https://www.ams.org/proc/1955-006-02/S0002-9939-1955-0069229-1/S0002-9939-1955-0069229-1.pdf

The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities

: The reliability of a complex system that is repaired (but not replaced) upon failure will often depend on the system chronological age. If only minimal repair is made so that the intensity (instantaneous rate) of system failure is not disturbed, than a nonhomogeneous Poisson process may be used to model this age-dependent reliability. This paper considers the theoretical and practical implications of the nonhomogeneous Poisson process model for reliability, and gives estimation, hypotheses testing, comparison and goodness of fit procedures when the process has a Weilbull intensity function. Applications of the Weilbull model in the field of reliability and in other areas are discussed. (Author)

/pdf/reliability-analysis-for-complex-repairable-systems-43hopz6oyc.pdf

Reliability Analysis for Complex, Repairable Systems

Tests based on the quantile regression process can be formulated like the classical Kolmogorov-Smirnov and Cramer-von-Mises tests of goodness-of-fit employing the theory of Bessel processes as in Kiefer (1959). However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as "the Durbin problem" since it was posed in Durbin (1973), for parametric empirical processes. In this paper, we consider an approach to the Durbin problem involving a martingale transformation of the parametric empirical process suggested by Khmaladze (1981) and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and location-scale shift models that underlie much of classical econometric inference. The methods are illustrated in some limited Monte-Carlo experiments and with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania experiments, conducted in 1988-89, were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells.

Inference on the Quantile Regression Process

Tests of fit are given for the generalized Pareto distribution (GPD) based on Cramer–von Mises statistics. Examples are given to illustrate the estimation techniques and the goodness-of-fit procedures. The tests are applied to the exceedances over given thresholds for 238 river flows in Canada; in general, the GPD provides an adequate fit. The tests are useful in deciding the threshold in such applications; this method is investigated and also the closeness of the GPD to some other distributions that might be used for long-tailed data.

https://www.jstor.org/stable/info/1270819

Goodness-of-Fit Tests for the Generalized Pareto Distribution

The main purpose of this paper is to obtain the limiting distribution of certain statistics described in the title. It was suggested by the author in [1] that these statistics might be useful for testing the homogeneity hypothesis $H_1$ that $k$ random samples of real random variables have the same continuous probability law, or the goodness-of-fit hypothesis $H_2$ that all of them have some specified continuous probability law. Most tests of $H_1$ discussed in the existing literature, or at least all such tests known to the author before [1] in the case $k > 2$, have only been shown to have desirable consistency or power properties against limited classes of alternatives (see e.g., [2], [3], [4] for lists of references on these tests), while those suggested here are shown to be consistent against all alternatives and to have good power properties. Some test statistics whose distributions can be computed from known results are also listed.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177706261

K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-V. Mises Tests

The "goodness of fit" problem, consisting of comparing the empirical and hypothetical cumulative distribution functions (cdf's), is treated here for the case when an auxiliary parameter is to be estimated. This extends the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2]. The characteristic function of the limiting distribution of the test function is found by consideration of a Guassian stochastic process.

/pdf/the-cramer-smirnov-test-in-the-parametric-case-sm0auda25u.pdf

D. A. Darling

Papers

The Cramer-Smirnov Test in the Parametric Case