Vol 25, No 2 (2016)

A nonhomogeneous Poisson process (NHPP) plays an important role in a variety of applications as reliability of repairable systems, software reliability and actuarial studies. An NHPP is characterized by its intensity function m(t), which provides information on the time-dependent nature of the reliability of the system. Various intensity functions, which describe different behavior (from reliability decay to reliability growth along with monotonicity, convexity or concavity), have been suggested for NHPP’s for modeling repairable systems. Perhaps one of the most frequently utilized NHPP is the power lawprocess (PLP) in which m(t) is a power function of t. Inthis studywe present a general method for constructing new intensity functions for NHPP’s yielding new classes of NHPP’s. This method utilizes certain operators L_n, n ∈ N₀, acting on some suitable functions L₀ = f (termed base functions). We call these classesOBIF’s (operator-based intensity functions). These classes are represented in terms of three parameters of which one is an indexing parameter n ∈ N₀ and two others are scale and shape parameters. The fact that n ∈ N₀ is also a parameter provides a flexibility in the choice of the appropriate statistical model for NHPP’s data. In particular, we consider the exponential operator acting on the PLP intensity function f and realize that L_n’s, n ≥ 2, inherit properties similar to those of L₁ (convexity and concavity) and thus are suitable for modelling bathtub data. We also consider a more comprehensive treatment of OBIF classes where both, the operator and base functions, are general. All of the introduced operators are demonstrated with illustrative examples.

Tests for certain covariance structures, including sphericity, are presented when the data may be high-dimensional but not necessarily normal. The tests are formulated as functions of location-invariant estimators defined as U-statistics of higher order kernels. Under a few mild assumptions, the limit distributions of the tests are shown to be normal. The accuracy of the tests is demonstrated by simulations.

Let T(λ₁,...,λ_n) be the lifetime of a parallel system consisting of exponential components with hazard rates λ₁,...,λ_n, respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ₁, λ₂) majorizes (γ₁, γ₂), then, T(λ₁, λ₂) is larger than T(γ₁, γ₂) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ₁,...,λ_n) is d-larger than (γ₁,...,γ_n), then T(λ₁,...,λ_n) is larger than T(γ₁,...,γ_n) in likelihood ratio order.