On some sequential simultaneous confidence intervals procedures

Summary The purpose of this paper is two-fold: (i) to extend the simultaneous confidence intervals procedures (SCIP) of Healy [7] along the lines of Chow and Robbins [3] and (ii) to develop certain robust non-parametric SCIP based on the results of Sen [10] and Sen and Ghosh [11]; the allied efficiency results are also presented. Work supported by the National Institutes of Health, Grant GM-12868.     

On some tests of homogeneity of variances

Summary In their paper [1], Ansari and Bradley discussed a two-sample rank test for dispersions and suggested the desirability of extending their results to the problem of several samples. In this paper, besides generalizing their results, we provide a few additional non-parametric tests, which include, among others, the multi-sample analogues of the two-sample normal scores test of dispersion and the tests considered by Mood [6], Klotz [4], and Siegel and Tukey [10]. The asymptotic distributions of the proposed test statistics are derived by an application of the author’s theorem [7]. The asymptotic efficiencies of these tests relative to one another and theF test ([9], pp. 83–87) are computed in the standard fashion along the lines of the author’s paper [7]. The work represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under Sloan Foundation Grant for statistics and under U.S. Navy Contract Nonr-285 (38).     

On some tests of growth curve model under Behrens-Fisher stituation

In this article, we have considered the problem of testing equality of several growth curves under Behrens-Fisher situation. In this context, the robustness of the existing test criteria have been studied. Also, some exact test procedures have been considered and the exact and asymptotic noncentral distribution problems have been studied.     

On some nonparametric tests for profile analysis of several multivariate samples

For profile analysis of independent samples from several multivariate populations, a nonparametric analog of the hypothesis of parallelism of population profiles is formulated. A class of asymptotically distribution-free statistics is offered to test this hypothesis. These are based on generalized U statistics and are in some sense modifications of statistics offered previously by one of the authors for testing the homogeneity hypothesis. Consistency of these statistics is established for suitable alternatives and also asymptotic power is investigated.     

On the power of some adaptive tests in the one-sample problem

In this paper, we study adaptive tests for the one-sample problem. The limit of the normalized difference between the power of a given adaptive test and that of the most powerful test is found. This limit directly yields the Hodges-Lehmann asymptotic deficiency of corresponding adaptive tests. Here adaptation means that the efficient score function of the test is estimated from the sample. A Fourier series estimator is used for the score function. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

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 [17], Ann. Statist.4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. [7], 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.     

Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P_? depending on a one-dimensional parameter ? the problem of sequentially testing ? = 0 against ? > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.     

On conditions of chernoff local asymptotic optimality of some nonparametric symmetry tests

A general upper bound for the Chernoff indices of a sequence of statistics used for testing the symmetry hypothesis against parametric alternatives is found. The notion of Chernoff local optimality is defined. The corresponding properties of signed-rank tests are investigated. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 101–108, 1993. Translated by the authors     

Characterisation of some sparse binary sequential arrays

A periodic binary array is said to besequential if and only if every line of the array is occupied by a given periodic binary sequence or by some cyclic shift or reversal of this sequence. This paper extends earlier results for arrays built on the square grid, characterising further sequential arrays with two ones per period, and those with three consecutive ones per period.     

Implementation of some multioperations on sequential computers

Some typical array-manipulating operations implemented on M-20 computers are analyzed. Program realizations of these multioperations included in the IS-2 interpreter standard programs and in programs generated by ALGOL, FORTRAN, and AL'FA translators were traced by a special counter-tracer program in order to establish the corresponding implementation overheads. The main characteristics of object programs and of multioperation realizations are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 184–204, 1977.We would like to acknowledge the valuable assistance of N. B. Malyshev in analyzing the programs.     

Estimation of the acceleration coefficient of sequential system tests

This paper deals with some methods of constructing the relationship between lifetimes in normal and accelerated tests. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez pp. 120–131, Perm, 1991.     

Structure of sequential tests minimizing an expected sample size

Summary The testing problem is to decide on the basis of repeated independent observations which of the probability densitiesf andg is true. Given upper bounds on the probabilities of error, the object is to minimize the expected sample size if the densityp is true (allowed to differ fromf andg). A characterization of the structure of optimal tests is obtained which is particularly informative in the case wheref,g, andp belong to a Koopman-Darmois family. Ifp=f org, then the optimal tests are sequential probability ratio tests (SPRT's) and a new proof of the well-known optimality property of these tests is obtained as a corollary.Research supported by National Science Foundation grant MCS 76-08489     

On asymptotic efficiency and deficiency of some adaptive tests in the one-sample problem

In this paper, we consider adaptive tests for the one-sample problem and investigate the order of difference between the power of a given adaptive asymptotically efficient test and that of the most powerful test. Here adaptation means that the efficient score function of the test is estimated from the sample. A Fourier series estimator is used for the score function. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

On sequential density estimation

We consider the problem of sequential estimation of a density function f at a point x ₀ which may be known or unknown. Let T _n be a sequence of estimators of x ₀ . For two classes of density estimators f _n , namely the kernel estimates and a recursive modification of these, we show that if N(d) is a sequence of integer-valued random variables and n(d) a sequence of constants with N(d)/n(d) 1 in probability as d 0, then f _N(d) (T _N(d) -f(x ₀) is asymptotically normally distributed (when properly normed). We also propose two new classes of stopping rules based on the ideas of fixed-width interval estimation and show that for these rules, N(d)/n(d) 1 almost surely and EN(d)/n(d) 1 as d 0. One of the stopping rules is itself asymptotically normally distributed when properly normed and yields a confidence interval for f(x ₀) of fixed-width and prescribed coverage probability.