Testing Whether New is Better Than Used
01 Aug 1972-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 43, Iss: 4, pp 1136-1146
TL;DR: A U-statistic J sub N is proposed for testing the hypothesis H sub O that a new item has stochastically the same life length as a used item of any age against the alternative hypothesis HSub 1 that anew item has Stochastically greater life length.
Abstract: : A U-statistic J sub N is proposed for testing the hypothesis H sub O that a new item has stochastically the same life length as a used item of any age (i.e., the life distribution F is exponential), against the alternative hypothesis H sub 1 that a new item has stochastically greater life length (F(x) F(y) > or = F(x+y), for all x > or = 0, y > or = 0, where F = 1-F). J sub n is unbiased; in fact, under a partial ordering of H sub 1 distributions, J sub n is ordered stochastically in the same way. Consistency against H sub 1 alternatives is shown, and asymptotic relative efficiencies are computed. Small sample null tail probabilities are derived, and critical values are tabulated to permit application of the test. (Author)
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TL;DR: In this article, the concept of increasing "conditional mean exceedance" provides a reasonable way of describing the heavy-tail phenomenon, and a family of Pareto distributions is shown to represent distributions for which this parameter is linearly increasing.
Abstract: Distributions with heavier-than-exponential tails are studied for describing empirical phenomena. It is argued that the concept of increasing “conditional mean exceedance” provides a reasonable way of describing the heavy-tail phenomenon, and a family of Pareto distributions is shown to represent distributions for which this parameter is linearly increasing. A test is developed and modified so as to be suitable for testing heavy-tailedness, and some graphical procedures are also suggested.
198 citations
TL;DR: In this paper, the authors developed tests for alternatives representing decreasing mean residual life and the property "new better than used in expectation" and obtained consistent and asymptotic relative efficiency results for the tests based on V* and K*.
Abstract: SUMMARY In this paper we develop tests for alternatives representing decreasing mean residual life and the property 'new better than used in expectation'. The decreasing mean residual life test statistic, V*, is new, and critical constants and a large-sample approximation are obtained to make the test readily applicable. The new better than used in expectation statistic, K*, is shown to be equivalent to the total time on test statistic; the latter is ordinarily viewed as a test statistic for alternatives of increasing failure rate. Consistency and asymptotic relative efficiency results are obtained for the tests based on V* and K*. These results lead to a reinterpretation of the total time on test statistic as a test statistic for classes of alternatives larger than the increasing failure rate class and including the 'new better than used in expectation' class.
188 citations
TL;DR: In this article, the authors extended the results obtained by Esary, Marshall and Proschan to a nonhomogeneous Poisson process and derived bounds on the moments of the life of a device subject to a given number of shocks.
176 citations
TL;DR: In this paper, the aging properties IFR, IFRA, NBUE and DMRL can be translated to properties of ϕF(t), and the power of some tests is estimated by simulation for some alternatives when the sample size is n = 10 or n = 20.
Abstract: Let F be a life distribution with survival function [Fbar] = 1 - F and finite mean The scaled total time on test transform was introduced by Barlow and Campo (1975) as a tool in the statistical analysis of life data. The aging properties IFR, IFRA, NBUE and DMRL can be translated to properties of ϕF(t). Guided by these properties of ϕF(t) we suggest some test statistics for testing exponentiality against IFR, IFRA, NBUE and DMRL, respectively. The IFR and IFRA tests are new; the NBUE and DMRL tests have already been proposed and stu- died by Hollander and Proschan (1975) . The exact and asymptotic distributions of the statistics are derived and the asymptotic efficiencies of the tests are studied. The power of some tests is estimated by simulation for some alternatives when the sample size is n = 10 or n = 20.
99 citations
Cites background or methods from "Testing Whether New is Better Than ..."
...We can here give the reason why we got the test statistics V* and K* of Hollander and Proschan (1975) by considering the TTT-plot....
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...We note that M^ is proportional to the test statistic V* proposed by Hollander and Proschan (1975) for testing against DMRL (IMRL)....
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...For V* and K* they are tabulated in Hollander and Proschan (1975) arid Barlo w et al. (1972), respectively....
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...For V* and K* they are tabulated in Hollander and Proschan (1975) arid Barlo w et al....
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...13) K*- ? <|-ii*£)t<j)/Sn was introduced by Hollander and Proschan (1975) for testing against the NBUE (NWUE) alternative....
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TL;DR: A wide selection of tests for exponentiality is discussed and compared in this article, where power computations, using simulations, were done for each procedure, and the score test presented in Cox and Oakes (1984) appears to be the best if one does not have a particular alternative in mind.
Abstract: A wide selection of tests for exponentiality is discussed and compared. Power computations, using simulations, were done for each procedure. Certain tests (e.g. Gnedenko (1969), Lin and Mudholkar (1980), Harris (1376), Cox and Oakes (1384), and Deshpande (1983)) performed well for alternative distributions with non-monotonic hazard rates, while others (e.g. Deshpande (1983), Gail and Gastwirth (1978), Kolmogorov-Smirnov (LillViefors (1969)), Hahn and Shapiro (1967), Hollander and Proschan (1972), and Cox and Oakes (1984)) fared well for monotonic hazard rates. Of all the procedures compared, the score test presented in Cox and Oakes (1984) appears to be the best if one does not have a particular alternative in mind.
94 citations
References
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TL;DR: In this article, the authors considered the problem of estimating a U-statistic of the population characteristic of a regular functional function, where the sum ∑″ is extended over all permutations (α 1, α m ) of different integers, 1 α≤ (αi≤ n, n).
Abstract: Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)
2,439 citations
TL;DR: In this article, the life distribution of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities of not surviving the first $k$ shocks.
Abstract: The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$ As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure We extend our results to continuous wear processes Applications of interest in renewal theory are obtained Total positivity theory is used in deriving many of the results
592 citations
TL;DR: In this paper, the hazard rate is derived from its probabilistic interpretation: if, for example, F is a life distribution, q(x)dx is the conditional probability of death in (x, x + dx) given survival to age x.
Abstract: : Properties of distribution functions F (or their densities f) are related to properties of the corres onding hazard rate q defined by q(x) equals f(x)/ 1 - F(x) . Interest in the hazard rate is derived from its probabilistic interpretation: if, for example, F is a life distribution, q(x)dx is the conditional probability of death in (x, x + dx) given survival to age x. Because of this interpretation f is assumed to be the density of a positive random variable, although for many of the results this is not necessary. The hazard rate is important in a number of applications, and is known by a variety of names. It is used by actuaries under the name of force of mortality to compute mortality tables, and its reciprocal is known to statisticians as Mill's ratio. In the analysis of extreme value distributions it is called the intensity function, and in reliability theory it is usually referred to as the failure rate. A number of general results are obtained, but particular attention is paid to densities with monotone hazard rate. (Autho )
421 citations
TL;DR: In this article, the concept of "aging" is discussed in terms of the entity's survival time distribution, and a set of seven criteria for aging is established, based on these quantities, and the chain of implications among the criteria is developed.
Abstract: The concept of “aging,” or progressive shortening of an entity's residual lifetime, is discussed in terms of the entity's survival time distribution. Quantities defined to describe the aging phenomenon include the “specific aging factor,” “hazard rate,” “hazard rate average,” and “mean residual lifetime.” A set of seven criteria for aging is established, based on these quantities, and a chain of implications among the criteria is developed. The hazard rate average and mean residual lifetime are noted as being particularly useful for empirical studies. An application of these two quantities is illustrated for a set of empirical survival time data.
341 citations