Ordering conditional lifetimes of coherent systems

Consider a system of n components that has the property that there exists a number r  (r<n)$(r<n)$, such that if it is known that at most r components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the rth component. The system is still working then, and we are interested in its residual life. In this paper we obtain some results which stochastically compare the residual lives of such systems with the same type, or with different types, of components. Some applications are given. In particular, we derive upper and lower bounds on the expected residual lives of such systems given that the warning light has not come up yet, and given that the component hazard rate functions are bounded from below or from above by a known constant.     

On testing more IFRA ordering

In this paper, first we consider the problem of testing that two unknown distributions are identical against the alternative that one is more IFRA than the other and propose a new test that is asymptotically normal and consistent. Next, we prove that beta family of distributions is ordered according to more IFRA ordering. The empirical power of the proposed test is simulated for some specific families of distributions like beta, gamma and Weibull that are ordered with respect to more IFRA order. Finally, we apply our test to some real data sets in the context of reliability.     

On testing more IFRA ordering-II

ABSTRACT

Suppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ? _*G) if G^{? 1}F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H₀: F = _*G against H₁: F ? _*G and F ≠ _*G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.