Inaccuracy and information measures based on cumulative residual entropy are quite useful and have attracted considerable attention in many fields including reliability theory. Using a point process martingale approach and a compensator version of Kumar and Taneja's generalized inaccuracy measure of two nonnegative continuous random variables, we define here an inaccuracy measure between two coherent systems when the lifetimes of their common components are observed. We then extend the results to the situation when the components in the systems are subject to failure according to a double stochastic Poisson process.

Environmental stress screening (ESS) of manufactured items is used to reduce the occurrence of future failures that are caused by latent defects by eliminating the items with these defects. Some practical descriptions of the relevant ESS procedures can be found in the literature; however, the appropriate stochastic modeling and the corresponding thorough analysis have not been reported. In this paper we develop a stochastic model for the ESS, analyze the effect of this operation on the population characteristics of the screened items, and also consider the relevant optimization issues.

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

We propose a nonhomogeneous Poisson hidden Markov model for a time series ofclaim counts that accounts for both seasonal variations and random fluctuations in the claims intensity. It assumes that the parameters of the intensity function for the nonhomogeneous Poisson distribution vary according to an (unobserved) underlying Markov chain. This can apply to natural phenomena that evolve in a seasonal environment. For example, hurricanes that are subject to random fluctuations (El Niño-La Niña cycles) affect insurance claims. The Expectation-Maximization (EM) algorithm is used to calculate the maximum likelihood estimators for the parameters of this dynamic Poisson hidden Markov model. Statistical applications of this model to Atlantic hurricanes and tropical storms data are discussed.

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

The notion of minimal repair is generalized to the case when the lifetime distribution function is a continuous or a discrete mixture of distributions (heterogeneous population). The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding stochastic intensities are defined and analyzed for the point processes generated by both types of minimal repair. Some generalizations are discussed. Several simple examples are considered.

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).