Helmut Finner

Some general results on least favorable parameter configurations with special reference to equivalence testing and the range statistic

Abstract The problem of hypotheses testing where equivalence is stated as the alternative hypothesis is investigated. With respect to a class of absolutely continuous probability distributions with location parameter ϑ ∈ R , k≧2 , equivalence is defined in terms of the range of the components of ϑ. For nonrandomized tests having certain reasonable invariance properties, general results for the extremal points of the power function over both the hypothesis of non-equivalence and subsets of the alternative are provided. The power problem is completely solved, among other things, for tests based on the range statistic. Finally, the equicorrelated normal case is treated in detail and sample sizes for controlling both error levels in equivalence testing are given.

Closed subset selection procedures for selecting good populations

Abstract The problem of selecting a subset containing all good treatments is considered. It is shown that under mild conditions the class of procedures deciding correctly with at least a prespecified probability P * is equivalent in some sense to the class of all consonant and coherent multiple tests at multiple level 1 − P * for a certain system of hypotheses. The equivalence relationship can be used to construct the so-called closed subset selection procedures by applying the well-known principle of closed test procedures. These procedures are more powerful than their single-step counterparts, the so-called natural selection procedures, in the sense that the selected subset of a closed selection procedure is never larger but often smaller than the selected subset of the corresponding natural decision rule. For a sampling statistic having a Lebesgue density with location parameter, analytical results are given concerning the critical values necessary to carry out the closed subset selection procedure. The normal case with both known and unknown common variance is treated in detail.

Some New Inequalities for the Range Distribution, with Application to the Determination of Optimum Significance Levels of Multiple Range Tests

Abstract A problem arising in multiple range tests is the monotonicity of the defining critical values, which ws first treated by Lehmann and Shaffer (1977). This article proves a general inequality for the distribution function of the range, which guarantees the monotonicity of a special set of critical values. It also discusses the relation to some other inequalities. The importance of these inequalities is pointed out in considering some concepts of admissibility and optimality of multiple range tests.

Distribution functions and log-concavity

This paper presents a collection of log-concavity results of one-dimensional cumulative distribution functions (cdfs) F(x,ϑ) and the related functions . in both x∈R or x∈Z and θ ∈ Θ. where R denotes the real line and Z the set of integers. We give a review of results available in the literature and try to fill some gaps in this field. It is well-known that log-concavity properties in x of a density f carry over to F. [Fbar]. and Jc in the continuous and discrete case. In addition, it will be seen that the log-concavity of g(y) = f(ey ) in y for a Lebesgue density f with f(x) = 0 for x < 0 implies the log-concavity of F. This criterion applies to many common densities. Moreover, a convex statistic T defined on R is shown to have a log-concave cdf whenever the underlying n-dimensional Lebesgue density h is log-concave. A slight generalization of the approach in Das Gupta & Sarkar (1984) is used to establish a connection between log-concavity in x of probability densities f or cdf s F and log-concavity of ...

On the modified s-method and directional errors

Scheffe (1970) introduced a modified S-method for testing all linear contrasts, which is a stagewise rejective two stage test procedure. If a contrast hypothesis αμ = 0 is rejected using this method, the problem arises whether or not one can decide for αμ 0 without violating the multiple level α. We give a general proof that the modified S-method still controls the experimentwise level of significance α. Furthermore, we show that this procedure can also be used as a very powerful peocedure for the special case of comparing three means with unequal sample size, which is of great practical importance. Finally we note that a similar result holds true for various closed directional range procedures for the comparison of three means.

Duality between multiple testing and selecting

Abstract Selection problems are considered for which it is possible to formulate a correct selection via a so-called correct selection indicator. The correct selection indicator can be used to define a related multiple hypotheses testing problem. It is shown that there exists a bijective mapping from the class of all selection rules which guarantee a specified confidence level P∗ for a correct selection into a subfamily of all multiple tests at multiple level 1 − P∗ . First of all, this correspondence may be of interest from a pure theoretical viewpoint. Moreover, methods developed for multiple hypotheses testing, such as the closure principle, sometimes allow considerable improvements of existing rules. Last but not least, the duality between multiple testing and selecting may yield a better understanding of the structure of the underlying decision problem. Three selection problems will be considered in more detail: subset selection of Guptas type, best-or-all selection, and unrestricted subset selection. Application of the closure principle to Guptas selection problem indicates that Guptas rule cannot be improved. In fact, Guptas rule turns out to be admissible in a certain sense. It is shown how the best-or-all selection problem can be treated by applying the duality results, and its connection to Bechhofers indifference zone approach is pointed out in detail. The idea of stepwise multiple tests is adopted for the unrestricted subset selection problem. The primary task in the derivation of step-down procedures is the determination of least favorable parameter configurations with respect to certain hypotheses. The validity of the suggested step-up selection procedures depends on the validity of a conjecture concerning the critical values for these procedures. The step-up approach can be considered as a more informative extension of the best-or-all selection rule.

Abgeschlossene multiple Spannweitentests

In dieser Arbeit wird ein abgeschlossener multipler Spannweitentest zum paarweisen Vergleich von k Lokationsparametern vorgestellt, der auf einer konsequenten Anwendung des Abschlusprinzips beruht. Im Gegensatz zu dem von Begun/Gabriel (1981) vorgestellten Abschlus des NewmanKeuls Tests im ANOVA-Modell werden fur die Partitionshypothesen Maxima von Spannweitenstatistiken verwendet. Durch eine Reihe von theoretischen Ergebnissen wird die Zahl der zu prufenden Hypothesen so reduziert, das sich Probleme mit bis zu 20 Lokationsparametern bequem losen lassen. Speziell fur das ANOVA-Modell wurden umfangreiche Tafeln fur die verwendeten Verteilungen und ein Programm zur Durchfuhrung der Testprozedur erstellt.

Asymptotic comparison of the critical values of step-down and step-up multiple comparison procedures

Abstract We consider the problem of comparing step-down and step-up multiple test procedures for testing n hypotheses when independent p -values or independent test statistics are available. The defining critical values of these procedures for independent test statistics are asymptotically equal, which yields a theoretical argument for the numerical observation that the step-up procedure is mostly more powerful than the step-down procedure. The main aim of this paper is to quantify the differences between the critical values more precisely. As a by-product we also obtain more information about the gain when we consider two subsequent steps of these procedures. Moreover, we investigate how liberal the step-up procedure becomes when the step-up critical values are replaced by their step-down counterparts or by more refined approximate values. The results for independent p -values are the basis for obtaining corresponding results when independent real-valued test statistics are at hand. It turns out that the differences of step-down and step-up critical values as well as the differences between subsequent steps tend to zero for many distributions, except for heavy-tailed distributions. The Cauchy distribution yields an example where the critical values of both procedures are nearly linearly increasing in n .

Multiple Tests und Fehler III. Art

Es werden sequentiell verwerfende Testprozeduren fur Hypothesen mit zweiseitigen Alternativen betrachtet. Bis heute ist nur in wenigen Spezialfallen bekannt, welche dieser Testprozeduren auch einseitige Entscheidungen liefern, ohne dadurch das multiple Niveau α zu verletzen. Es wird gezeigt, das der modifizierte Scheffe—Test in Normalverteilungsmodellen mit bekannter Varianz diese Eigenschaft hat.

LFC-Results for Tests in k-Variate Location Models

Let X = (X 1,..., X k ) be a k-variate random variable with values in IR k . We assume that the distribution P μ of X has a Lebesgue-density f μ (x) = f(x — μ) depending on a location parameter μ ∈ Θ = IR k . If one is interested in testing hypotheses concerning the parameter μ, often the problem of determining least favourable parameter configurations (LFC’s) occurs. A typical hypothesis is the homogeneity hypothesis H: μ 1 = ··· = μ k , or, if testing for material significance (cf. Hodges & Lehmann, 1954) is of interest, e.g. the hypotheses H: max1≤i≤j≤k |μ 1 — μ j | ≤ δ, H: (H:{max _{1 leqslant i leqslant j leqslant k}}left| {{mu _1}{mu _j}} right| leqslant delta ,H:sumnolimits_{i = 1}^k {{{left( {{mu _1} - bar mu .} right)}^2} leqslant } delta ,or H:sumnolimits_{i = 1}^k {left| {{mu _1} - bar mu .} right| leqslant } delta ,), respectively, are appropriate. If ≤ δ is replaced by ≥ δ, we are in the situation of testing for equivalence.