On additive and multiplicative damage models and the characterizations of linear and logarithmic exponential families

Rao (1963) has formulated a damage model which we call an additive damage model. A suitable damage model, which we call a multiplicative damage model, has been considered by Krishnaji (1970) for income-related problems. In these models, an original observation is subjected to damage, e.g., death or under-reporting, according to a specified probability law. Within the framework of an additive damage model, with a special form of damage, characterizations of the linear and logarithmic exponential families are formulated using regression properties of the damaged part on the undamaged part. The characterizations of the gamma and Pareto distributions that have been found of some use in the theory of income distributions, are obtained as special cases. Similar results are investigated within the framework of the multiplicative damage model.     

On random variate generation for the generalized hyperbolic secant distributions

We give random variate generators for the generalized hyperbolic secant distribution and related families such as Morris's skewed generalized hyperbolic secant family and a family introduced by Laha and Lukacs. The rejection method generators are uniformly fast over the parameter space and are based upon a complex function representation of the distributions due to Harkness and HarknessThe author's research was sponsored by NSERC Grant A3456 and FCAR Grant 90-ER-0291.     

ANALYSIS OF DOUBLY CENSORED DATA AND THE HIV INCUBATION PERIOD DISTRIBUTION IN WESTERN AUSTRALIA

The paper analyses the distribution of times from HIV seroconversion to the first AIDS defining illness for a subcohort of the Western Australian HIV Cohort Study for whom the seroconversion date is known to fall within a calendar time window. The analysis is based on a generalised gamma model for the incubation times and a piecewise constant distribution for the conditional times of seroconversion given the seroconversion windows. This allows flexible hazard shapes and also allows comparison of goodness of fit of the gamma and Weibull distributions which are often used for modelling incubation times. Computational issues are discussed. In these data, neither age at seroconversion, nor calendar time of seroconversion, nor the identification of a seroconversion illness appears to afFect incubation distributions. The Weibull distribution appears to provide a reasonable fit. The distribution of times from seroconversion to an HIV-related death is also briefly considered.     

Inequalities for linear combinations of gamma random variables

We study the behavior of the tail probabilities of weighted averages of certain independently and identically distributed random variables as the weights are varied. We show that the upper and lower tails are smallest when all the weights are equal. Our results apply to exponential, chi-squared, gamma, and Weibull random variables.     

Operational Slack and Venture Survival

Slack can act as a double‐edged sword. While it can buffer against environmental threats to help ensure business continuity, slack can also be costly and reduce profitability. In this study we focus on operational slack, the form related to the firm's production processes. We investigate the role of operational slack on firm survival during its venture stage when its survival is significantly challenged by environmental threats. Specifically, we explore how change in three types of environmental uncertainty, namely dynamism, complexity, and lack of munificence, affect the relationship between operational slack and venture survival. Results suggest that with an increase in environmental uncertainty, operational slack lowers the likelihood of venture failure.     

Gamma failure‐time mixture models: yet another way to establish efficacy

Using a Yamaguchi‐type generalized gamma failure‐time mixture model, we analyse the data from a study of autologous and allogeneic bone marrow transplantation in the treatment of high‐risk refractory acute lymphoblastic leukaemia, focusing on the time to recurrence of disease. We develop maximum likelihood techniques for the joint estimation of the surviving fractions and the survivor functions. This includes an approximation to the derivative of the survivor function with respect to the shape parameter. We obtain the maximum likelihood estimates of the model parameters. We also compute the variance‐covariance matrix of the parameter estimators. The extended family of generalized gamma failure‐time mixture models is flexible enough to include many commonly used failure‐time distributions as special cases. Yet these models are not used in practice because of computational difficulties. We claim that we have overcome this problem. The proposed approximation to the derivative of the survivor function with respect to the shape parameter can be used in any statistical package. We also address the issue of lack of identifiability. We point out that there can be a substantial advantage to using the gamma failure‐time mixture models over nonparametric methods. Copyright © 2003 John Wiley & Sons, Ltd.     

Generalized Laplacian Distributions and Autoregressive Processes

Generalized Laplacian distribution is considered. A new distribution called geometric generalized Laplacian distribution is introduced and its properties are studied. First- and higher-order autoregressive processes with these stationary marginal distributions are developed and studied. Simulation studies are conducted and trajectories of the process are obtained for selected values of the parameters. Various areas of application of these models are discussed.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

On Sharp Jensen's Inequality and Some Unusual Applications

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.