A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

Multivariate discrete reversed hazard rates

In the present paper, we define and study four versions of multivariate discrete reversed hazard rates, namely scalar reversed hazard rate, vector reversed hazard rate, alternative reversed hazard rate, and conditional reversed hazard rate. Various properties of these functions are studied. Interrelationships between these reversed hazard rates are explored. We also present characterization of discrete distributions using these reversed hazard rates.     

Alternative expectation formulas for real-valued random vectors

Abstract

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.     

Inequalities involving expectations of selected functions in reliability theory to characterize distributions

Recently, authors have studied inequalities involving expectations of selected functions, viz. failure rate, mean residual life, aging intensity function, and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time (X ? x, instead of X > x) and their applications. In the present work we consider reversed hazard rate, expected inactivity time, and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.     

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood‐based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.     

Hazard rate ordering of the second-order statistics from multiple-outlier PHR samples

In this paper, we compare the hazard rate functions of the second-order statistics arising from two sets of independent multiple-outlier proportional hazard rates (PHR) samples. It is proved that the submajorization order between the sample size vectors together with the supermajorization order between the hazard rate vectors imply the hazard rate ordering between the corresponding second-order statistics from multiple-outlier PHR random variables. The results established here provide theoretical guidance both for the winner's price for the bid in the second-price reverse auction in auction theory and fail-safe system design in reliability. Some numerical examples are also provided for illustration.     

A Mixture Model of Proportional Reversed Hazard Rate

This article investigates properties of mixture model of proportional reversed hazard rate. Firstly, the mixing random variable and the overall population variable are proved to be positively likelihood dependent. Secondly, lower bounds for the distribution function as well as the conditional distribution are established in the case that the mixing variable belongs to certain nonparametric classes. Finally, some stochastic orders on the mixing (baseline) variables are proved to be translated to the corresponding overall population variables.     

Gaussian copula distributions for mixed data,with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.     

Generalizations of a contamination model for continuous type random variables

In 1965, Stanley Warner (Warner, 1965) introduced a model for contaminating discrete type random variables. He presented this scheme as being potentially useful in survevs where sensitive in-formation is being gathered. Since that time much research has been conducted and many papers written on the development of these discrete type randomized response models. More recently, atten-tion has been focused on the application of randomized response type models for preservation of confidentiality in existing data files (Boruch 1971 and 1972, Ranney 1975, Felligi 1974, and Inge-marsson 1975). In 1974, Poole (Poole, 1974) introduced a randomized response model for a positive continuous type random variable which was basically a continuous variable analog of the discrete variable Warner model. In this paper the results of the 1974 paper are extended to a lt-dimensional continuous type random variable in k-dimensional Euclidean space.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

A coefficient of multiple association based on ranks

A generalization of Kendall's tau is formulated for describing the association between a dependent variable and a collection of independent variables. The coefficient may be defined in terms of the proportional reduction in prediction errors obtained by predicting the ordering of pairs of observations on the dependent variable based on orderings of the pairs on the independent variables. The coefficient is formulated both for continuous and discrete variables. Approximate large-sample distributions are considered for both cases. Some of the properties of this coefficient are discussed and compared with those of other multiple measures of association based on ranks.     

Multivariate survival mixed models for genetic analysis of longevity traits

A class of multivariate mixed survival models for continuous and discrete time with a complex covariance structure is introduced in a context of quantitative genetic applications. The methods introduced can be used in many applications in quantitative genetics although the discussion presented concentrates on longevity studies. The framework presented allows to combine models based on continuous time with models based on discrete time in a joint analysis. The continuous time models are approximations of the frailty model in which the baseline hazard function will be assumed to be piece-wise constant. The discrete time models used are multivariate variants of the discrete relative risk models. These models allow for regular parametric likelihood-based inference by exploring a coincidence of their likelihood functions and the likelihood functions of suitably defined multivariate generalized linear mixed models. The models include a dispersion parameter, which is essential for obtaining a decomposition of the variance of the trait of interest as a sum of parcels representing the additive genetic effects, environmental effects and unspecified sources of variability; as required in quantitative genetic applications. The methods presented are implemented in such a way that large and complex quantitative genetic data can be analyzed. Some key model control techniques are discussed in a supplementary online material.     

Competing risk and the Cox proportional hazard model

We propose a heuristic for evaluating model adequacy for the Cox proportional hazard model by comparing the population cumulative hazard with the baseline cumulative hazard. We illustrate how recent results from the theory of competing risk can contribute to analysis of data with the Cox proportional hazard model. A classical theorem on independent competing risks allows us to assess model adequacy under the hypothesis of random right censoring, and a recent result on mixtures of exponentials predicts the patterns of the conditional subsurvival functions of random right censored data if the proportional hazard model holds.     

Reduced-Form Versus Structural Models of Water Demand Under Nonlinear Prices

Increasing-block prices are common in markets for water, cellular phone service, and retail electricity. This study estimates demand models under block prices and conducts a Monte Carlo experiment to test the small-sample bias of structural and instrumental variables (IV) estimators. We estimate the price and income elasticity of water demand under increasing-block prices using a structural discrete/continuous choice (DCC) model, as well as random effects and IV. Elasticity estimates are sensitive to the modeling framework. The Monte Carlo experiment suggests that IV and DCC models estimate both price and income elasticity with bias, with no clear best choice among estimators.     

On characterizations and variance bounds of discrete α-unimodality

Characterizations of α-unimodality for integer-valued random variables about a specific mode are established in terms of their probability mass functions, distribution functions and characteristic functions. Using these characterizations variance lower bounds in terms of α and the mode are derived. For α=1 all these results are reduced to ordinary unimodality. The new variance lower bounds for discrete unimodality is sharper than its continuous counterpart. An upper bound for the variance of discrete unimodal distribution defined on a finite support is discussed.     

On Discrete Gamma Distribution

A new discrete counterpart of gamma distribution for modelling discrete life data is defined based on similar mathematical form and properties of the continuous version. The main statistical and reliability properties of this distribution are derived and it is shown that this model can deal with both over and under-dispersed data. Geometric variables and finite sum of geometric variables, i.e., negative binomial are shown to be special cases of the proposed discrete gamma. Also, the size-biased discrete gamma distribution is derived and discussed. Moreover, different estimation methods of the underlying parameters of this distribution are utilized and comparisons of their performance have been made. Finally, an application in real-life data is used to elucidate the earlier results of this article.     

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.     

Approaching in Distribution with Applications to Resampling of Stochastic Processes

We introduce the notion of weak approaching and conditionally weak approaching sequences of random processes. This notion generalizes the conventional weak convergence, and has been proposed for real valued random variables in Belyaev (1995). Some of the standard tools for an investigation of the behaviour of weak approaching sequences of random elements in metric spaces are developed. The spaces of smoothed and right-continuous functions with left-hand limits are considered. This technique allows us to use the resampling approach for an evaluation of distributions of continuous functionals on realizations of sum of an increasing number of independent random processes. Two numerical examples are presented for such functionals as supremum and number of level crossings.     

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.     

Mark-specific additive hazards regression with continuous marks

For survival data, mark variables are only observed at uncensored failure times, and it is of interest to investigate whether there is any relationship between the failure time and the mark variable. The additive hazards model, focusing on hazard differences rather than hazard ratios, has been widely used in practice. In this article, we propose a mark-specific additive hazards model in which both the regression coefficient functions and the baseline hazard function depend nonparametrically on a continuous mark. An estimating equation approach is developed to estimate the regression functions, and the asymptotic properties of the resulting estimators are established. In addition, some formal hypothesis tests are constructed for various hypotheses concerning the mark-specific treatment effects. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a data set from the first HIV vaccine efficacy trial is provided.