Composite Hypothesis Testing Research Articles

Tests of Composite Hypotheses for the Multivariate Exponential Family

This paper is concerned with testing the hypothesis that the parameter in a multivariate exponential distribution lies in a linear subspace of the natural parameter space. Our main result characterizes a complete class of tests which is independent of the particular exponential distribution. This class is, in fact, complete relative to the stronger ordering among tests which compares conditional power, given a certain statistic, pointwise. The conclusion holds without any restriction on the exponential distribution. Many of the tests are admissible, but examples show that although the class is essentially the smallest class complete relative to all exponential distributions, it is not in general minimally complete. Some special cases where the class is minimally complete are discussed.

STUDY OF LINEARITY AND HOMOSCEDASTICITY OF TEST SCORES IN THE CHANCE RANGE

ETS Research Bulletin SeriesVolume 1966, Issue 2 p. 1-16 ArticleFree Access STUDY OF LINEARITY AND HOMOSCEDASTICITY OF TEST SCORES IN THE CHANCE RANGE Robert F. Boldt, Robert F. Boldt Statistical Analysis Division, ETSSearch for more papers by this author Robert F. Boldt, Robert F. Boldt Statistical Analysis Division, ETSSearch for more papers by this author First published: December 1966 https://doi.org/10.1002/j.2333-8504.1966.tb00980.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat REFERENCES Angoff, W. H. The validity of negative scores in the scholastic aptitude test. Unpublished memorandum, 1964. Cliff, Rosemary. The predictive value of chance level scores. Educational and Psychological Measurement, 1958, 18, 607– 616. Levine, R. D., & Lord, F. M. An index of the discriminating powers of a test at different parts of the score range. Educational and Psychological Measurement, 1959, 19, 497– 500. Lord, F. M. An empirical study of item-test regression, Psychometrika, 1965, 30, 373– 376. Mood, A. M. Introduction to the theory of statistics. New York: McGraw-Hill, 1950. Smith, E. S. Binomial, normal, and Poisson probabilities. Bel Air, Md.: Author, 1953. Snedecor, G. W. Statistical methods. Ames, Iowa: Iowa State College Press, 1956. Wilks, S. S. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 1938, 9, 60– 62. Volume1966, Issue2December 1966Pages 1-16 ReferencesRelatedInformation

On optimal asymptotic tests of composite hypotheses with several constraints

A locally asymptotically most powerful test for composite hypotheses with t independent linear constraints on the parameters has been developed for the case where the observed random variables {X nk , k=1,2,...,n} are independently but not necessarily identically distributed. However, their distributions depend on two vector parameters, one ξ = (ξ 1, ξ 2, ..., ξ t) being under test, and the other θ = (θ 1, θ 2, ..., θ s) being the nuisance parameter.

Asymptotically Optimal Tests of Composite Hypotheses for Randomized Experiments with Noncontrolled Predictor Variables

Abstract The paper is concerned with randomized experiments with one treatment. Two randomization schemes are considered: randomized pairs and unrestricted randomization. If effective at all, the treatment is supposed to affect the conditional distribution of the “experimental” variable Y given another variable X, called “predictor”. The distribution of X is not affected by the treatment. Using the general theory published elsewhere, the paper deduces the locally asymptotically optimal test of the hypothesis that the treatment has no effect. Apart from the usual difficulties connected with asymptotic tests (how large must N be?), the theory is easily applicable in many “live” cases even though the conditional distribution of Y given X may contain nuisance parameters and be of unusual form.

Asymptotically Optimal Tests of Composite Hypotheses for Randomized Experiments with Noncontrolled Predictor Variables

Asymptotically Optimal Tests of Composite Hypotheses for Randomized Experiments with Noncontrolled Predictor Variables

Problems of Statistical Inference for Birth and Death Queuing Models

A large sample theory for birth and death queuing processes which are ergodic and metrically transitive is applied to make inferences about arrival and service rates. Likelihood ratio tests and maximum likelihood estimators are derived for simple models. Composite hypotheses, for example that the arrival rate does not vary with the number in the system, are considered. A numerical example illustrating these results, based on data generated by simulating a known model, is presented, which includes the testing of both true and false composite hypotheses.

Asymptotically Optimal Tests for Multinomial Distributions

Tests of simple and composite hypothesis for multinomial distributions are considered. It is assumed that the size $\alpha_N$ of a test tends to 0 as the sample size $N$ increases. The main concern of this paper is to substantiate the following proposition: If a given test of size $\alpha_N$ is "sufficiently different" from a likelihood ratio test then there is a likelihood ratio test of size $\leqq\alpha_N$ which is considerably more powerful than the given test at "most" points in the set of alternatives when $N$ is large enough, provided that $\alpha_N \rightarrow 0$ at a suitable rate. In particular, it is shown that chi-square tests of simple and of some composite hypotheses are inferior, in the sense described, to the corresponding likelihood ratio tests. Certain Bayes tests are shown to share the above-mentioned property of a likelihood ratio test.

On the Computation of the Likelihood Ratio for Gaussian Processes with a Rational Spectrum

Previous article Next article On the Computation of the Likelihood Ratio for Gaussian Processes with a Rational SpectrumV. F. PisarenkoV. F. Pisarenkohttps://doi.org/10.1137/1110034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. F. Pisarenko and , Yu. A. Rozanov, On certain problems for stationary processes leading to integral equations related to Wiener-Hopf equations, Problemy Peredači Informacii No., 14 (1963), 113–135, (In Russian.) MR0192564 Google Scholar[2] V. F. Pisarenko, A Wiener-Hopf type equation for higher-dimensional random processes with a rational spectral density, Dokl. Akad. Nauk SSSR, 149 (1963), 776–779, (In Russian.) MR0153056 0131.16305 Google Scholar[3] V. F. Pisarenko, Masters Thesis, Some statistical problems for multi-dimensional processes with a rational spectrum, Candidate's theses, Vil'nyus, 1963, (In Russian.) Google Scholar[4] Jaroslav Hájek, On linear statistical problems in stochastic processes, Czechoslovak Math. J., 12 (87) (1962), 404–444 MR0152090 0114.34504 Google Scholar[5] I. M. Gel'fand and , A. M. Yaglom, Computation of the amount of information about a stochastic function contained in another such function, Uspehi Mat. Nauk (N.S.), 12 (1957), 3–52, (In Russian.) MR0085153 Google Scholar[6] Walter F. Freiberger, An approximate method in signal detection, Quart. Appl. Math., 20 (1962/1963), 373–378 MR0139498 0111.14404 CrossrefGoogle Scholar[7] Ulf Grenander and , Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958vii+245 MR0094840 0080.09501 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Statistical inference for stationary linear models with tapered dataStatistics Surveys, Vol. 15, No. none Cross Ref Absolute Continuity29 September 2014 Cross Ref Absolute Continuity15 August 2006 Cross Ref On Sufficient Statistics of Gaussian Processes with Rational Spectral Density FunctionM. Arató28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (900 KB)On Parameter Estimation in the Presence of NoiseM. Arató17 July 2006 | Theory of Probability & Its Applications, Vol. 29, No. 3AbstractPDF (638 KB)Remarks on the Approximation of the Likelihood Function of a Stationary Gaussian ProcessJ. Coursol and D. Dacunha-Castelle17 July 2006 | Theory of Probability & Its Applications, Vol. 27, No. 1AbstractPDF (428 KB)Tests of Composite Hypotheses for Random Variables and Stochastic ProcessesK. O. Dzhaparidze17 July 2006 | Theory of Probability & Its Applications, Vol. 22, No. 1AbstractPDF (1375 KB)On Evaluation of the Likelihood Ratio for a Generalized Gaussian Process with Rational Spectral DensityK. O. Dzaparidze and G. I. Marr28 July 2006 | Theory of Probability & Its Applications, Vol. 19, No. 2AbstractPDF (357 KB)On the Estimation of the Spectral Parameters of a Gaussian Stationary Process with Rational Spectral DensityK. O. Dzaparidze17 July 2006 | Theory of Probability & Its Applications, Vol. 15, No. 3AbstractPDF (827 KB)Radon-Nikodym derivatives in case of rational spectral densities Cross Ref On the Densities of Gaussian Distributions and Wiener-Hopf Integral EquationsYu. A. Rozanov17 July 2006 | Theory of Probability & Its Applications, Vol. 11, No. 1AbstractPDF (858 KB) Volume 10, Issue 2| 1965Theory of Probability & Its Applications History Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1110034Article page range:pp. 299-303ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Asymptotically Optimum Sequential Inference and Design

In recent years the study of sequential procedures which are asymptotically optimum in an appropriate sense as the cost $c$ per observation goes to zero has received considerable attention. On the one hand, Schwarz (1962) has recently given an interesting theory of the asymptotic shape, as $c \rightarrow 0$, of the Bayes stopping region relative to an a priori distribution $F$, for testing sequentially between two composite hypotheses $\theta \leqq \theta_1$ and $\theta \geqq \theta_2$ concerning the real parameter $\theta$ of a distribution of exponential (Koopman-Darmois) type, with indifference region the open interval $(\theta_1, \theta_2)$. (An example of Schwarz's considerations is described in connection with Figure 4.) One aim of the present paper is to generalize Schwarz's results to the case where (with or without indifference regions) the distributions have arbitrary form and there can be more than two decisions (Sections 2, 3, 4). In this general setting we obtain, under mild assumptions, a family $\{\delta_c\}$ of procedures whose integrated risk is asymptotically the same as the Bayes risk. (In fact, extending Schwarz's result, a family $\{\delta'_c\}$ can be constructed so as to possess this asymptotic Bayes property relative to all a priori distributions with the same support as $F$, or even with smaller indifference region support than $F$.) Procedures like our $\{\delta_c\}$ have already been suggested by Wald (1947) for use in tests of composite hypotheses (e.g., the sequential $t$-test), but his concern was differently inspired. At the same time, we show how such multiple decision problems can be treated by using simultaneously a number of sequential tests for associated two-decision problems. A second aim is to extend, strengthen, and somewhat simplify the asymptotic sequential considerations originated by Chernoff (1959) and further developed by Albert (1961) and Bessler (1960) (Section 5). Our point of departure here is a device utilized by Wald (1951) in a simpler estimation setting, and which in the present setting amounts to taking a preliminary sample with predesignated choice of designs and such that, as $c \rightarrow 0$, the size of this preliminary sample tends to infinity, while its ratio to the total expected sample size tends to zero. The preliminary sample can then be used to guess the true state of nature and thus to choose the future pattern once and for all rather than to have to reexamine the choice of after subsequent observations. (In Wald's setting the only design problem was to pick the size of the second sample of his two-sample procedure.) The properties of the resulting procedure can then be inferred from the considerations of Sections 2, 3, and 4, where there is no problem but where most of the work in this paper is done; using Wald's idea, we thereby obtain procedures for the problem fairly easily, once we have the (non-design) sequential inference structure to build upon. The family $\{\delta^\ast_c\}$ so obtained has the same asymptotic Bayes property as that described above for the family $\{\delta_c\}$ of the non-design problem. Furthermore, a family $\{\delta^{\ast\ast}_c\}$ can be constructed so that, like $\{\delta'_c\}$ in the non-design problem, it is asymptotically Bayes for all a priori distributions with the same support. The value of the asymptotic Bayes risk of such a family is closely related to the lower bound which was obtained by Chernoff et al for the risk function of certain procedures, and which gives another form for the optimality statement. The role of the sequential procedures considered by Donnelly (1957) and Anderson (1960) for hypothesis testing with an indifference region is indicated at the end of Section 1. Asymptotic solutions to the problem of Kiefer and Weiss (1957) are given. An Appendix contains proofs of certain results on fluctuations of partial sums of independent random variables, which are used in the body of the paper.

Inference in Stochastic Processes. I

In this paper we consider a number of questions regarding the existence of various optimal criteria for testing composite hypotheses on probability measures corresponding to random processes of one or another class.

The Use of Least Favorable Distributions in Testing Composite Hypotheses

The usual method of finding a most powerful size $\alpha$ test of a composite hypothesis against a simple alternative is the guessing of a Least Favorable Distribution (LFD)--introduced at various levels of generality by Neyman and Pearson [6], Wald [7], Lehmann [4], and Lehmann and Stein [5]--and testing the mixture of the distributions of the hypothesis over this LFD against the alternate using the Neyman-Pearson Fundamental Lemma. In guessing LFD's statisticians have looked for a mixture which is like the alternate. In this paper, the notion of Uniformly Least Favorable Mixture (ULFM) is introduced. In Section 2, we show that a ULFM is a point in the convex set of mixtures of the hypothesis which is closest (in the sense of the $\mathfrak{L}^1$ norm) to the alternate. The condition is not sufficient. More generally, any LFM corresponds to a point which is closest to the alternate in some expansion or contraction of this set of mixtures. A sufficient condition for ULFM's is, essentially, that the nuisance parameter can take on the same values in the alternate as in the hypothesis. In Section 3, we consider the case where no ULFM exists. We show, inter alia, that any distribution is least favorable for a closed set of $\alpha$'s. (A pathological example shows that this closed set need not be the union of a finite number of closed intervals.)

Approximation to the Distribution of the Sample Size for Sequential Tests.: II. Tests of Composite Hypotheses

Approximation to the Distribution of the Sample Size for Sequential Tests.: II. Tests of Composite Hypotheses

Approximation to the distribution of the sample size for sequential tests: II. Tests of composite hypotheses

Approximation to the distribution of the sample size for sequential tests: II. Tests of composite hypotheses

Significance Level and Power

Significance testing, as described in most textbooks, consists in fixing a standard significance level $\alpha$ such as .01 or .05 and rejecting the hypothesis $\theta = \theta_0$ if a suitable statistic $Y$ exceeds $C$ where $P_{\theta_0}\{Y > C\} = \alpha$. Such a procedure controls the probability of false rejection (error of the first kind) at the desired level $\alpha$ but leaves the power of the test and hence the probability of an error of the second kind to the mercy of the experiment. It seems more natural when deciding on a significance level (and this suggestion is certainly not new) to take into account also what power can be achieved with the given experiment. In Section 3 a specific suggestion will be made as to how to balance $\alpha$ against the power $\beta$ obtainable against the alternatives of interest. The adoption of this or some similar rule for choosing a significance level has important consequences for the theory of testing composite hypotheses, where nuisance parameters are present. Since the quantity $\alpha$ is then potentially a function of the nuisance parameter $\vartheta$, the classical rule of a fixed significance level leads to the condition that the tests be exact or similar, that is, that $\alpha(\vartheta)$ equal the preassigned value $\alpha$ for all $\vartheta$. On the other hand, the power $\beta$ that can be attained against any alternative $\theta = \theta_1$ frequently depends on $\vartheta$. The requirement that $\alpha(\vartheta)$ and $\beta(\vartheta)$ be in a certain balance thus leads to tests which are not similar and hence do not agree with the standard solutions. To obtain a suitable setting for this discussion, we consider first a minimal complete class of tests for testing the hypothesis $H:\theta \leqq \theta_0$ in a multiparameter exponential family (Section 2). The proposed $\alpha, \beta$-relation is discussed in Section 3, and in Section 4 is applied to the exponential family. Section 5 gives some illustrations of the theory.

Characterizations of Complete Classes of Tests of Some Multiparametric Hypotheses, with Applications to Likelihood Ratio Tests

For the problem of testing a simple hypothesis on a density function of the form $f_\theta(e) = \exp \{\psi_0(\theta) + \sum^k_1 \psi_i(\theta)t_i(e) + t_0(e)\}$, explicit characterizations are given of a minimal essentially complete class of tests, the minimal complete class, and the closure of the class of Bayes' solutions, under certain assumptions. Applications are made to discrete distributions of the above form and to some problems of testing composite hypotheses. The likelihood ratio tests of these hypotheses are characterized and shown to be admissible under certain assumptions.

On a property of the sequentialt-test

Abstract 1. Introduction. In one of his papers [1], and later in his book on sequential analysis [2], Wald introduced a general method for constructing sequential tests of composite hypotheses and applied the method to construct a sequential t-test. Since Wald devoted a considerable amount of space and mathematics on his t-test, it has been taken for granted that he proved certain optimum properties of the test. It is the purpose of this note to show that the test cannot possess one of the properties thought to hold for it.

A Class of Minimax Tests for One-sided Composite Hypotheses

Fixed sample-size procedures are considered for testing a one-sided composite hypothesis concerning a real, one-dimensional parameter of an exponential distribution (1.1). In particular, conditions are studied such that the minimax tests have a critical region which is a semi-infinite interval on the real line.

Sequential tests for composite hypotheses

AbstractA method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.

A Property of Some Tests of Composite Hypotheses

A Property of Some Tests of Composite Hypotheses

On Optimum Tests of Composite Hypotheses with One Constraint

This paper is concerned with optimum tests of certain composite hypotheses. In section 2 various aspects of a theorem of Scheffe concerning type $B_1$ tests are discussed. It is pointed out that the theorem can be extended to cover uniformly most powerful tests against a one-sided set of alternatives. It is also shown that the method for determining explicitly the optimum test region may in certain cases be reduced to a simple formal procedure. These results are used in section 3 to obtain optimum tests for the composite hypothesis specifying the value of the circular serial correlation coefficient in a normal distribution. A surprising feature of this example is the fact that for the simple hypothesis obtained by specifying values for the nuisance parameters no test with the corresponding optimum properties exists. In section 4 the totality of similar regions is obtained for a large class of probability laws which admit a sufficient statistic. Some composite hypotheses concerning exponential and rectangular distributions are treated in section 5. It is proved that the likelihood ratio tests of these hypotheses have various optimum properties.