Henry W. Block

A Continuous Bivariate Exponential Extension

Abstract Two derivations are given for an absolutely continuous bivariate extension of the exponential distribution. This distribution turns out to be the absolutely continuous part of the bivariate exponential distribution of Marshall and Olkin and a variant of the bivariate exponential extension of Freund. One derivation utilizes the loss of memory property (LMP) which Marshall and Olkin used to derive their bivariate exponential distribution. Distributional properties, reliability considerations and estimation for this distribution are discussed. Further, the LMP is characterized for absolutely continuous bivariate random variables (X, Y) through the independence of min (X, Y) and X – Y.

The Reversed Hazard Rate Function

In this paper we discuss some properties of the reversed hazard rate function. This function has been shown to be useful in the analysis of data in the presence of left censored observations. It is also natural in discussing lifetimes with reversed time scale. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these concepts can often, although not always, lead to anomalies. For example, one result gives that if the reversed hazard rate function is increasing, its interval of support must be (—∞, b ) where b is finite. Consequently nonnegative random variables cannot have increasing reversed hazard rates. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification. Reversed hazard rates are also important in the study of systems. Hazard rates have an affinity to series systems; reversed hazard rates seem more appropriate for studying parallel systems. Several results are given that demonstrate this. In studying systems, one problem is to relate derivatives of hazard rate functions and reversed hazard rate functions of systems to similar quantities for components. We give some results that address this. Finally, we carry out comparisons for k -out-of- n systems with respect to the reversed hazard rate ordering.

A Decomposition for Multistate Monotone Systems.

A decomposition theorem for multistate structure functions is proven. This result is applied to obtain bounds for the system performance function. Another application is made to interpret the multistate structures of Barlow and Wu. Various concepts of multistate importance and coherence are also discussed.

Tail Behavior of the Failure Rate Functions of Mixtures

The tail behavior of the failure rate of mixtures of lifetime distributions is studied. A typical result is that if the failure rate of the strongest component of the mixture decreases to a limit, then the failure rate of the mixture decreases to the same limit. For a class of distributions containing the gamma distributions this result can be improved in the sense that the behavior of the failure rate of the mixture asymptotically mirrors that of the strongest component in whether it decreases or increases to a limit.

SHOCK MODELS WITH NBUE SURVIVAL

Conditions are given on a process so that a shock model governed by the process will have NBUE survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.

REPAIR REPLACEMENT POLICIES

In this paper we introduce the concept of repair replacement. Repair replacement is a maintenance policy in which items are preventively maintained when a certain time has elapsed since their last repair. This differs from age replacement where a certain amount of time has elapsed since the last replacement. If the last repair was a complete repair, repair replacement is essentially the same as age replacement. It is in the case of minimal repair that these two policies differ. We make comparison between various types of policies in order to determine when and under which condition one type of policy is better than another.

A Concept of Negative Dependence Using Stochastic Ordering.

A concept of negative dependence called negative dependence by stochastic ordering is introduced. This concept satisfies various closure properties. It is shown that three models for negetive dependence satisfy it and that it implies the basic negative orthant inequalities. This concept is also satisfied by the multinomial, multivariate hypergeometric. Dirichlet and Dirichlet compound multinomial distributions. Furthermore, the joint distribution of ranks of a sample and the multivariate normal with nonpositive pairwise correlations also satisfy this condition. The positive dependence analog of this condition is also studied.

A Criterion for Burn-in That Balances Mean Residual Life and Residual Variance

Optimum burn-in times have been determined for a variety of criteria such as mean residual life and conditional survival. In this paper we consider a residual coefficient of variation that balances mean residual life with residual variance. To study this quantity, we develop a general result concerning the preservation ofbathtub distributions. Using this result, we give a condition so that the residual coefficient of variation is bathtub-shaped. Furthermore, we show that it attains its optimum value at a time that occurs after the mean residual life function attains its optimum value, but not necessarily before the change point of the failure rate function.

Metrics on permutations useful for positive dependence

Abstract 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.

L-SUPERADDITIVE STRUCTURE FUNCTIONS

Structure functions relate the level of operations of a system as a function of the level of the operation of its components. In this paper structure functions are studied which have an intuitive property, called L-superadditive (L-subadditive). Such functions describe whether a system is more series-like or more parallel-like. L-superadditive functions are also known under the names supermodular, quasimonotone and superadditive and have been studied by many authors. Basic properties of both discrete and continuous (i.e., taking a continuum of values) L-superadditive structure functions are studied. For binary structure functions of binary values, El-Neweihi (1980) showed that L-superadditive structure functions must be series. This continues to hold for binary-valued structure functions even if the component values are continuous (see Proposition 3.1). In the case of non-binary-valued structure functions this is no longer the case. We consider structure functions taking discrete values and obtain results in various cases. A conjecture concerning the general case is made.