A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

Some reliability properties of order statistics

Let X_i:j denote the i^th order statistic of a random sample of size j from a continuous life distribution. We show that if X_k:n, is IFR, IFRA, NBU, or DMRL, so are X_k+1:n, X_k+1:n?1 and X_k+1:n+1. Further we show that, in the first three cases, X_k+1:n+2 also shares the corresponding property if k ≤ (n+3)/2. We also present dual results for DFR, DFRA and NWU classes.     

Improved Asymptotics of a Decreasing Mean Residual Life Estimator

The mean residual life of a life distribution, X, with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. Kochar et al. (2000 Kochar, S.C., Mukerjee, H., Samaniego, F.J. (2000). Estimation of a monotone mean residual life. Ann. Stat. 28: 905–921.[Crossref], [Web of Science ®] , [Google Scholar]) provided an estimator of M when it is assumed to be decreasing. They showed that its asymptotic distribution was the same as that of the empirical estimate, but only under very stringent analytic and distributional assumptions. We provide a more general asymptotic theory, and under much weaker conditions. We also provide improved asymptotic confidence bands.     

On Upshifted Reversed Mean Residual Life Order

Recently, the concept of reversed mean residual life order based on the mean of the random variable X _t  = (t ? X | X ≤ t), t > 0, called the reversed residual life, defined for the nonnegative random variable X, has been introduced in the literature. In this paper, a stochastic order based on the shifted version of the reversed mean residual life is proposed, based on the reversed mean residual life function for a random variable X with support (l _X , ∞), where l _X may be negative infinity, and its properties are studied. Closure under the Poisson shock model and properties for spare allocation are also discussed.     

On hypothesis tests in misspecified change-point problems for a Poisson process

Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g_* to an upper function h_* at some unknown point ?_* that has to be identified. We consider two known continuous functions g and h such that g_*(t) ? g(t) < h(t) ? h_*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.     

On Properties of Reversed Mean Residual Life Order for Weighted Distributions

In this article, first, in order to compare X and X ^w (the weighted version of X with weight function w(·)) according to reversed mean residual life order, we provide an equivalent condition. We then try to provide conditions under which the reversed mean residual life order is preserved by weighted distributions. For this end, we obtain several independent results. Finally, the problem of preservation of increasing reversed mean residual life class under weighting is investigated, as well. Some examples are also given to illustrate the results.     

A simple approach for testing constant failure rate against different ageing classes for discrete data

In this paper, we develop simple non-parametric test based on U-statistics for testing constant failure rate against IFR, IFRA, DMRL, NBU and NBUE alternatives. The asymptotic properties of the test statistics are studied. In particular, the test statistics are shown to be asymptotically normal and consistent against the relevant alternatives. Some numerical results are presented to demonstrate the performance of the proposed tests.     

Tests of non-monotonic stochastic aging notions in reliability theory

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study we restrict ourselves to the problem of testing exponentiality against non-monotonic aging notions. We model non-monotonic aging using the notions of bathtub failure rate, increasing and then decreasing mean residual life and new worse then better than used in expectation classes. The different tests of exponentiality against these alternatives are discussed in detail.     

Precise large deviations for the difference of two sums of WUOD and non identically distributed random variables with dominatedly varying tails

In this article, we study large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j, where {X_1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F_1j(x), j ? 1}, {X_2j, j ? 1} is a sequence of independent identically distributed random variables, n₁(t) and n₂(t) are two positive integer-valued functions, and {N_i(t), t ? 0}²_{i = 1} with EN_i(t) = λ_i(t) are two counting processes independent of {X_ij, j ? 1}²_{i = 1}. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

A moment-based empirical likelihood ratio test for exponentiality using the probability integral transformation

ABSTRACT

A simple and efficient goodness-of-fit test for exponentiality is developed by exploiting the characterization of the exponential distribution using the probability integral transformation. We adopted the empirical likelihood methodology in constructing the test statistic. The proposed test statistic has a chi-square limiting distribution. For small to moderate sample sizes Monte-Carlo simulations revealed that our proposed tests are much more superior under increasing failure rate (IFR) and bathtub decreasing-increasing failure rate (BFR) alternatives. Real data examples were used to demonstrate the robustness and applicability of our proposed tests in practice.     

Inequalities involving expectations of selected functions in reliability theory to characterize distributions

Recently, authors have studied inequalities involving expectations of selected functions, viz. failure rate, mean residual life, aging intensity function, and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time (X ? x, instead of X > x) and their applications. In the present work we consider reversed hazard rate, expected inactivity time, and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.     

Nonparametric two‐step regression estimation when regressors and error are dependent

This paper considers estimation of the function g in the model Y_t = g(X_t ) + ?_t when E(?_t|Xt) ≠ 0 with nonzero probability. We assume the existence of an instrumental variable Z_t that is independent of ?_t, and of an innovation ηt = X_t — E(X_t|Z_t). We use a nonparametric regression of X_t on Z_t to obtain residuals η_t, which in turn are used to obtain a consistent estimator of g. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.     

Reliability Properties of Reversed Residual Lifetime

Abstract

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable X _t  = (t ? X|X ≤ t), for a fixed t > 0, is known as `time since failure', which is analogous to the residual lifetime random variable used in reliability and survival analysis. The reversed hazard rate function, which is related to the random variable X _t , has received the attention of many researchers in the recent past [(cf. Shaked, M., Shanthikumar, J. G., 1994 Shaked, M. and Shanthikumar, J. G. 1994. Stochastic Orders and Their Applications New York: Academic Press.  [Google Scholar]). Stochastic Orders and Their Applications. New York: Academic Press]. In this paper, we define some new classes of distributions based on the random variable X _t and study their interrelations. We also define a new ordering based on the mean of the random variable X_t and establish its relationship with the reversed hazard rate ordering.     

A test of linearity against functional coefficient autoregressive models

Our object in this paper is to propose a powerful test for detecting a broad class of nonlinearity of time series as one application of the method by Matsuda (1998). Consider autoregresive models X_t=Φ₁X_t?1+…+Φ_ρX_t?ρ+ε_t and we propose a statistic for testing whether or not Φ_i depends on delayed values X_t?d for some i. We compare the power of our test with that of tests proposed by Luukkonen, Saikkonen and Teräsvirta (1988a) and Hjellvik and TjΦstheim (1995) by simulation studies and our test is shown to be satisfactory.     

Bivariate Aging Properties under Archimedean Dependence Structures

Let X  = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X _t  = [(X ? t, Y ? t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X _t , with t ≥ 0, were studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X _t and X _t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.