Jayant V Deshpande

Bounds for the hazard gradients in the competing risks set up

The joint survival function of the latent lifetimes in the dependent competing risks set up is nonidentifiable from the joint probability distribution of the observation (failure time, cause of failure). This paper concentrates on the hazard gradients associated with the joint survival function, for which we provide bounds in the general dependent case and improve these in case the type of dependence (e.g., RTI, RCSI, etc.) is known. This leads to bounds on net (marginal) hazard rates in terms of cause-specific hazard rates which are identifiable. The problem of estimation of these bounds is discussed at the end.

Testing of two sample proportional intensity assumption for non-homogeneous Poisson processes

Abstract For two independent non-homogeneous Poisson processes with unknown intensities we propose a test for testing the hypothesis that the ratio of the intensities is constant versus it is increasing on (0,t]. The existing test procedures for testing such relative trends are based on conditioning on the number of failures observed in (0,t] from the two processes. Our test is unconditional and is based on the original time truncated data which enables us to have meaningful asymptotics. We obtain the asymptotic null distribution (as t becomes large) of the proposed test statistic and show that the proposed test is consistent against several large classes of alternatives. It was observed by Park and Kim (IEEE. Trans. Rehab. 40 (1), 1992, 107–111) that it is difficult to distinguish between the power-law and log-linear processes for certain parameter values. We show that our test is consistent for such alternatives also.

TESTS FOR SOME STATISTICAL HYPOTHESES FOR DEPENDENT COMPETING RISKS–A REVIEW

The competing risks situation arises in life studies when a unit is subject to many, say k , modes of failure and the actual failure , when it occurs , can be ascribed to a unique mode. These k modes are also called the k risks to which the unit is exposed and as they all seemingly compete for the life of the unit, the term ’competing risks’ is used to describe it. Suppose that the continuous positive valued random variable T represents the lifetime of the unit and δ taking values 1, 2, . . . , k represent the risk which caused the failure of the unit. The joint probability distribution of (T, δ) is specified by the set of k distribution functions F (i, t) = P [T ≤ t, δ = i], (1)

Tests for Equality of Intensities of Failures of a Repairable System Under Two Competing Risks

Let a repairable system be subject to failure due to two competing risks. It is assumed that repairs are instantaneous.

NONPARAMETRIC DENSITY ESTIMATION

This chapter describes the background material related to the nonparametric density estimation. Techniques such as histograms (together with its extension, known as ASH, see Sect. 2.3), Parzen windows and k-nearest neighbors are at the core of the applications of nonparametric density estimation. For that reason, we decided to include a chapter describing these for the sake of completeness and to allow less experienced readers develop their intuitions in terms of the nonparametric estimation. Most of the material is presented taking into account only the univariate case; extending the results to cover more than one variable, however, is often a straightforward task. The chapter is organized as follows: Sect. 2.2 presents a short overview of the fundamental concepts related to histograms. Section2.3 is devoted to a description of a smart extension of certain well-known histograms aimed at avoiding some of their drawbacks. Section2.4 presents basic concepts related to the nonparametric density estimation. Section2.5 is devoted to the Parzen windows, while Sect. 2.6 to the k-nearest neighbors approach.