On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

Certain nonparametric classification rules and their asymptotic efficiencies

Two nonparametric classification rules for e-univariace populations are proposed, one in which the probability of correct classification is a specified number and the other in which one has to evaluate the probability of correct classification. In each case the classification is with respect to the Chernoff and Savage (1958) class of statistics, with possible specialization to populations having different location shifts and different changes of scale. An optimum property, namely the consistency of the classification procedure, is established for the second rule, when the distributions are either fixed or “near” in the Pitman sense and are tending to a common distribution at a specified rate. A measure of asymptotic efficiency is defined for the second rule and its asymptotic efficiency based on the Chernoff-Savage class of statistics relative to the parametric competitors ie the case of location shifts and scale changes is shown to be equal to the analogous Pitman efficiency.     

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     

An optimal sign test for one-sample bivariate location model using an alternative bivariate ranked-set sample

The aim of this paper is to find an optimal alternative bivariate ranked-set sample for one-sample location model bivariate sign test. Our numerical and theoretical results indicated that the optimal designs for the bivariate sign test are the alternative designs with quantifying order statistics with labels {((r+1)/2, (r+1)/2)}, when the set size r is odd and {(r/2+1, r/2), (r/2, r/2+1)} when the set size r is even. The asymptotic distribution and Pitman efficiencies of these designs are derived. A simulation study is conducted to investigate the power of the proposed optimal designs. Illustration using real data with the Bootstrap algorithm for P-value estimation is used.     

On testing exponentiality against new better than used of specified age

R.M. Hollander, D.H. Park and F. Proschan [A class of life distributions for aging, J. Amer. Statist. Assoc. 81 (1986) 91–95] introduced the concept of the larger class of life distributions called new better than used of specified age. In practice, one might be interested in the new better than used behaviour at an unknown but estimable age t₀. Here, we investigate the testing of new better than used of specified age t₀ (NBU-t₀) alternatives. A class of test statistics for testing NBU-t₀ (t₀ is known) based on a U-statistic whose kernel depends on sub-sample minima is proposed. A member of the class of tests proposed by N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] for this problem belongs to the class of tests proposed here. The distributional properties of the class of test statistics are studied. The performances of a few members of the proposed class of tests are studied in terms of Pitman asymptotic relative efficiency. The Pitman ARE values show that the members of the class perform well in comparison with the N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] tests. The proposed class of tests is shown to be consistent for NBU-t₀ alternatives.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.     

Goodness-of-fit tests for the Pareto distribution based on its characterization

A new characterization of the Pareto distribution is proposed, and new goodness-of-fit tests based on it are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, it is similar to the classical statistics \(\omega _n^1\). The second one is a Kolmogorov type statistic. We show that the kernels of our statistics are non-degenerate. The limiting distribution and large deviations asymptotics of the new statistics under null hypothesis are described. Their local Bahadur efficiency for parametric alternatives is calculated. This type of efficiency is mostly appropriate for the solution of our problem since the Kolmogorov type statistic is not asymptotically normal, and the Pitman approach is not applicable to this statistic. For the second statistic we evaluate the critical values by using Monte-Carlo methods. Also conditions of local optimality of new statistics in the sense of Bahadur are discussed and examples of such special alternatives are given. For small sample size we compare the power of those tests with some common goodness-of-fit tests.     

Non Null Asymptotic Distributions of Some Criteria in Censored Exponential Regression

In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n^{? 1/2} of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.     

One-and two-sample sign tests for ranked set sample selective designs

Statistical inference based on a ranked set sample depends very much on the location of the quantified observations. A selective design which determines the location of the quantified observations in a ranked set sample is introduced. The paper investigates the effects of selective designs on one and two sample sign test statistics. The Pitman efficiencies of one- and two sample sign tests are calculated for selective designs and compared with ranked set samples of the same size. If the design quantifies observations at the center points, then the proposed procedure is superior to a ranked set sample of the same size in the sense of Pitman efficiency. Some practical problems are addressed for the two-sample sign test.     

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.     

On the relation between Hodges-Lehmann efficiency and Pitman efficiency

The relation between Hodges–Lehmann efficiency and Pitman efficiency is studied in the context of testing one-sided hypotheses about a real-valued parameter. It is first shown that for tests based on sums of independently and identically distributed observations local Hodges-Lehmann efficiency is equivalent to Pitman efficiency. Then, it is proved that this equivalence also carries over to tests based on two broad classes of M -estimators for the location problem. In all cases considered explicit formulas of the Hodges-Lehmann efficacies are obtained.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

An Affine-Invariant Generalization of the Wilcoxon Signed-Rank Test for the Bivariate Location Problem

This paper proposes an affine‐invariant test extending the univariate Wilcoxon signed‐rank test to the bivariate location problem. It gives two versions of the null distribution of the test statistic. The first version leads to a conditionally distribution‐free test which can be used with any sample size. The second version can be used for larger sample sizes and has a limiting χ₂² distribution under the null hypothesis. The paper investigates the relationship with a test proposed by Jan & Randles (1994). It shows that the Pitman efficiency of this test relative to the new test is equal to 1 for elliptical distributions but that the two tests are not necessarily equivalent for non‐elliptical distributions. These facts are also demonstrated empirically in a simulation study. The new test has the advantage of not requiring the assumption of elliptical symmetry which is needed to perform the asymptotic version of the Jan and Randles test.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

SELECTING "DEMONSTRABLY BEST" OR "DEMONSTRABLY WORST" EXPONENTIAL POPULATIONS

Consider k independent observations Y_i (i= 1,., k) from two-parameter exponential populations _i with location parameters μ and the same scale parameter If the μ_i are ranked as consider population as the “worst” population and II_p(k) as the “best” population (with some tagging so that p{) and p(k) are well defined in the case of equalities). If the Y_i are ranked as we consider the procedure, “Select provided Y_R(k) Y_r(k) is sufficiently large so that is demonstrably better than the other populations.” A similar procedure is studied for selecting the “demonstrably worst” population.     

An improved ranked set two‐sample mann‐whitney‐wilcoxon test

The authors present an improved ranked set two‐sample Mann‐Whitney‐Wilcoxon test for a location shift between samples from two distributions F and G. They define a function that measures the amount of information provided by each observation from the two samples, given the actual joint ranking of all the units in a set. This information function is used as a guide for improving the Pitman efficacy of the Mann‐Whitney‐Wilcoxon test. When the underlying distributions are symmetric, observations at their mode(s) must be quantified in order to gain efficiency. Analogous results are provided for asymmetric distributions.     

TESTING NEW BETTER THAN RENEWAL USED LIFE DISTRIBUTIONS BASED ON U-TEST

In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.     

A joint test for parametric specification and independence in nonlinear regression models

This paper develops a testing procedure to simultaneously check (i) the independence between the error and the regressor(s), and (ii) the parametric specification in nonlinear regression models. This procedure generalizes the existing work of Sen and Sen [“Testing Independence and Goodness-of-fit in Linear Models,” Biometrika, 101, 927–942.] to a regression setting that allows any smooth parametric form of the regression function. We establish asymptotic theory for the test procedure under both conditional homoscedastic error and heteroscedastic error. The derived tests are easily implementable, asymptotically normal, and consistent against a large class of fixed alternatives. Besides, the local power performance is investigated. To calibrate the finite sample distribution of the test statistics, a smooth bootstrap procedure is proposed and found work well in simulation studies. Finally, two real data examples are analyzed to illustrate the practical merit of our proposed tests.     

THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.