Catálogo de publicaciones - libros

Let X _(1)<...< X _ (n) be the order statistics from n independent nonidentically distributed exponential random variables. We investigate the dependence structure of these order statistics, and provide a distributional identity that facilitates their simulation and the study of their moment properties. Next, we consider the partial sum T _i=∑ _ j=i +1 ^n X _( j ), 0≥ i ≥ n −1. We obtain an explicit expression for the cdf of T _ i , exploiting the memoryless property of the exponential distribution. We do this for the identically distributed case as well, and compare the properties of T _ i under the two settings.

( 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.

This paper consists of three sections. The first section gives an overview of the basic information functions, their interpretations, and dynamic information measures that have been recently developed for lifetime distributions. The second section summarizes the information features of univariate Pareto distributions, tabulates transformations of a Pareto random variable under which information measures of numerous distributions can be obtained, and gives a few characterizations of the generalized Pareto distribution. The final section summarizes information measures for order statistics and tabulates the expressions for Shannon entropies of order statistics for numerous distributions.

Non-parametric “sign” intervals for a parent median based on order statistics have the important property of being generally valid. With small sample sizes, the available confidence coefficients (CCs) are sparse, however, and it is natural to try to interpolate between adjacent sign intervals to attain intermediate levels. This chapter provides the CC associated with weighted means of adjacent sign intervals over some interesting classes of parent distributions, including: (a) all distributions, (b) all symmetric distributions, and (c) all symmetric and unimodal distributions. The behavior of these CCs as functions of the weight is simple but intuitively quite surprising, with certain discontinuities and intervals of constancy. Some unexpected domination relations among weighted means of adjacent sign intervals follow from these results. The resulting nondominated intervals constitute a considerable extension of the sign intervals, with substantially more confidence-coefficient levels; and they are valid under no or weak shape assumptions about the parent distribution.

In this chapter, we give conditional representations for families of statistics based on higher-order spacings and spaing frequencies. This allows us to compute accurate approximations to the distribution of such statistics, including tail probabilities and critical values. These results generalize those discussed in ( 1999 ) and are essential in using such statistics in various testing contexts.

We present two optimal bounds on the expectations of arbitrary L -statistics based on i.i.d. samples with a bounded support expressed in the support length units. One depends on the location of the population mean in the support interval, and the other is general. The results are explicitly described in the special cases of single-order statistics and their differences.

Mixtures of distributions of lifetimes occur in many settings. In engineering applications, it is often the case that populations are heterogeneous, often with a small number of subpopulations. In survival analysis, selection effects can often occur. The concept of a failure rate in these settings becomes a complicated topic, especially when one attempts to interpret the shape as a function of time. Even if the failure rates of the subpopulations of the mixture have simple geometric or parametric forms, the shape of the mixture is often not transparent. Recent results, developed by the author (with Joe, Li, Mi, Savits, and Wondmagegnehu) in a series of papers, are presented. These results focus on general results concerning the asymptotic limit and eventual monotonicity of a mixture, and also the overall behavior for mixtures of specific parametric families. An overall picture is given of different things that influence the behavior of the failure rate of a mixture.

The signature of a system of components with independent and identically distributed (iid) lifetimes is a probability vector whose i th element represents the probability that the i th component failure causes the system to fail. ( 1985 ) introduced the concept and used it to characterize the class of systems that have increasing failure rates (IFR) when the components are iid IFR. ( 1999 ) showed that when signatures are viewed as discrete probability distributions, the stochastic, hazard rate or likelihood ratio ordering of two signature vectors implies the same ordering of the lifetimes of the corresponding systems in iid components. In this paper, these latter results are extended in a variety of ways. For example, conditions on system signatures are identified that are not only sufficient for such orderings of lifetimes to hold, but are also necessary. More generally, given any two coherent systems whose iid components have survival functions S _ i (t) and failure rates r _ i (t) , respectively, for i =1,2, the number and locations of crossings of the systems’ survival functions or failure rates in (0, ∞) can be fully specified in terms of the two system signatures. One is thus able to deduce how these systems compare to each other in real time, in contrast to the asymptotic comparisons one finds in the literature.

The life lengths of some possibly dependent components in a system can be modelled by a multivariate distribution. In this paper, we suppose that the joint distribution of the units is a symmetric multivariate Gumbel exponential distribution (GED). Hence, the components are exchangeable and have exponential (marginal) distributions. For this model, we obtain basic reliability properties for k -out-of- n systems (order statistics) and, in particular, for series and parallel systems. We pay special attention to systems with two components. Some results are extended to coherent systems with n exchangeable components.

This paper considers the step-stress accelerated life tests (ALT) on an exponential population with mean θ. The MLEs of θ are studied for different data structures that include grouped data and censored data. Here, by grouped data we mean that instead of observing the exact failure times, only numbers of failures in some predetermined subintervals are available. Applying the tampered failure rate (TFR) model, we show the existence, uniqueness, strong consistency, and the asymptotic normality of the MLE of θ. An upper bound of the MLE of θ based on the grouped data is also derived.