Chernoff distance for doubly truncated distributions

In a recent paper, Nair et al. [Stat Pap 52:893–909, 2011] proposed Chernoff distance measure for left/right-truncated random variables and studied their properties in the context of reliability analysis. Here we extend the definition of Chernoff distance for doubly truncated distributions. This measure may help the information theorists and reliability analysts to study the various characteristics of a system/component when it fails between two time points. We study some properties of this measure and obtain its upper and lower bounds. We also study the interval Chernoff distance between the original and weighted distributions. These results generalize and enhance the related existing results that are developed based on Chernoff distance for one-sided truncated random variables.     

Chernoff distance for truncated distributions

In the present paper we extend the definition of Chernoff distance considered in Akahira (Ann Inst Stat Math 48:349–364, 1996) for truncated distributions and examine its properties. The relationship of this measure with other discrimination measures is examined. We study Chernoff distance between the original and weighted distributions. We also provide a characterization result for the proportional hazards model using the functional form of Chernoff distance.     

Measures of inaccuracy in record values

In this paper , we consider a measure of inaccuracy between distributions of the nth record value and parent random variable. We also propose the measure of residual inaccuracy of record values and study characterization results of dynamic cumulative residual inaccuracy measure. We discuss some properties of the proposed measures.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

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.     

Characterizations of Bivariate Models Using Some Dynamic Conditional Information Divergence Measures

In this article, we study some relevant information divergence measures viz. Renyi divergence and Kerridge’s inaccuracy measures. These measures are extended to conditionally specified models and they are used to characterize some bivariate distributions using the concepts of weighted and proportional hazard rate models. Moreover, some bounds are obtained for these measures using the likelihood ratio order.     

Acceptance sampling plans based on truncated life tests for weighted exponential distribution

Gupta and Kundu proposed a new class of weighted exponential distributions using the idea of Azzalini. In this article, we develop an acceptance sampling plan for the weighted exponential distribution under a truncated life test. For various acceptance numbers, consumer’s confidence levels and values of the ratio of the experimental time to the specified mean lifetime, the minimum sample size necessary to ensure a certain mean lifetime are obtained. The operating characteristic function values and the associated producer’s risks are also presented. A numerical example is provided to illustrate the acceptance sampling plan.     

Influence analysis in truncated distributions

Conditional bias and asymptotic mean sensitivity curve (AMSC) are useful measures to assess the possible effect of an observation on an estimator when sampling from a parametric model. In this paper we obtain expressions for these measures in truncated distributions and study their theoretical properties. Specific results are given for the UMVUE of a parametric function. We note that the AMSC for the UMVUE in truncated distributions verifies some of the most relevant properties we got in a previous paper for the AMSC of UMVUE in the NEF-QVF case, main differences are also established. As for the conditional bias, since it is a finite sample measure, we include some practical examples to illustrate its behaviour when the sample size increases.     

The weighted log-rank class under truncated binomial design: saddlepoint p-values and confidence intervals

The randomization design used to collect the data provides basis for the exact distributions of the permutation tests. The truncated binomial design is one of the commonly used designs for forcing balance in clinical trials to eliminate experimental bias. In this article, we consider the exact distribution of the weighted log-rank class of tests for censored data under the truncated binomial design. A double saddlepoint approximation for p-values of this class is derived under the truncated binomial design. The speed and accuracy of the saddlepoint approximation over the normal asymptotic facilitate the inversion of the weighted log-rank tests to determine nominal 95% confidence intervals for treatment effect with right censored data.     

A class of weighted normal distributions and its variants useful for inequality constrained analysis

This article develops a class of the weighted normal distributions for which the probability density function has the form of a product of a normal density and a weight function. The class constitutes marginal distributions obtained from various kinds of doubly truncated bivariate normal distributions. This class of distributions strictly includes the normal, skew–normal and two-piece skew–normal and is useful for selection modelling and inequality constrained normal mean analysis. Some distributional properties and Bayesian perspectives of the class are given. Probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify distribution and to implement computation, with output readily adapted for required analysis. Necessary theories and illustrative examples are provided.     

Behavior of agreement measures in the presence of zero cells and biased marginal distributions

Kappa and B assess agreement between two observers independently classifying N units into k categories. We study their behavior under zero cells in the contingency table and unbalanced asymmetric marginal distributions. Zero cells arise when a cross-classification is never endorsed by both observers; biased marginal distributions occur when some categories are preferred differently between the observers. Simulations studied the distributions of the unweighted and weighted statistics for k=4, under fixed proportions of diagonal agreement and different patterns off-diagonal, with various sample sizes, and under various zero cell count scenarios. Marginal distributions were first uniform and homogeneous, and then unbalanced asymmetric distributions. Results for unweighted kappa and B statistics were comparable to work of Muñoz and Bangdiwala, even with zero cells. A slight increased variation was observed as the sample size decreased. Weighted statistics did show greater variation as the number of zero cells increased, with weighted kappa increasing substantially more than weighted B. Under biased marginal distributions, weighted kappa with Cicchetti weights were higher than with squared weights. Both statistics for observer agreement behaved well under zero cells. The weighted B was less variable than the weighted kappa under similar circumstances and different weights. In general, B's performance and graphical interpretation make it preferable to kappa under the studied scenarios.     

Moments of truncated normal/independent distributions

In this work we have considered the problem of finding the moments of a doubly truncated member of the class of normal/independent distributions. We obtained a general result and then use it to derive the moments in the case of doubly truncated versions of Pearson type VII distribution, slash distribution, contaminated normal distribution, double exponential distribution and variance gamma distribution. We also give an application of some actuarial data.     

The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling

The minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.     

Some results on the truncated multivariate t distribution

The use of truncated distributions arises often in a wide variety of scientific problems. In the literature, there are a lot of sampling schemes and proposals developed for various specific truncated distributions. So far, however, the study of the truncated multivariate t (TMVT) distribution is rarely discussed. In this paper, we first present general formulae for computing the first two moments of the TMVT distribution under the double truncation. We formulate the results as analytic matrix expressions, which can be directly computed in existing software. Results for the left and right truncation can be viewed as special cases. We then apply the slice sampling algorithm to generate random variates from the TMVT distribution by introducing auxiliary variables. This strategic approach can result in a series of full conditional densities that are of uniform distributions. Finally, several examples and practical applications are given to illustrate the effectiveness and importance of the proposed results.     

On the Bayesian estimation of the weighted Lindley distribution

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.     

The Diffusion of Innovations: A Methodological Reappraisal

Studies of diffusion have traditionally relied on specific distributions, primarily the logistic, to characterize and estimate those processes. We argue that such an approach gives rise to serious problems of comparability and interpretation and may result in large biases in the estimates of the parameters of interest. We propose instead the Gini's expected mean difference as a measure of diffusion speed. We discuss its advantages over the traditional approach, present a nonparametric estimation procedure, and tackle with it the problems of truncated processes and of intergroup comparisons. We also elaborate on the use of the hazard rate and suggest various extensions. The diffusion of computed tomography scanners is presented as an illustration.     

Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models

Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equidispersed distributions and is limited in its ability to capture bimodality. The Conway–Maxwell–Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for over- and underdispersion. In this work, we propose a mixture of CMPs for capturing a wide range of truncated discrete data, which can exhibit unimodal and bimodal behavior. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M step to estimate multiple parameters. We examine computational and theoretical issues. The methods are illustrated for modeling ordered rating data as well as truncated count data, using simulated and real examples.     

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.     

A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.     

Confidence intervals for discrete distributions based on a single observation

Confidence intervals are developed for the mode of a discrete unimodal distribution in the case where only a single observation is available. These intervals are centered on either the observation, X, or a weighted average of X with a constant, b, chosen by the investigator. Intervals are derived for nonrestricted unimodal distributions, for unimodal distributions with a symmetry property, and for a family of two-sided truncated geometric distributions.