A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.     

Testing for exponentiality against monotone failure rate average alternatives

We propose a test for exponentiality against the class of non-exponential distributions having monotone failure rate averages. The test statistic, which is a U-statistic and hence asymptotically normally distributed, is much simpler than its competitors yet compares favorably with them in efficiency and power comparisons.     

Tests for bivariate mean residual life

In this paper we have developed tests for bivariate exponentiaIity against the ‘bivariate decreasing mean residual life (BDMRL)’ and ‘bivariate new better than used in expectation (BNBUE)’ classes of non-exponentia1 probability distributions. We have also obtained a large-sample approximation to make the test readily applicable.     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

A class of distribution-free tests for the two-sample slippage problem

A class of distribution-free tests for the two-sample slippage problem, when the random variables take only nonnegative values, is proposed. These tests are consistent and unbiased against the general slippage alternative. Recurrence relations for generating small sample significance points are given. The tests have been compared with the Savage test, the Wilcoxon test and the appropriate locally most powerful rank test by considering Pitman asymptotic relative efficiencies for several alternative hypotheses. Some of these tests exhibit considerable robustness in terms of efficiency for the various alternative hypotheses which are considered.     

Multivariate generalizations of jonckheere's test for ordered alternatives

In many dose-response studies, each of several independent groups of animals is treated with a different dose of a substance. Many response variables are then measured on each animal. The distributions of the response variables may be nonnormal, and Jonckheere's (1954) test for ordered alternatives in the one-way layout is sometimes used to test whether the level of a single variable increases with increasing dose. In some applications, however, it is important to consider a set of response variables simultaneously. For instance, an increase in each of certain enzymes in the blood serum may suggest liver damage. To test whether these enzyme levels increase with increasing dose, it may be preferable to consider these enzymes as a group, rather than individually.

I propose two multivariate generalizations of Jonckheere's univariate test. Each multivariate test statistic is a function of coordinate-wise Jonckheere statistics—one a sum, the other a quadratic form. The sum statistic can be used to test the alternative hypothesis that each variable is stochastically increasing with increasing dose. The quadratic form statistic is designed for the more general alternative hypothesis that each variable is stochastically ordered with increasing dose.

For each of these two alternatives, I also propose a multivariate generalization of a normal theory test described by Puri (1965). I examine the asymptotic distributions of the four test statistics under the null hypothesis and under translation alternatives and compare each distribution-free test to the corresponding normal theory test in terms of asymptotic relative efficiency.

The multivariate Jonckheere tests are illustrated using does-response data from a subchronic toxicology study carried out by the National Toxicology Program. Four groups of ten male rats each were treated with increasing doses of vinylidene flouride, and the serum enzymes SDH, SGOT, and SGPT were measured. A comparison of univariate Jonckheere tests on each variable, bivariate tests on SDH and SGOT, and multivariate tests on all three variables gives insight into the behavior of the various procedures.     

A family of tests for exponentiality against IFR alternatives

In this paper we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a class of test statistics are constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distributions of the test statistics are obtained and the test statistics are proved to be consistent. The Pitman efficiency has also been studied.     

Aspects of the mean residual life order for weighted distributions

In this paper, we first provide conditions for preservation of the mean residual life (mrl) order under weighting. Then we apply the obtained results to establish our results about preservation of the decreasing mrl class by weighted distributions. In addition, we present some results for comparing the original random variable to its weighted version in terms of the mrl order. Also, some examples are given to illustrate the results.     

Two tests for monotone failure rate with incomplete data

Two statistics are proposed for testing for the exponential distribution against monotone failure rate alternatives when ran-domly right censored data are available. One of them is a general-ization of the Billmann, Antle and Bain test based on the MLE of the shape parameter of the Weibull distribution. The second has the advantage of being given in closed form. For this test the asymptotic null distribution is given. Consistency of the two tests is proved starting from an expected value inequality characterizing monotone failure rate.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

Comparisons of some tests for validation of stochastic simulation models

Properties of proposed parametric and nonparametric tests for the problem of validating stochastic simulation models are discussed. Asymptotic efficiencies for various types of alternatives are developed for some of the tests. Numerical procedures along with simulation methods are used to evaluate the adequacy of normal approximations to the null distributions of the statistics and the small-sample power of the tests.     

A class of nonparametric tests for location and scale parameters

A class of nonparametric two-sample tests for testing identity of distributions versus alternatives containing both location and scale parameters is proposed and some properties are derived. A recursion formula for the exact distribution under the hypothesis is presented and, the asymptotic distribution is given under both the hypothesis and a contiguous sequence of alternatives. Some asymptotic optimality properties are deduced for particular tests of the class and finally, the asymptotic efficiency is found.     

Some tests for uniformity of circular distributions powerful against multimodal alternatives

The author introduces new statistics suited for testing uniformity of circular distributions and powerful against multimodal alternatives. One of them has a simple expression in terms of the geometric mean of the sample of chord lengths. The others belong to a family indexed by a continuous parameter. The asymptotic distributions under the null hypothesis are derived. We compare the power of the new tests against Stephens's alternatives with those of Ajne, Watson, and Hermans‐Rasson's tests. Some of the new tests are the most powerful when the alternative has three or four modes. A heuristic justification of this feature is given. An application to the analysis of archaeological data is provided. The Canadian Journal of Statistics 38:80–96; 2010 © 2010 Statistical Society of Canada     

Smooth tests of fit for censored gmma samples

In this paper properties of two estimators of Cpm are investigated in terms of changes in the process mean and variance. The bias and mean squared error of these estimators are derived. It can be shown that the estimate of Cpm proposed by Chan, Cheng and Spiring (1988) has smaller bias than the one proposed by Boyles (1991) and also has a smaller mean squared error under certain conditions. Various approximate confidence intervals for Cpm are obtained and are compared in terms of coverage probabilities, missed rate and average interval width.     

Efficiency of tests based on weighted rankings for equality of treatment effects in a complete randomized blocks layout

Quade (1972,1979) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout.

In this paper we obtain an expression for the Pitman asymp-totic relative efficiency of these tests with respect to the Friedman test.     

Tests for properties of residual life

The lOOα -percentile (0 < α < 1) residual life function at time x is defined to be the lOOα -percentile of the remaining life given survival up to time x. Joe and Proschan (1982b) develop tests for testing the alternatives representing decreasing 100α-percentile residual life (DPRL-α ) and the property ‘new better than used with respect to the lOOα -percentile’ (NBUP-α ). In this paper, tests are developed for DPRL[α, l) and NBUP[α, l) alternatives, where DPRL[α, l) is the class of life distributions which are DPRL-β distributions for all a ≤ β < 1 if 0 ≤ α < 1 and for all 0 < β < 1 if α = 0, and NBUP[α, l) is similarly defined. When α = 0, the DPRL[α, l) class of life distributions corresponds to the increasing failure rate class and the NBUP[α, l) class of life distributions corresponds to the new better than used class, and the test statistics are respectively asymptotically equivalent to the Hollander and Proschan (1975) test statistics for decreasing mean residual life and new better than used in expectation alternatives.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

A general class of non parametric tests for comparing scale parameters

In this paper, a general class of non parametric tests is proposed for the two-sample scale problem. Testing of the scale parameter is very useful in real-life situations commonly faced in engineering, trade, cultivation, industries, medicine, etc. In all these fields, one will prefer the method that gives more consistent results. Thus, it is worthwhile to test the equality of scale parameters. The distribution of the proposed test is established. To assess the performance of the proposed test, the asymptotic efficacies are studied for some underlying distributions and the results are interpreted with useful information. To see the working of the proposed test, an illustrative example for the real-life data set is provided. The simulation study is also carried out to find the asymptotic power of the proposed test. An extension of the general class of tests to the multiple-sample problem is also discussed.     

The power of the exponential scores test for an ordinary renewal process against trend alternatives

The power properties of a statistic based on the use of exponential scores which may be used for testing whether a series of events occurring in time form an ordinary renewal process against trend alternatives are examined. Small sample power comparisons under a Lehmann trend alternative are made with an alternative nonparametric test based on a rank trend statistic and with the parametric test when the intervals are exponentially distributed. Finally, some asymptotic efficiency results are developed for limiting trend alternatives.