Pitman Efficiency Research Articles

A relation between the Chernoff index and the Pitman efficacy

A relation between the Chernoff index and the Pitman efficacy given implicitly in Chernoff (1952) for sums of i.i.d. random variables is extended to broad classes of M-estimators. This relation establishes the local equivalence of Chernoff efficiency and Pitman efficiency.

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.

Distribution Free Tests for Stochastic Ordering in the Competing Risks Model

The competing risks set-up is considered where an individual is subject to failure due to two independent competing risks. The available data consist of observed times to failure and the causes of failure. On the basis of this information, distribution-free tests are proposed for testing the equality of the two failure distributions against location, scale and general stochastic ordering alternatives. Locally most powerful rank tests are derived and a generalization of the Wilcoxon test has been proposed. Exact critical points are provided for the newly proposed tests, and Pitman efficiency comparisons made.

On the relation between Hodges‐Lehmann efficiency and Pitman efficiency

AbstractThe 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.

Some rank tests for one–sided many–one comparisons

We compare some nonparametric tests for the (/+ 1)–sample problem with additive effects under the constraint that in every sample the treatment effect is not less than that in the first sample, i.e. of some control. The behavior of the Pitman efficiency of the respective tests (essentially tests of a Kruskal–Wallis–, Wilcoxon–, Fligner–Wolfe–, Steel–, and Nemenyi–type) is discussed which turns out to depend on the level and power of the tests as well as on the directions, from which the alternative tends to the hypothesis. It will be shown that none of the tests under consideration is uniformly superior to the others.

Testing for superadditivity of the mean value function of a non-homogeneous poisson process

Consider a non-homogeneous Poisson process, N(t), with mean value function Λ(t) and intensity function λ(t). A conditional test of the hypothesis that the process is homogeneous, versus alternatives for which Λ(t) is superadditive, was proposed by Hollander and Proschan (1974). A new test for superadditivity of Λ(t), which is based on a linear combination of the occurrence times of the process N{t) is suggested in this paper. Though this test has the same Pitman efficiency as the Hollander-Proschan test, it is shown by Monte-Carlo simulation that our test has more power for many important alternatives. Tables for the exact null distribution of the test statistic have been given.

Modification of the mann-whitney test

A modification of the Mann–Whitney test was introduced by Lemmer (1987) in hopes of gaining power. A Monte Carlo investigation was performed, but important questions were left unanswered. We widen the investigation to a whole class of modifications, and then provide some answers. some answers are theoretical and some are from simulation. We also propose in the context of this problem how null hypothesis simulation can lead to estimates of Pitman efficiency against contiguous alternatives; this seems to be an important new idea.

Bahadur Efficiency of Two-Sample Tests Based on One-Sample Sign Statistics

SUMMARY This paper discusses the two-sample test of location based on the comparison of two distribution-free one-sample confidence intervals derived from sign statistics. This test procedure, first introduced by Hettmansperger (1984), rejects the null hypothesis of equal population medians when the two intervals are disjoint. He presents three different ways to select the two one-sample intervals and one choice leads to Mood's test. All solutions have the same Pitman efficiency. This paper shows that the choices can be distinguished on the basis of Bahadur's efficiency. We formulate the problem in terms of (asymptotically) fixed width confidence intervals. In this context various median tests (including Mood's test) arise as special cases and they yield different performance. The solution that specifies equal asymptotic lengths for the one-sample intervals (which is different from Mood's test) is recommended.

Estimation of sample sizes in case-control studies with multiple controls per case: dichotomous data.

In planning case-control studies with matched sets, the calculation of exact sample sizes is difficult, because this calculation depends on some nuisance parameters that are usually unknown in practice. Using the Pitman efficiency of Miettinen's test relative to McNemar's test, Schlesselman and Stolley (Case-control studies: design, conduct, analysis. Oxford: Oxford University Press, 1982:144-70) derived an approximate sample size formula which requires the assumption that the difference in exposure rates between cases and controls is small. Furthermore, on the basis of an assumption similar to that used in Schlesselman and Stolley's approach, Taylor (Stat Med 1986;5:29-36) proposed another approximation formula. In this paper, an alternative and explicit formula that does not require the exposure difference to be small between case and control groups has been derived. Monte Carlo studies are given for comparing the accuracy of these three procedures. The results indicate that when odds ratios of exposure between cases and controls are small (less than or equal to 4) and there is more than one matched control per case, the formula derived in this paper seems to be the best. When odds ratios are large (greater than or equal to 5), however, Taylor's more conservative estimate is recommended, unless the exposure prevalence in the general population is large (0.9).

The Bahadur Efficiency of Tests of Some Joint Hypotheses

Abstract This article examines the relative efficiency of finite induced and Wald-like (or “infinite induced”) tests of some commonly encountered joint hypotheses. One two-sided testing problem and two types of two-sided testing problems concerning two parameters and normally distributed estimates are considered. The exact slopes of six test statistics are derived, and from these the Bahadur relative efficiency of tests based on any pair of the statistics can be easily computed. For the problems considered, Bahadur relative efficiency is equal to limiting Pitman efficiency, whereas Pitman relative efficiency coincides with exact finite sample relative efficiency. I present numerical results showing that Bahadur relative efficiency generally approximates Pitman efficiency rather poorly, although the patterns exhibited by these two efficiency criteria bear a reasonable similarity in most cases.

The Bahadur Efficiency of Tests of Some Joint Hypotheses

Abstract This article examines the relative efficiency of finite induced and Wald-like (or “infinite induced”) tests of some commonly encountered joint hypotheses. One two-sided testing problem and two types of two-sided testing problems concerning two parameters and normally distributed estimates are considered. The exact slopes of six test statistics are derived, and from these the Bahadur relative efficiency of tests based on any pair of the statistics can be easily computed. For the problems considered, Bahadur relative efficiency is equal to limiting Pitman efficiency, whereas Pitman relative efficiency coincides with exact finite sample relative efficiency. I present numerical results showing that Bahadur relative efficiency generally approximates Pitman efficiency rather poorly, although the patterns exhibited by these two efficiency criteria bear a reasonable similarity in most cases.

Linear rank tests for radomazed block design

A class of locally most powerful linear rank tests for no treatment effect for various randomized block designs are presented. Closed form expressions of the test statistic are derived for the complete block design, the balanced incomplete block design and a special design in which there are missing observations on one treatment only. In addition, a test statistic is UCLiVCU LUL LUC unbalanced incomplete block design. The maximin efficiency robust test is given for a randomized block design* This test maximizes the minimum Pitman efficiency over a class of possible error distributions. Multiple comparison procedures are presented

On Local and Nonlocal Measures of Efficiency

General results on the limiting equivalence of local and nonlocal measures of efficiency are obtained. Why equivalence occurs in so many testing and estimation problems is clarified. Uniformity of the convergence is a key point. The concepts of Frechet- and Hadamard-type differentiability, which imply uniformity, play an important role. The theory is applied to tests based on linear rank statistics, showing equivalence of the local limit of exact Bahadur efficiency and Pitman efficiency. As a second application, the relation between the inaccuracy rate and the asymptotic variance of $L$-estimators is investigated.

Two-sample inference based on one-sample Wilcoxon signed rank statistics

A two-sample test is studied which rejects the null hypothesis of equal population medians when two Wilcoxon distribution free confidence intervals are disjoint. A confidence interval for the difference in population medians is constructed by subtracting the endpoints of two one-sample confidence intervals. Two different ways to select the one-sample intervals are presented. A solution that specifies equal confidence coefficients for the one-sample intervals is recommended. All solutions are shown to have the same asymptotic (Pitman) efficiency as the Mann-Whitney two-sample test.

Linear Rank Statisics For The Orderes Alternatives Problem

Linear rank statistics for the ordered alternatives prblem are considered. Pitman efficiencies between several procedures are studied. A new minimax procedure is suggested and shown to have good efficiency propertieds. Additionally, it is the only considered with relative efficiency bounded away from zero

The likelihood ratio test for the change point problem for exponentially distributed random variables

Let x 1,…, x n+1 be independent exponentially distributed random variables with intensity λ 1 for i ⩽ τ and λ 2 for i > τ, where τ as well as λ 1 and λ 2 are unknown. By application of theorems concerning the normed uniform quantile process it is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing λ 1 = λ 2 (or, equivalently, τ = 0 or n + 1) is an extreme value distribution. Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the ‘strong invariance principle’. Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications. The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test.

On the limiting Pitman efficiency of some rank tests of independence

The limiting (as the significance level approaches 0) Pitman efficiency of a new “regression-based” rank test of independence to Kendall's tau and Spearman's rho is derived. The result is based on a version of Wieand's theorem (1976, Ann. Statist. 4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. Kiefer and Wolfowitz (1958, Trans. Amer. Math. Soc. 87, 173–186) result is applied to verify the main assumption of Wieand's paper. This approach is shown to be useful in some other situations, also.

Relative efficiencies of goodness of fit procedures for assessing univariate normality

Efficiency properties of the Cramer-von Mises, Anderson-Darling, Watson, and DeWet-Venter statistics for assessing normality are investigated. For these statistics, the approximate slopes are determined, and the equivalence of ratios of limiting approximate slopes to limiting Pitman efficiencies is established. From relative efficiency comparisons, the Cramer-von Mises and Watson statistics perform rather poorly; choice between the Anderson-Darling and DeWet-Venter statistics should be made on the basis of anticipated alternatives.

LINEAR TREND TEST VERSUS GLOBAL TEST: A COMPARISON

AbstractFor the I–sample problem, the classical global tests on treatment differences are compared with the respective trend tests both in the parametric and nonparametric case using a comparison criterion closely related to the concept of asymptotic relative Pitman efficiency.

Finite Sample Properties and Asymptotic Efficiency of Monte Carlo Tests

Since their introduction by Dwass (1957) and Barnard (1963), Monte Carlo tests have attracted considerable attention. The aim of this paper is to give a unified approach that covers the case of an arbitrary null distribution in order to study the statistical properties of Monte Carlo tests under the null hypothesis and under the alternative. For finite samples we obtain bounds for the power of the Monte Carlo test with the original test that allow determination of the required simulation effort. Furthermore the concept of asymptotic (resp. local asymptotic) relative Pitman efficiency (ARPE, resp. LARPE) is adapted to Monte Carlo tests for the study of their asymptotic behaviour. The normal limit case is investigated in more detail, leading to explicit formulas for ARPE and LARPE.