Arthur Cohen

Two stage conditionally unbiased estimators of the selected mean

The problem is to estimate the mean of the selected population. The selection rule is to choose the population with the largest sample mean when such sample means are calculated from the first stage sample. An estimator of the selected mean is unbiased if its expected value equals the expected value of the selected mean. We seek conditionally unbiased estimators of the selected mean given the ordering of the set of sample means based on the first stage sample. Conditionally unbiased estimators are of course unconditionally unbiased. For several distributions such as the normal, with unknown mean, and binomial, no conditionally unbiased estimators exist based on a one stage sample. We propose a two stage sample where observations at stage two are taken from the selected population only. Such a procedure has the advantage of yielding conditionally unbiased estimators and enables, possibly a better allocation of available sample points. We find the uniformly minimum variance conditionally unbiased estimators (UMVCUE) for the normal case when the variance is known or when a common unknown variance is present. We also find the UMVCUE for the gamma case and indicate that the method is suitable for many other cases as well.

ESTIMATION OF THE LARGER OF TWO NORMAL MEANS

Let Xi1, Xi2, …, Xin, i = 1, 2, be a pair of random samples from populations which are normally distributed with means θi, and common known variance τ2. The problem is to estimate the function φ(θ1, θ2) = maximum (θ1, θ2). In this paper we consider five different estimators (or sets of estimators) for φ(θ1, θ2) and evaluate their biases and mean square errors. The estimators are (i) φ(X1, X2), where Xi is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of φ(θ1, θ2) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X1 + X2)/2 when |X1 – X2| is small, and estimate by φ(X1, X2) when |X1 − X2| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, ...

Asymptotically Optimal Methods of Combining Tests

Abstract A criterion for the asymptotic optimality of combined independent tests is given. Methods other than Fishers are shown to satisfy the criterion. Certain problems call for a weighted combination of tests. It is shown that many weighted procedures are optimal. In particular, a weighted procedure based on gamma transforms is optimal. This weighted procedure has the additional advantage that critical values are easily determined. Goods weighting procedure is shown not to be asymptotically optimal.

Improved Confidence Intervals for the Variance of a Normal Distribution

Abstract In considering the problem of confidence estimation for the variance of a normal distribution, Tate and Klett [5] raise two questions. “Does the interval of shortest length based on (, S2) depend only on ” (S2 denotes the sum of squared deviations.) “Among those intervals based only on S2 is the interval of shortest length necessarily of the form ” The answer to the first question is no; to the second, yes. The results are given for the general linear hypothesis model of rank p. The amount of improvement of new intervals is discussed.

Decision theory results for one-sided multiple comparison procedures

A resurgence of interest in multiple hypothesis testing has occurred in the last decade. Motivated by studies in genomics, microarrays, DNA sequencing, drug screening, clinical trials, bioassays, education and psychology, statisticians have been devoting considerable research energy in an effort to properly analyze multiple endpoint data. In response to new applications, new criteria and new methodology, many ad hoc procedures have emerged. The classical requirement has been to use procedures which control the strong familywise error rate (FWE) at some predetermined level a. That is, the probability of any false rejection of a true null hypothesis should be less than or equal to a. Finding desirable and powerful multiple test procedures is difficult under this requirement. One of the more recent ideas is concerned with controlling the false discovery rate (FDR), that is, the expected proportion of rejected hypotheses which are, in fact, true. Many multiple test procedures do control the FDR. A much earlier approach to multiple testing was formulated by Lehmann [Ann. Math. Statist. 23 (1952) 541-552 and 28 (1957) 1-25]. Lehmanns approach is decision theoretic and he treats the multiple endpoints problem as a 2 k finite action problem when there are k endpoints. This approach is appealing since unlike the FWE and FDR criteria, the finite action approach pays attention to false acceptances as well as false rejections. In this paper we view the multiple endpoints problem as a 2 k finite action problem. We study the popular procedures single-step, step-down and step-up from the point of view of admissibility, Bayes and limit of Bayes properties. For our model, which is a prototypical one, and our loss function, we are able to demonstrate the following results under some fairly general conditions to be specified: (i) The single-step procedure is admissible. (ii) A sequence of prior distributions is given for which the step-down procedure is a limit of a sequence of Bayes procedures. (iii) For a vector risk function, where each component is the risk for an individual testing problem, various admissibility and inadmissibility results are obtained. In a companion paper [Cohen and Sackrowitz, Ann. Statist. 33 (2005) 145-158], we are able to give a characterization of Bayes procedures and their limits. The characterization yields a complete class and the additional useful result that the step-up procedure is inadmissible. The inadmissibility of step-up is demonstrated there for a more stringent loss function. Additional decision theoretic type results are also obtained in this paper.

ESTIMATING ORDERED LOCATION AND SCALE PARAMETERS

Assume independent random samples are drawn from K populations whose distributions are location, scale, or location-scale famil ies. Let T^ be an estimator which is admissible for the parameter corresponding to the f i r s t population. Next assume that the parameters are ordered. The question addressed is does Tĵ remain admissible? For various special cases of the model we exhibit estimators which are better than Tp The models include estimating a location parameter with squared error loss, estimating a scale parameter with normalized squared error loss, finding a confidence interval for a location parameter, and finding a confidence interval for a normal variance. In these latter models universal domination as defined by Hwang [4] is shown. *) Research supported by NSF Grant DMS-84-18416 AMS 1980 Subject classifications: Primary 62 G 05, 62 F 10 Secondary 62 C 15

Testing hypotheses about the common mean of normal distributions

Abstract An overview of hypothesis testing for the common mean of independent normal distributions is given. The case of two populations is studied in detail. A number of different types of tests are studied. Among them are a test based on the maximum of the two available t -tests, Fishers combined test, a test based on Graybill–Deals estimator, an approximation to the likelihood ratio test, and some tests derived using some Bayesian considerations for improper priors along with intuitive considerations. Based on some theoretical findings and mostly based on a Monte Carlo study the conclusions are that for the most part the Bayes-intuitive type tests are superior and can be recommended. When the variances of the populations are close the approximate likelihood ratio test does best.

Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure

The problem of multiple endpoint testing for k endpoints is treated as a 2 k finite action problem. The loss function chosen is a vector loss function consisting of two components. The two components lead to a vector risk. One component of the vector risk is the false rejection rate (FRR), that is, the expected number of false rejections. The other component is the false acceptance rate (FAR), that is, the expected number of acceptances for which the corresponding null hypothesis is false. This loss function is more stringent than the positive linear combination loss function of Lehmann [Ann. Math. Statist. 28 (1957) 1-25] and Cohen and Sackrowitz [Ann. Statist. (2005) 33 126-144] in the sense that the class of admissible rules is larger for this vector risk formulation than for the linear combination risk function. In other words, fewer procedures are inadmissible for the vector risk formulation. The statistical model assumed is that the vector of variables Z is multivariate normal with mean vector μ and known intraclass covariance matrix E. The endpoint hypotheses are H i : μ i = 0 vs K i : μ i > 0, ι = 1,..., k. A characterization of all symmetric Bayes procedures and their limits is obtained. The characterization leads to a complete class theorem. The complete class theorem is used to provide a useful necessary condition for admissibility of a procedure. The main result is that the step-up multiple endpoint procedure is shown to be inadmissible.

Effective directed tests for models with ordered categorical data

Summary This paper offers a new method for testing one-sided hypotheses in discrete multivariate data models. One-sided alternatives mean that there are restrictions on the multidimensional parameter space. The focus is on models dealing with ordered categorical data. In particular, applications are concerned with R×C contingency tables. The method has advantages over other general approaches. All tests are exact in the sense that no large sample theory or large sample distribution theory is required. Testing is unconditional although its execution is done conditionally, section by section, where a section is determined by marginal totals. This eliminates any potential nuisance parameter issues. The power of the tests is more robust than the power of the typical linear tests often recommended. Furthermore, computer programs are available to carry out the tests efficiently regardless of the sample sizes or the order of the contingency tables. Both censored data and uncensored data models are discussed.

HYPOTHESIS TESTS AND OPTIMALITY PROPERTIES IN DISCRETE MULTIVARIATE ANALYSIS

Publisher Summary This chapter presents the hypothesis tests and optimality properties in discrete multivariate analysis. The level of the LRT test is obtained as a limit when one parameter is set equal to zero and all other parameters tend to +∞. Linear combinations of the ωs can be tested provided the coefficient vector of the linear combination contains both positive and negative elements. The chapter describes admissibility of tests for Poisson sampling. A chi-square test for independence in an r × s contingency table with multinomial sampling is admissible as it has convex acceptance sections. The chapter also discusses the admissibility of likelihood ratio, Pearson chi-square, and other tests.