S.N.U.A. Kirmani

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations. In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original di...

Closure and Monotonicity Properties of Nonhomogeneous Poisson Processes and Record Values

Interconnections between occurrence times of nonhomogeneous Poisson processes, record values, minimal repair times, and the relevation transform are explained. A number of properties of the distributions of occurrence times and interoccurrence times of a nonhomogeneous Poisson process are proved when the mean-value function of the process is convex, starshaped, or superadditive. The same results hold for upper record values of independently identically distributed random variables from IFR, IFRA, and NBU distributions.

A measure of discrimination between two residual life-time distributions and its applications

A measure of discrepancy between two residual-life distributions is proposed on the basis of Kullback-Leibler discrimination information. Properties of this measure are studied and the minimum discrimination principle is applied to obtain the proportional hazards model.

Some results on normalized spacings from restricted families of distributions

It is well known that the normalized spacings of a random sample from a DFR (IFR) distribution are stochastically increasing (decreasing). In this note we strengthen this result to show that if the parent distribution is DFR, the successive normalized spacing increase in the failure rate ordering (which implies stochastic ordering) sense. We also study the joint distribution of the normalized spacings when the parent observations are not necessarily identical. It is shown that when the observations are independent with (possibly different) log-convex densities, the joint distribution of the normalized spacings is arrangement increasing.

Residual coefficient of variation and some characterization results

Abstract In life-testing situations, the additional lifetime that a component has survived until time t is called the residual life function of the component. In this paper, we study the residual coefficient of variation, γ(t), and show that γ(t) characterizes the distribution in the univariate as well as the bivariate case. Some examples are presented to illustrate the procedure. Finally, conditions on the failure rate and the mean residual life functions are investigated which ensure the monotonicity of γ(t).

On order relations between reliability measures

It is well known that the three reliability measures, namely, the survival function, the failure rate function, and the mean residual life function are equivalent in the sense that knowing any one of them, the other two can be determined. This paper deals with the question of whether an ordering of two life distributions with respect to a reliability measure implies the same ordering with respect to another reliability measure. Implications of various orderings are discussed from the point of view of some properties of the life distributions involved.

On the Proportional Odds Model in Survival Analysis

The proportional odds (PO) model with its property of convergent hazard functions is of considerable value in modeling survival data with non-proportional hazards. This paper explores the structure, implications, and properties of the PO model. Results proved include connections with geometric minima and maxima, ageing characteristics, and bounds on mean and variance of survival times.

Some Characterization of Distributions by Functions of Failure Rate and Mean Residual Life

Abstract In reliability studies, it is well known that the failure rate, the mean residual life function, and their product characterize the distribution. In this article, we show that their ratio also characterizes the distribution. This result is then utilized to show that the second residual moment characterizes the distribution. A general uniqueness theorem is proved to show that the variance residual life function characterizes the distribution. An example is presented to illustrate the results. Finally, some applications, of the results established earlier, to nonhomogeneous Poisson processes are provided.

On the Proportional Mean Residual Life Model and its Implications

The proportional mean residual life (PMRL) model and its implications are studied. Its relation with the proportional hazard model is explored. Some results concerning ageing properties in reliability are investigated for the PMRL model and conditions under which these classes remain closed are examined. Finally, bounds on the residual moments and residual variance are obtained and some examples are furnished.

ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.