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.     

On blest's measure of rank correlation

Blest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.     

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.     

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.     

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.     

Non-parametric tests for several sample location problem based on subsample extrema

In this paper non-parametric tests for homogeneity of several populations against locationtype alternatives are proposed. For this all possible subsamples of fixed size are drawn from each sample and their maxima and minima are computed One class of tests is obtained using these subsample minima whereas other class of tests involves use of sub sample maxima. Tests belonging t o these two classes have been compared with many of the presently available tests in terms of their Pitman asymptotic relative efficiency . Some of the members of these proposed classes of tests prove to robust in terms of efficiency.     

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.     

A class of distribution-free tests for two-way layout

In this article a class of distribution-free tests for the hypothesis of no row (treatment) effect in a two-way layout design, with several observations per cell, is proposed. The tests are based on U-statistics, constructed by considering minima of all possible subsamples of same size from each cell.The proposed class of tests is compared with the parametric test, Mack and Skillings test and Yate's test for two-way layout, in terms of Pitman ARE sense. It is seen that for the case of equal number of observations per cell, the proposed tests have better efficiency for exponential and uniform error distributions.     

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.     

Comparing tests judged asymptotically equally efficient

For comparing two competing testing procedures whose power functions cannot be calculated exactly, the Pitman asymptotic relative efficiency is commonly used. This measure offers the advantages of simplicity in computation and interpretation, but is asymptotic in nature. In particular, it cannot distinguish between certain pairs of tests, namely, those with an asymptotic relative efficiency of 1. In this paper, we offer an alternative basis for making finite sample comparisons between tests judged asymptotically equally efficient. The criteria for comparison is based on the ratio of efficacies. Using this new criterion, we consider a series of examples in which two tests with an ARE of 1 are compared for finite n. In each case, either exact or simulated power curves are used to evaluate the effectiveness of this criterion in properly discriminating between the competing test procedures. Lastly, conditions are specified where the proposed criterion should be most effective.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

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.     

Moment inequalities for HNBRUE with hypothesis testing application

The paper deals with life distributions which are harmonic new better than renewal used in expectation (HNBRUE). The goal is to derive moment inequalities and use them for further analysis of the class HNBRUE. We establish new characterization of exponentiality versus HNBRUE. Pitman’s asymptotic relative efficiency is employed to assess the performance of the proposed test with respect to other available tests. Finally, we carried out numerical simulation to produce table for the critical values of the test.     

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.     

Distribution-free tests for independence against positive quadrant dependence: A generalization

A class of distribution-free tests based on two matched pairs is considered for testing independence against positive quadrant dependence. The class of tests proposed by Kochar and Gupta (1990) is a member of the proposed class. The performance of the proposed class is evaluated in terms of Pitman asymptotic relative efficiency for Block-Basu (1974) model and Woodworth family of distributions. The small sample performance of few members of the proposed class is also studied by finding empirical powers. Unbiasedness and consistency of the proposed class of tests have been established.     

Bounds on efficiencies for some two-sample nonparametric statistics

Bounds on the Pitman efficiency for two-sample scale and loca-tion statistics are presented along with densities for which these bounds are sharp.     

An affine‐invariant multivariate sign test for cluster correlated data

The author presents a multivariate location model for cluster correlated observations. He proposes an affine‐invariant multivariate sign statistic for testing the value of the location parameter. His statistic is an adaptation of that proposed by Randles (2000). The author shows, under very mild conditions, that his test statistic is asymptotically distributed as a chi‐squared random variable under the null hypothesis. In particular, the test can be used for skewed populations. In the context of a general multivariate normal model, the author obtains values of his test's Pitman asymptotic efficiency relative to another test based on the overall average. He shows that there is an improvement in the relative performance of the new test as soon as intra‐cluster correlation is present Even in the univariate case, the new test can be very competitive for Gaussian data. Furthermore, the statistic is easy to compute, even for large dimensional data. The author shows through simulations that his test performs well compared to the average‐based test. He illustrates its use with real data.     

A WEIGHTED RANK MEASURE OF CORRELATION

Spearman's rank correlation coefficient is not entirely suitable for measuring the correlation between two rankings in some applications because it treats all ranks equally. In 2000, Blest proposed an alternative measure of correlation that gives more importance to higher ranks but has some drawbacks. This paper proposes a weighted rank measure of correlation that weights the distance between two ranks using a linear function of those ranks, giving more importance to higher ranks than lower ones. It analyses its distribution and provides a table of critical values to test whether a given value of the coefficient is significantly different from zero. The paper also summarizes a number of applications for which the new measure is more suitable than Spearman's.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

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.