Improved Ordering Results for Fail-Safe Systems with Exponential Components

Let X_{2: n} and Y_{2: m} be the second order statistics from n independent exponential variables with hazards λ₁, …, λ_n, and an independent exponential sample of size m with hazard change to λ, respectively. When m ? n, we obtain necessary and sufficient conditions for comparing X_{2: n} and Y_{2: m} in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λ_i’s and λ. The established results show how one can compare an (n ? 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m ? 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.     

On optimality of extremal schemes in progressive type II censoring

In a progressively type II censored life-testing experiment intact units may be removed from the experiment after every failure. If the initial number of units in the experiment and the total number of failures are fixed, the experimenter may choose between different censoring schemes. By specifying an optimality criterion, one may improve the outcome of the experiment by choosing the respective optimal scheme. We establish a simple property of a general optimality criterion that yields optimality of certain extremal schemes. Applications to some criteria that measure the total time of the experiment and its variability illustrate the approach. The results are based on stochastic orderings of generalized order statistics.     

Tests for Stochastic Orders and Mean Order Statistics

The aim of this article is to emphasize the fact, not observed previously in the literature, that many discrepancy measures used in tests related to different stochastic orders can be expressed as expectations of order statistics. In this way, we provide a new meaning to the corresponding test statistics which allows us to understand better, and potentially improve, the testing procedures. As illustration, we consider tests to detect overdispersion with respect to a specific probability model. In this setting, a test for the Weibull distribution is discussed in detail.     

An exact test for nbue class of survival functions

Let F(x) be a life distribution. An exact test is given for testing H₀ F is exponential, versusH₁Fε NBUE (NWUE); along with a table of critical values for n=5(l)80, and n=80(5)65. An asymptotic test is made available for large values of n, where the standardized normal table can be used for testing.