The ageing curve and partial orderings of life distributions

In this paper we define a new measure to describe the process of ageing of lifetime distributions. In terms of this measure we will define three partial orderings of lifetime distributions. The relationships between these orderings and other classical orderings are studied.     

Distance measures in a one-parameter class of density functions

In a class of density functions with parameter λ > 0 , which includes Weibull and Gamma distributions, three distance measures are examined. Several properties of these measures and relationships between them are proved. Furthermore the exponential distribution is characterized in a one- parameter class of Weibull distributions.     

Relationship between the weighted distributions and some inequality measures

In this paper, we study the relationships between the weighted distributions and the parent distributions in the context of Lorenz curve, Lorenz ordering and inequality measures. These relationships depend on the nature of the weight functions and give rise to interesting connections. The properties of weighted distributions for general weight functions are also investigated. It is shown how to derive and to determine characterizations related to Lorenz curve and other inequality measures for the cases weight functions are increasing or decreasing. Some of the results are applied for special cases of the weighted distributions. We represent the reliability measures of weighted distributions by the inequality measures to obtain some results. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context by concentration curves. We also review and extend the problem of stochastic orderings and aging classes under weighting. Finally, the relationships between the weighted distribution and transformations are discussed.     

Delineating bird populations using ring recoveries

We delineate bird populations using cluster analysis to group ringing sites based on pairwise comparisons of recoveries. Clustering provides a quantitative (but non-unique) grouping that can be used to examine the relationships of bird distributions at both local and regional geographic scales. Clustering is based on similarity matrices composed of pairwise comparisons of recovery distributions from ringing sites. We demonstrate the method using mallard ( Anas platyrhynchos ) ring recoveries to group ringing sites in south-central Canada, and discuss the possibilities for these analyses for non-hunted species with few recoveries.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.     

A class of weighted multivariate elliptical models useful for robust analysis of nonnormal and bimodal data

A class of weighted elliptical models useful for analyzing nonnormal and bimodal multivariate data is introduced. It is obtained from the marginal distribution of a centrally truncated multivariate elliptical distribution. As a special case, a finite mixture of weighted multinormal distribution is examined in detail, establishing connections with the multinormal and the finite mixture of multinormal. The special class of distributions is studied from several aspects such as weighting of probability density functions, association with centrally truncated distributions, and a finite scale mixture scheme. The relationships among these aspects are given, and various properties of the class are also discussed. For the inference of the class, an MCMC procedure and its numerical example are provided.     

Series evaluation of Tweedie exponential dispersion model densities

Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.     

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

Properties of Entropies of Record Values in Reliability and Life Testing Context

In this article, the entropies of record value distributions from some continuous probability models are computed and their properties are investigated, both analytically when feasible, and numerically. In an attempt to establish relationships between entropies of the parent and the corresponding record value distributions, the entropies of record value distributions associated with the uniform, exponential, Weibull, classical Pareto, normal, gamma, beta, and Cauchy distributions are considered in this article. The entropy of record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, have tractable closed forms. The entropies of record value distributions associated with the normal, gamma, beta, and Cauchy distributions do not have tractable closed forms and need further investigations. Some general conclusions are drawn in the final section.     

Stochastic ordering and a class of multivariate new better than used distributions

A new characterization for the univariate class of new better than used ‘NBU’ distributions in terms of stochastic ordering is introduced. A multivariate version of this characterization is then used to define a multivariate class of NBU distributions. Basic properties of this class are derived. Comparisons and relationships of this new class with earlier classes are developed. Two multivariate new worse than used (NWU) classes of life distributions are defined and compared and their basic properties are studied.     

D-distribution and its applications

We derive the distribution of the sum ofn independent doubly truncated Poisson variables not necessarily identically distributed. This distribution is called a D-distribution. Its p.d.f. can be expressed in terms of a D-number and an incomplete exponential function which are both defined in this paper. We investigate the relationship between these numbers and distributions, and use these relationships to derive recurrence relations and other properties of the D-distribution. A minimum variance unbiased estimates of the p.d.f. of this D-distribution is also obtained. Some example are included at the end to illustrate the use of this D-distribution.     

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.     

Lorenz ordering of order statistics from log-logistic and related distributions

This paper obtains Lorenz ordering relationships among order statistics from log-logistic samples of possibly different sizes. Some results extend other families including the Lomax, Burr III and Burr XII distributions.     

A Class of Multivariate Bilateral Selection t Distributions and Its Properties

This article proposes a class of multivariate bilateral selection t distributions useful for analyzing non-normal (skewed and/or bimodal) multivariate data. The class is associated with a bilateral selection mechanism, and it is obtained from a marginal distribution of the centrally truncated multivariate t. It is flexible enough to include the multivariate t and multivariate skew-t distributions and mathematically tractable enough to account for central truncation of a hidden t variable. The class, closed under linear transformation, marginal, and conditional operations, is studied from several aspects such as shape of the probability density function, conditioning of a distribution, scale mixtures of multivariate normal, and a probabilistic representation. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

An Improved Range Estimator of Sigma for Determining Sample Sizes

The use of a range estimator of the population standard deviation, sigma (σ), for determining sample sizes is discussed in this study. Standardized mean ranges (d_n's), when divided into the ranges of sampling frames, provide estimates of the standard deviation of the population. These estimates can be used for determining sample sizes. The d_n's are provided for seven different distributions for sampling frame sizes that range from 2 to 2000, For each of the seven distributions, functional relationships are developed such that d_n = f(n_SF) where n_SF is the size of the sample frame. From these functions, d_n's can be estimated for sampling frame sizes which are not presented in the study.     

Some Properties of Weighted Distributions in the Context of Repairable Systems

ABSTRACT

In this article we introduce some structural relationships between weighted and original variables in the context of maintainability function and reversed repair rate. Furthermore, we prove some characterization theorems for specific models such as power, exponential, Pareto II, beta, and Pearson system of distributions using the relationships between the original and weighted random variables.     

Characterizations through reliability measures from weighted distributions

In this paper we obtain general characterizations of probability distributions from relationships between failure rate and mean residual life from the origin distribution and associated weighted distribution. Our characterization properties extend particular results given by Gupta and Keating (1986), Jain et al. (1989) and Asadi (1998). Using the theoretical results we obtain characterizations of some usual distributions. Received: December 7, 1998; revised version: May 10, 2000     

Multiple Imputation of Missing or Faulty Values Under Linear Constraints

Many statistical agencies, survey organizations, and research centers collect data that suffer from item nonresponse and erroneous or inconsistent values. These data may be required to satisfy linear constraints, for example, bounds on individual variables and inequalities for ratios or sums of variables. Often these constraints are designed to identify faulty values, which then are blanked and imputed. The data also may exhibit complex distributional features, including nonlinear relationships and highly nonnormal distributions. We present a fully Bayesian, joint model for modeling or imputing data with missing/blanked values under linear constraints that (i) automatically incorporates the constraints in inferences and imputations, and (ii) uses a flexible Dirichlet process mixture of multivariate normal distributions to reflect complex distributional features. Our strategy for estimation is to augment the observed data with draws from a hypothetical population in which the constraints are not present, thereby taking advantage of computationally expedient methods for fitting mixture models. Missing/blanked items are sampled from their posterior distribution using the Hit-and-Run sampler, which guarantees that all imputations satisfy the constraints. We illustrate the approach using manufacturing data from Colombia, examining the potential to preserve joint distributions and a regression from the plant productivity literature. Supplementary materials for this article are available online.     

A Bayesian hierarchical mixture model for platelet-derived growth factor receptor phosphorylation to improve estimation of progression-free survival in prostate cancer

Summary.  Advances in understanding the biological underpinnings of many cancers have led increasingly to the use of molecularly targeted anticancer therapies. Because the platelet-derived growth factor receptor (PDGFR) has been implicated in the progression of prostate cancer bone metastases, it is of great interest to examine possible relationships between PDGFR inhibition and therapeutic outcomes. We analyse the association between change in activated PDGFR (phosphorylated PDGFR) and progression-free survival time based on large within-patient samples of cell-specific phosphorylated PDGFR values taken before and after treatment from each of 88 prostate cancer patients. To utilize these paired samples as covariate data in a regression model for progression-free survival time, and be cause the phosphorylated PDGFR distributions are bimodal, we first employ a Bayesian hierarchical mixture model to obtain a deconvolution of the pretreatment and post-treatment within-patient phosphorylated PDGFR distributions. We evaluate fits of the mixture model and a non-mixture model that ignores the bimodality by using a supnorm metric to compare the empirical distribution of each phosphorylated PDGFR data set with the corresponding fitted distribution under each model. Our results show that first using the mixture model to account for the bimodality of the within-patient phosphorylated PDGFR distributions, and then using the posterior within-patient component mean changes in phosphorylated PDGFR so obtained as covariates in the regression model for progression-free survival time, provides an improved estimation.     

关于斜Laplace分布与Levy偏稳定分布的性质

目前有关重尾或偏态数据的统计分析和理论模型相对较少,基于传统的Laplace分布,提出一种处理偏态和重尾数据的新模型——斜Laplace分布,以研究其参数估计方法。利用数理统计知识推导出该分布与一些常见分布(如正态分布、指数分布)间的统计关系,并给出一种可通过设置不同参数值得到不同分布的Levy偏稳定分布及其稳定性。