4. Modelling Survival Data in Medical Research

In this article, we present a concise review of developments on discrete multivariate distributions. We first present some basic definitions and notations. Then, we present several important discrete multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
stirling numbers;
regression;
inflated distributions;
truncated forms;
compound distributions;
multinomial;
negative multinomial;
multivariate poisson;
multivariate hypergeometric;
multivariate Polya–Eggenberger;
multivariate discrete exponential;
multivariate power series;
multivariate hermite;
multivariate occupancy;
multivariate weighted;
dirichlet;
multivariate run-related distributions

Discrete Multivariate Distributions

(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

https://link.springer.com/content/pdf/10.1057%2Fjors.1984.224.pdf

Comparison Methods for Queues and Other Stochastic Models

(2007). Stochastic Ageing and Dependence for Reliability. Technometrics: Vol. 49, No. 2, pp. 222-222.

Stochastic Ageing and Dependence for Reliability

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable X t  = (t − X|X ≤ t), for a fixed t > 0, is known as `time since failure', which is analogous to the residual lifetime random variable used in reliability and survival analysis. The reversed hazard rate function, which is related to the random variable X t , has received the attention of many researchers in the recent past [(cf. Shaked, M., Shanthikumar, J. G., 1994). Stochastic Orders and Their Applications. New York: Academic Press]. In this paper, we define some new classes of distributions based on the random variable X t and study their interrelations. We also define a new ordering based on the mean of the random variable Xt and establish its relationship with the reversed hazard rate ordering.

Reliability Properties of Reversed Residual Lifetime

Nonparametric classes of life distributions are usually based on the pattern of aging in some sense. The common parametric families of life distributions also feature monotone aging. In this paper we consider the class of log-concave distributions and the subclass of concave distributions. The work is motivated by the fact that most of the common parametric models of life distributions (including Weibull, Gamma, log-normal, Pareto, and Gompertz distributions) are log-concave, while the remaining life of maintained and old units tend to have a concave distribution. The classes of concave and log-concave distributions do not feature monotone aging. Nevertheless, these two classes are shown to have several interesting and useful properties. We examine the closure of these classes under a number of reliability operations, and provide sharp reliability bounds for nonmaintained and maintained units having life distribution belonging to these classes. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 419–433, 1999

Log-concave and concave distributions in reliability

Some results on generalized residual entropy

Recently, the reversed hazard rate (RHR) function, defined as the ratio of the density to the distribution function, has become a topic of interest having applications in actuarial sciences, forensic studies and similar other fields. Here we establish results with respect to RHR ordering between the exponentiated random variables. We also address the ordering results between component redundancy and system redundancy. Both the cases of matching spares and non-matching spares are discussed. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system.

Some results on reversed hazard rate ordering

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.

Asok K. Nanda

Papers

Reliability Properties of Reversed Residual Lifetime

Log-concave and concave distributions in reliability

Some results on generalized residual entropy

Some results on reversed hazard rate ordering

The hazard rate and the reversed hazard rate orders, with applications to order statistics