On some generalized ageing orderings

Some partial orderings that compare probability distributions with the exponential distribution are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which describe the same kind of phenomenon of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.     

Dynamic mixing probability measures of mixtures

ABSTRACT

Lifetime of heterogeneous population can be modeled as mixture of a family of lifetime distributions according to a mixing probability measure. With the help of dynamic mixing measure, the hazard rate of the mixture can also be expressed as the mixture of the hazard rates of the lifetime distributions. Various local stochastic orderings are defined in this article. Applying these local stochastic orderings, we can explore the behavior of the dynamic mixing measures locally and then compare the hazard rates of two heterogeneous populations in both the local and global ways.     

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.     

Rényi entropy of m-generalized order statistics

Abstract

Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.     

Proportional hazards model for discrete data: Some new developments

ABSTRACT

Many times, a product lifetime can be described through a non negative integer valued random variable. In this article, we propose a proportional hazards model for discrete data analogous to the version for continuous data. Some ageing properties of the model are discussed. Stochastic comparison of pair of random variables that follow the model are also made. A new test based on U-statistics is developed for testing that the proportionality parameter in the proposed model is 1. The asymptotic properties of the proposed test are studied. We present some numerical results to asses the performance of the test procedure.     

Some ordering properties of series and parallel systems with dependent component lifetimes

Abstract

In this paper, we consider series systems and parallel systems with the dependence between the component lifetimes modelled by an Archimedean copulas. We obtain sufficient and necessary conditions of relative ageing orders between series (parallel) systems with different component numbers, which partially generalize some main results of Misra and Francis. When the component lifetimes follow the scale model, we also characterize the ordering properties between the series systems and (n–1)-out-of-n systems (parallel systems and 2-out-of-n systems) by mixture distribution.     

Nonparametric inference on mean residual life function with length-biased right-censored data

Abstract

In survival or reliability studies, the mean residual life (MRL) function is an important characteristic in understanding the survival or ageing process. In this article, we consider the problem of nonparametric MRL function estimation with length-biased right-censored data. Two nonparametric estimators of the MRL are proposed and their weak convergence is presented. In order to evaluate the performance of these estimators, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided.     

Metrics on permutations useful for positive dependence

For two permutations i and j of the integers 1, …, n, a number of different metrics measuring the distance between i and j are studied in a unified fashion. Attention is focused on two new metrics and on two previously studied metrics. One of the new metrics is related to the more associated partial ordering between pairs of bivariate random vectors. Results are also given connecting these metrics to a number of previously considered partial orderings on the set of all permutations of 1, …, n where these partial orderings were used to describe when one bivariate empirical distribution function was more positively dependent than another.     

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ?$?$. In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.     

Robust Estimators for the Parameters of Multivariate Lognormal Distribution

Abstract

In this paper, we introduce a class of location and scale estimators for the p-variate lognormal distribution. These estimators are obtained by applying a log transform to the data, computing robust Fisher consistent estimators for the obtained Gaussian data and transforming those estimators for the lognormal using the relationship between the parameters of both distributions. We prove some of the properties of these estimators, such as Fisher consistency, robustness and asymptotic normality.     

Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles

Yanagimoto and Okamoto (1969) introduced a stochastic ordering that generalizes a concept of monotone regression dependence introduced by Lehmann (1966). In this paper, we define and examine the properties of three new orderings which imply that of Yanagimoto and Okamoto. One of these orderings is seen to extend Shaked's (1977) notion of DTP(0, 1), and another includes Lehmann's concept of positive likelihood-ratio dependence as a special case. The proposed orderings are also compared with the TP₂ positive-dependence ordering defined by Kimeldorf and Sampson (1987).     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

Testing the Equality of Multivariate Distributions Using the Bootstrap and Integrated Empirical Processes

We introduce some projected integrated empirical processes for testing the equality of two multivariate distributions. The bootstrap is used for determining the approximate critical values. We show that the bootstrap test is consistent. A number-theoretic method is used for efficient computation of the bootstrap critical values. Some simulation results are also given.     

On zero-distorted generalized geometric distribution

ABSTRACT

We propose a new generalized geometric distribution which permits inflation/deflation of the zero count probability and study some of its properties. We also present an actuarial application of this distribution and fit it to three datasets used by other researchers. It is observed that the proposed distribution fits reasonably well to these data. Further, in a regression setup, the performance of this distribution is studied vis–a–vis other competing distributions used for explaining variability in a response variable.     

Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

The echelon Markov and Chebyshev inequalities

Abstract

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.     

Kurtosis comartsions of the cauchy and double exponential distributions

We first describe a class of quantile-based kurtosis orderings on symmetric distributions that use density matching to match the scales of distributions before kurtosis comparisons are made. We then use the orderings to give a meaningful comparison of the kurtosis properties of the Cauchy and Double Exponential distributions. Since these distributions are often used as models for heavy-tailed distributions and there appears some confusion about their properties such a comparison should be useful.     

Saddlepoint approximations for subordinator processes

ABSTRACT

We develop the saddlepoint approximations in obtaining the transition functions for general subordinator processes. We derive explicit expressions of the first- and second-order approximations. Specifically, we consider some particular classes of subordinators including the Poisson processes, the Gamma processes, the α-stable subordinators, and the Poisson random integrals. We test this technique on the Poisson and Gamma processes, which have closed-form transition functions. Outcomes show that the approximate expressions are consistent with the true transition functions. We then use this method to predict transition density functions for the α-stable subordinator processes. Finally, we calculate approximated transition densities for some Poisson random integrations. Numerical analysis shows the perfect ability of the saddlepoint approximations to predict the transition densities of the α-stable processes and the Poisson random integrations.     

A general family of NBU classes of life distributions

In this paper we introduce and study a family of new better than used ageing notions parameterized by a function h. The new better than used , the new better than used of a specified age and the new better than used in total time on test transform order , are special cases of this new family. We study some properties of the new family, and we give some applications of it in actuarial science, reliability theory and statistics.