Two–step sequential sampling

Choosing the sample size in advance is a familiar problem: often, additional observations appear to be desirable. The final sample size then becomes a random variable, which has rather serious consequences.
Two such sample extension situations will be considered here. In the first situation, the observed sample variance determines whether or not to double the original sample size. In the second situation, the variances observed in two independent samples are compared; their ratio determines the number of additional observations.     

Two-stage point estimation with a shrinkage stopping rule

Consider the problem of estimating a mean vector in ap-variate normal distribution under two-stage sequential sampling schemes. The paper proposes a stopping rule motivated by the James-Stein shrinkage estimator, and shows that the stopping rule and the corresponding shrinkage estimator asymptotically dominate the usual two-stage procedure under a sequence of local alternatives forp3. Also the results of Monte Carlo simulation for average sample sizes and risks of estimators are stated.     

Locally most powerful rank tests for several sample problems

Summary This paper develops for the several sample problem the notions of a parametric family of non-parametric alternatives. In section two we give a detailed characterization of the class of all locally most powerful rank order tests for testing the hypothesis of equality of several continuous distributions. In section three we show that the class of all locally most powerful rank order tests coincides with the class of all locally least powerful tests. In section four we consider a multiple decision problem with non-parametric slippage alternatives and characterize the class of all locally best symmetric decision procedures based on rank orders.

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Zusammenfassung In dieser Arbeit wird für dask-Stichproben-Problem der Begriff einer parametrischen Familie nicht-parametrischer Alternative-Hypothesen entwickelt. Der zweite Paragraph enth?lt eine ausführliche Charakterisierung der Klasse aller lokal trennscharfen Rang-Tests, welche die Hypothese der Gleichheit verschiedener stetiger Verteilungen prüft. Im dritten Paragraphen wird gezeigt, da? die Klasse der lokal trennscharfen Rang-Tests mit der Klasse der lokal am wenigsten trennscharfen Rang-Tests übereinstimmt. Im Paragraphen vier betrachten wir ein nicht-parametrischesk-Stichproben-Ausrei?erproblem und charakterisieren die Klasse aller lokal optimalen symmetrischen auf Rang beruhenden Entscheidungsverfahren.

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This research was supported by the Air Force Office of Scientific Research of the Office of Aerospace Research, under Contract No. AF 49(638)–1057.     

Nonparametric methods for paired samples

The small sample and asymptotic properties of nonparametric tests for paired sampled are examined. Linear rank statistics are compared with the paired t-test and the Wilcoxon-signed-rank test in simulation studies. From a minimax point of view the linear rank statistics turn out to be the best. Moreover, it is illustrated that the Wilcoxon-signed-rank test should not be used if it is not clear that the differences of the pairs have a symmetric distribution.     

Rearrangements and sequential rank order dominance

Distributive analysis typically involves comparisons of distributions where individuals differ in more than just one attribute. In the particular case where there are two attributes and where the distribution of one of these two attributes is fixed, one can appeal to sequential rank order dominance for comparing distributions. We show that sequential rank order domination of one distribution over another implies that the dominating distribution can be obtained from the dominated one by means of a finite sequence of favourable permutations, and conversely. We provide two examples where favourable permutations prove to have interesting implications from a normative point of view.     

Local comparison of linear rank tests,in the Bahadur sense

Summary As continuation ofKremer [1979a] a theory of asymptotic comparison based on local Bahadur efficiency is derived for general linear rank tests of the one-sample symmetry problem and thek-sample problem (k2). The results are similar to former considerations based on Pitman efficiency but hold under weaker conditions on the scores-generating functions or local alternatives.     

Calculating Cumulants of a Taylor Expansion of a Multivariate Function

A method, which we believe is simpler and more transparent than the one due to McCullagh (1984) , is described for obtaining the cumulants of a scalar multivariate stochastic Taylor expansion. Its generalisation is also suggested. An important feature, previously not reported, is that the expansion of every cumulant of order ≥ 2 is made up of separate subseries.
In order to handle certain frequently occurring sums over permutations of members of compound index sets, we introduce a new notation  [ m ]_*,  where   m   is a positive integer.     

One-sided and multi-sided rank tests against trend in the case of concordant observers

Summary Optimal rank tests are derived for testing homogeneity ofk populations observed bym independent andconcordant observers against oriented respectively non-oriented contiguity alternatives with respect to a location parameter.By means of the theorems obtained appropriate rank tests are proposed for testing against a one-sided respectively a multi-sided trend.     

On rankings and top choices in random utility models with dependent utilities

Different aggregate preference orders based on rankings and top choices have been defined in the literature to describe preferences among items in a fixed set of alternatives. A useful tool in this framework is constituted by random utility models, where the utility of each alternative, or object, is represented by a random variable, indexed by the object, which, for example, can capture the variability of preferences over a population. Applications are derived in diverse research fields, including computer science, management science and reliability. Recently, some stochastic ordering conditions have been provided for comparing alternatives by means of some aggregate preference orders in the case of independent random utility variables by Joe (Math Soc Sci 43:391–404, 2002). In this paper we provide new conditions, based on some joint stochastic orderings, for aggregate preference orders among the alternatives in the case of dependent random utilities. We also provide some examples of application in different research fields.      

The Borda class: An axiomatic study of the Borda rule on top-truncated preferences

The Borda rule, originally defined on profiles of individual preferences modelled as linear orders over the set of alternatives, is one of the most important voting rules. But voting rules often need to be used on preferences of a different format as well, such as top-truncated orders, where agents rank just their most preferred alternatives. What is the right generalisation of the Borda rule to such richer models of preference? Several suggestions have been made in the literature, typically considering specific contexts where the rule is to be applied. In this work, taking an axiomatic perspective, we conduct a principled analysis of the different options for defining the Borda rule on top-truncated preferences.     

Test of Bernoulli success probability in inverse sampling for nearer alternatives using adaptive allocation

This paper provides a test procedure for the problem of testing Bernoulli success probability in the case of costly trials when an inverse sampling is carried out. The proposed test is based on a two population adaptive sampling scheme used in clinical trials. Some exact and asymptotic results related to the test are studied. The proposed procedure is applicable where the alternatives are not too far from the null hypothetical value.     

THE ASYMPTOTIC BEHAVIOR OF SIMPLE LINEAR RANK STATISTICS

The concept of a simple linear rank statistic is essential in nonparametric theory. These statistics may be used for testing the null-hypothesis that all observations are independent and identically distributed against classes of alternatives indicated by the choice of regression constants and scores. Special cases are the two-sample Linear rank statistics and Spearman's rank correlation coefficient rho under the null-hypothesis of independence.
In this paper we shall give a survey of the asympotic distribution theory of simple linear rank statistics.     

A Sequential distribution-free two-sample grouped test with three possible decisions*

Summary A sequential two-sample grouped rank test with two possible decisions (onesided testing) developed by WILCOXON, RHODES and BRADLEY (1963) is generalized to a sequential test with three possible decisions (two-sided testing) using a method given by SOBEL and WALD (1949). An auxiliary table with some moments concerning LEHMANN'S alternatives for the normal distribution, is given in order to facilitate the choice of the alternative hypotheses.     

Neyman–Pearson Lemma for Fuzzy Hypotheses Testing with Vague Data

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which both hypotheses and observations are imprecise. This paper tries to develop a new approach for testing fuzzy hypothesis when the available data are fuzzy, too. First, some definitions are provided, such as: fuzzy sample space, fuzzy-valued random sample, and fuzzy-valued random variable. Then, the problem of fuzzy hypothesis testing with vague data is formulated. Finally, we state and prove a generalized Neyman–Pearson Lemma for such problem. The proposed approach is illustrated by some numerical examples.     

Distribution theory for the studentized mean for long,short, and negative memory time series

We consider the problem of estimating the variance of the partial sums of a stationary time series that has either long memory, short memory, negative/intermediate memory, or is the first-difference of such a process. The rate of growth of this variance depends crucially on the type of memory, and we present results on the behavior of tapered sums of sample autocovariances in this context when the bandwidth vanishes asymptotically. We also present asymptotic results for the case that the bandwidth is a fixed proportion of sample size, extending known results to the case of flat-top tapers. We adopt the fixed proportion bandwidth perspective in our empirical section, presenting two methods for estimating the limiting critical values—both the subsampling method and a plug-in approach. Simulation studies compare the size and power of both approaches as applied to hypothesis testing for the mean. Both methods perform well–although the subsampling method appears to be better sized–and provide a viable framework for conducting inference for the mean. In summary, we supply a unified asymptotic theory that covers all different types of memory under a single umbrella.     

LINEAR TREND TEST VERSUS GLOBAL TEST: A COMPARISON

For 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.     

Meta-analytic methods to test relative efficacy

In the context of multiple treatments for a particular problem or disorder, it is important theoretically and clinically to investigate whether any one treatment is more effective than another. Typically researchers report the results of the comparison of two treatments, and the meta-analytic problem is to synthesize the various comparisons of two treatments to test the omnibus null hypothesis that the true differences of all particular pairs of treatments are zero versus the alternative that there is at least one true nonzero difference. Two tests, one proposed by Wampold et al. (Psychol. Bull. 122:203–215, 1997) based on the homogeneity of effects, and one proposed here based on the distribution of the absolute value of the effects, were investigated. Based on a Monte Carlo simulation, both tests adequately maintained nominal error rates, and both demonstrated adequate power, although the Wampold test was slightly more powerful for non-uniform alternatives. The error rates and power were essentially unchanged in the presence of random effects. The tests were illustrated with a reanalysis of two published meta-analyses (psychotherapy and antidepressants). It is concluded that both tests are viable for testing the omnibus null hypothesis of no treatment differences.     

A nonsymmetric sequential test

A one-sided testing problem based on an i.i.d. sample of observations is considered. The usual one-sided sequential probability ratio test would be based on a random walk derived from these observations. Here we propose a sequential test where the random walk is replaced by Lindleys random walk which starts anew at zero as soon as it becomes negative. We derive the asymptotics of the expected sample size and the error probabilities of this sequential test. We discuss the advantages of this test for certain nonsymmetric situations.Acknowledgement. The authors thank the referee for helpful comments and suggestions. Their research was supported by the German Research Foundation (DFG) and the Russian Foundation for Basic Research (RFBR).