E. L. Lehmann

Testing Statistical Hypothesis.

Testing statistical hypothese , Testing statistical hypothese , کتابخانه دیجیتال جندی شاپور اهواز

The Efficiency of Some Nonparametric Competitors of the t-Test

Consider samples from continuous distributions F(x) and F(x — θ). We may test the hypothesis θ = 0 by using the two-sample Wilcoxon test. We show in Section 1 that its asymptotic Pitman efficiency, relative to the f-test, never falls below 0.864. This result also holds for the Kruskal-Wallis test compared with the jF-test, and for testing the location parameter of a single symmetric distribution.

Rank Methods for Combination of Independent Experiments in Analysis of Variance

It is now coming to be generally agreed that in testing for shift in the two-sample problem, certain tests based on ranks have considerable advantage over the classical t-test. From the beginning, rank tests were recognized to have one important advantage: their significance levels are exact under the sole assumption that the samples are randomly drawn (or that the assignment of treatments to subjects is performed at random), whereas the t-test in effect is exact only when we are dealing with random samples from normal distributions. On the other hand, it was felt that this advantage had to be balanced against the various optimum properties possessed by the t-test under the assumption of normality. It is now being recognized that these optimum properties are somewhat illusory and that, under realistic assumptions about extreme observations or gross errors, the t-test in practice may well be less efficient than such rank tests as the Wilcoxon or normal scores test [6], [7].

The Fisher, Neyman-Pearson theories of testing hypotheses: One theory or two?

Abstract The Fisher and Neyman-Pearson approaches to testing statistical hypotheses are compared with respect to their attitudes to the interpretation of the outcome, to power, to conditioning, and to the use of fixed significance levels. It is argued that despite basic philosophical differences, in their main practical aspects the two theories are complementary rather than contradictory and that a unified approach is possible that combines the best features of both. As applications, the controversies about the Behrens-Fisher problem and the comparison of two binomials (2 × 2 tables) are considered from the present point of view.

The use of Previous Experience in Reaching Statistical Decisions

Instead of minimizing the maximum risk it is proposed to re-strict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount. Subject to this restriction one may wish to minimize the average risk with respect to some guessed a priori distribution suggested by previous experience. It is shown how Wald’s minimax theory can be modified to yield analogous results concerning such restricted Bayes solutions. A number of examples are discussed, and some extensions of the above criterion are briefly considered.

A Theory of Some Multiple Decision Problems. II

A class of multiple decision procedures is described and its members are shown to possess uniformly minimum risk among all procedures that are unbiased with respect to a certain loss function. This provides a justification for a number of procedures considered by Tukey, Duncan, and others, for certain classes of point estimates, and for some nonparametric decision procedures based on sample cumulative distribution functions and related to tests of the Kolmo-goroff-Smimoff type.

Robust Estimation in Analysis of Variance

In linear models with several observations per cell, estimates of all contrasts are given whose small and large sample behaviour is analogous to that of the estimate of a shift parameter proposed in [2]. In particular, the asymptotic efficiency of these estimates relative to the standard least squares estimates, as the number of observations in each cell gets large, is shown to be the same as the Pitman efficiency of the Wilcoxon test relative to the t-test.

Descriptive Statistics for Nonparametric Models II. Location

Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, R-estimators and M-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and certain other weighted quantiles) are the only ones which are both robust and whose estimators have guaranteed high efficiency relative to the mean X for all underlying distributions.

On the Theory of Some Non-Parametric Hypotheses

For two types of non-parametric hypotheses optimum tests are derived against certain classes of alternatives. The two kinds of hypotheses are related and may be illustrated by the following example: (1) The joint distribution of the variables Xi, • • •, Xm, Yi, • • •, Yn is invariant under all permutations of the variables; (2) the variables are independently and identically distributed. It is shown that the theory of optimum tests for hypotheses of the first kind is the same as that of optimum similar tests for hypotheses of the second kind. Most powerful tests are obtained against arbitrary simple alternatives, and in a number of important cases most stringent tests are derived against certain composite alternatives. For the example (1), if the distributions are restricted to probability densities, Pitman’s test based on ȳ — x is most powerful against the alternatives that the X’s and F’s are independently normally distributed with common variance, and that E(Xi) = ξ, E(Yi) =η where η > ξ If η — ξ may be positive or negative the test based on | ȳ — x| is most stringent. The definitions are sufficiently general that the theory applies to both continuous and discrete problems, and that tied observations present no difficulties. It is shown that continuous and discrete problems may be combined. Pitman’s test for example, when applied to certain discrete problems, coincides with Fisher’s exact test, and when m = n the test based on | ȳ — x | is most stringent for hypothesis (1) against a broad class of alternatives which includes both discrete and absolutely continuous distributions.

Some Problems in Minimax Point Estimation

In the present paper the problem of point estimation is considered in terms of risk functions, without the customary restriction to unbiased estimates. It is shown that, whenever the loss is a convex function of the estimate, it suffices from the risk viewpoint to consider only nonrandomized estimates. For a number of specific problems the minimax estimates are found explicitly, using the squared error as loss. Certain minimax prediction problems are also solved.