the initial risk factors in a competitive scenario. In other words, what is recorded is only from the undamaged portion of the original number of risk factors.

This paper establishes the log-concavity property of several forms of univariate and multivariate skew-normal distributions. This property is then used to prove the monotonicity of the hazard as well as reversed hazard functions. The log-concavity and monotonicity of the hazard and reversed hazard functions of series and parallel systems of components is then discussed. The corresponding results for the multivariate normal distribution follow readily as special cases.

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.

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.

For general step-stress experiments with arbitrary baseline distributions, wherein the stress levels change immediately after having observed pre-specified numbers of observations under each stress level, a sequential order statistics model is proposed and associated inferential issues are discussed. Maximum likelihood estimators (MLEs) of the mean lifetimes at different stress levels are derived, and some useful properties of the MLEs are established. Joint MLEs are also derived when an additional location parameter is introduced into the model, and estimation under order restriction of the parameters at different stress levels is finally discussed.

In this paper, we discuss a general class of skew two-piece skew-normal distributions, denoted by GSTPSN(λ₁, λ₂, ρ). We derive its moment generating function and discuss some simple and interesting properties of this distribution. We then discuss the modes of these distributions and present a useful representation theorem as well. Next, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and discuss some of its properties. Finally, three well-known examples are used to illustrate the practical usefulness of this family of distributions.

A new class of entropy functions of discrete systems, the class of concave entropies, is introduced. Each concave entropy function satisfies the fundamental axioms of general entropies. The Shannon and trigonometric entropies belong to the class of concave entropies. Each of the classes of Renyi and polynomial entropies contains a subclass of concave entropies. A sufficient condition is given, under which the total entropy tends to ∞, when the number N of probability components approaches ∞.

We discuss here the problem of bivariate random scaling. Both direct and inverse problems of bivariate random scaling are solved by two methods. While the first method is a distributional one, the second method is an indirect one associated with bivariate Mellin transform. Finally, we use bivariate random scaling for some statistical and simulational applications.

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.

The univariate extreme-value distributions consist of types 1 (Gumbel), 2 (Fréchet), and 3. The three types can be transformed to each other. The type 3 distribution of (−X) is the usual Weibull distribution. In the bivariate context, marginals are of secondary interest compared with the dependence structure.

This paper considers the problem of multi-sample nonparametric comparison of mean functions of point processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct a class of nonparametric test statistics based on the integrated weighted differences between the estimated mean functions of the point processes. The asymptotic distributions of the proposed statistics are rigorously derived when the monotonicity assumptions for weight processes are removed, and their finite-sample properties are examined through Monte Carlo simulations. The simulation results show that the proposed methods are good for practical use and are slightly powerful than the existing tests. A set of panel count data from a cancer study is analyzed and presented as an illustrative example.

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.

Confidence intervals for quantiles and tolerance intervals based on ordered ranked set samples (ORSS) are discussed in this paper. For this purpose, we first derive the cdf of ORSS and the joint pdf of any two ORSS. In addition, we obtain the pdf and cdf of the difference of two ORSS, viz. $$X_{s:N}^{ORSS}-X_{r:N}^{ORSS}$$ , 1 ≤ r < s ≤ N. Then, confidence intervals for quantiles based on ORSS are derived and their properties are discussed. We compare with approximate confidence intervals for quantiles given by Chen (Journal of Statistical Planning and Inference, 83, 125–135; 2000), and show that these approximate confidence intervals are not very accurate. However, when the number of cycles in the RSS increases, these approximate confidence intervals become accurate even for small sample sizes. We also compare with intervals based on usual order statistics and find that the confidence interval based on ORSS becomes considerably narrower than the one based on usual order statistics when n becomes large. By using the cdf of $$X_{s:N}^{ORSS}-X_{r:N}^{ORSS}$$ , we then obtain tolerance intervals, discuss their properties, and present some tables for two-sided tolerance intervals.

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.