Jayant V. Deshpande

SOME RESULTS ON THE RELATIVE AGEING OF TWO LIFE DISTRIBUTIONS

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

Testing for second order stochastic dominance

A test based on empirical distribution function has been devised for testing second order stochastic dominance of distribution functions. The test is useful for comparison of random ehonomic prospects for risk averters.

A distribution-free test for the equality of failure rates due to two competing risks

A distribution–free test has been proposed for testing the equality of two failure rates in the competing risks set up when the information available is only the causes of failure and the observed times of failure. The proposed test is consistent and unbiased. The test performs well as compared to sign test for a wide spectrum of alternatives, whereas for the proportional hazards model the sign test is seen to do better.

A linear combination of two u-statistics for testing new better than used

Let X be a nonnegative random variable with absolutely continuous distribution function F and survival function . Given a random sample X1,…,Xn from the distribution F, the problem considered is to test Ho (α being an unspecified positive constant) against the alternative for every x,y ≥ 0, that is F is NBU. If F is NBU, then for x,y ≥ 0. Let D(F) = EF[δ(x,y)]. Then D(F) can be taken as an overall measure of deviation for testing Ho against H1. Let S be the U-statistic associated with D(F). It can be seen that S = U − J, where U is the Ahmad statistic and J is the Hollander-proschan statistic already proposed in the literature. Test is to reject Ho for large values of S. The test has good ARE properties.

On exponential scores statistics for testing against positive aging

It has been shown in this paper that a test proposed by Barlow and Doksum (1972) based on the exponential scores statistic for testing exponentiality against increasing failure rate distributions is consistent for the much wider class of harmonic new better than used in expectation distributions.

A note on conditionally unbiased estimation after selection

We establish the equivalence of conditional unbiasedness and Lehmanns risk unbiasedness and a necessary and sufficient condition for the existence of such estimators. This condition brings out transparently the reason for the nonexistence of such estimators in most cases based on single stage sampling.

A Lower Bound on the L-Class of Life Distributions and its Applications

Two life distributions F and G can be ordered with respect to their Laplace transforms. We say that F is Laplacelarger than G if ∫ o ∞   exp (   − s t )   F ¯ ( t )   d t   ⩾   ∫ o ∞   exp (   − s t )   G ¯ ( t )   d t for all s ⩾ 0, where F ¯ =   1   −   F . Let F be a distribution function with finite mean µ. Then F is said to belong to the L-class of life distributions if F is Laplace-larger than G, where G is exponential with the same mean µ. In this paper we obtain a lower bound on F ¯   ε   L in terms of µ. Some applications are indicated through examples.

Some bounds on reliability of coherent systems of IFRA components

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.

The response functions for competing risks

The competing risks data consist of a pair (T,δ), where T ≥ 0 can be interpreted as the failure time and δ as the identifier of the risk causing the failure. In many practical situations, the main interest lies in studying the inter-relationships between the probability structures of the failure time under various risks. Here, we consider two risks and propose the functional relationships between the probability structure of (T,δ = 1) and (T,δ = 2) by time-dependent scale and shape shifts. Also, a. model which captures the relative aging of a unit under the two risks is proposed. The necessary theory for confidence estimation of these shift functions is developed. These techniques are illustrated through several data sets available in the literature.

Nonparametric tests for the two sample problem with two independent competing risks

The problem of testing the equality of the joint distribution of two independent competing risks operating in one environment with their joint distribution while operating in another environment is considered in this paper. Two tests based on the Gehans modification of the Wilcoxon-Mann-Whitney statistic are proposed and their properties including consistency and asymptoic relative efficiency are investigated. A simulation study for estimating the power of the test is also carried out.