Some locally optimal subset selection rules for comparison with a control

Let π₀,π₁,…,π_k be k+1 independent populations. For i=0,1,…,k,π_i has the densit f(x,θ_i), where the (unknown) parameter θ_i belongs to an interval of the real line. Our goal is to select from π₁,… π_k (experimental treatments) those populations, if any, that are better (suitably defined) than π₀ which is the control population. A locally optimal rule is derived in the class of rules for which Pr(π_i is selected)γ_i, i=1,…,k, when θ₀=θ₁=?=θ_k. The criterion used for local optimality amounts to maximizing the efficiency in a certain sense of the rule in picking out the superior populations for specific configurations of θ=(θ₀,…,θ_k) in a neighborhood of an equiparameter configuration. The general result is then applied to the following special cases: (a) normal means comparison — common known variance, (b) normal means comparison — common unknown variance, (c) gamma scale parameters comparison — known (unequal) shape parameters, and (d) comparison of regression slopes. In all these cases, the rule is obtained based on samples of unequal sizes.     

Minimax subset selection for the multinomial distribution

Let (X₁,…,X_k) be a multinomial vector with unknown cell probabilities (p₁,?,p_k). A subset of the cells is to be selected in a way so that the cell associated with the smallest cell probability is included in the selected subset with a preassigned probability, P¹. Suppose the loss is measured by the size of the selected subset, S. Using linear programming techniques, selection rules can be constructed which are minimax with respect to S in the class of rules which satisfy the P¹-condition. In some situations, the rule constructed by this method is the rule proposed by Nagel (1970). Similar techniques also work for selection in terms of the largest cell probability.     

Allocating observations between two normal populations to maximize the half slope of the F-test

Consider two independent normal populations. Let R denote the ratio of the variances. The usual procedure for testing H₀: R = 1 vs. H₁: R = r, where r≠1, is the F-test. Let θ denote the proportion of observations to be allocated to the first population. Here we find the value of θ that maximizes the rate at which the observed significance level of the F-test converges to zero under H₁, as measured by the half slope.     

Subset selection of superior populations when the number of populations is large

The situation where k populations are partitioned into one inferior group and one superior group is considered. The statistical problem is to select a random size subset of superior populations while trying to avoid including any inferior populations. A selection procedure is assumed to satisfy the condition that the probability of selecting at least one superior population is bounded below by P¹<1. The performance of a procedure is measured by the probability of including an inferior population.The asymptotic performance, as k→∞ of Gupta's traditional maximum type procedure ψ^G is considered in the location-model. For normally distributed populations, ψ^G turns out to be asymptotically optimal, provided the size of the inferior group does not become infinitely larger than the size of the superior group.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

Goodness-of-fit tests and minimum distance estimation via optimal transformation to uniformity

A unified approach of parameter-estimation and goodness-of-fit testing is proposed. The new procedures may be applied to arbitrary laws with continuous distribution function. Specifically, both the method of estimation and the goodness-of-fit test are based on the idea of optimally transforming the original data to the uniform distribution, the criterion of optimality being an L2-type distance between the empirical characteristic function of the transformed data, and the characteristic function of the uniform (0,1)$(0,1)$ distribution. Theoretical properties of the new estimators and tests are studied and some connections with classical statistics, moment-based procedures and non-parametric methods are investigated. Comparison with standard procedures via Monte Carlo is also included, along with a real-data application.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

Confidence regions for the common mean vector of several multivariate normal populations

Suppose that there are independent samples available from several multivariate normal populations with the same mean vector m? but possibly different covariance matrices. The problem of developing a confidence region for the common mean vector based on all the samples is considered. An exact confidence region centered at a generalized version of the well-known Graybill-Deal estimator of m? is developed, and a multiple comparison procedure based on this confidence region is outlined. Necessary percentile points for constructing the confidence region are given for the two-sample case. For more than two samples, a convenient method of approximating the percentile points is suggested. Also, a numerical example is presented to illustrate the methods. Further, for the bivariate case, the proposed confidence region and the ones based on individual samples are compared numerically with respect to their expected areas. The numerical results indicate that the new confidence region is preferable to the single-sample versions for practical use.     

On a multiple endpoints testing problem

In a clinical trial comparing drug with placebo, where there are multiple primary endpoints, we consider testing problems where an efficacious drug effect can be claimed only if statistical significance is demonstrated at the nominal level for all endpoints. Under the assumption that the data are multivariate normal, the multiple endpoint-testing problem is formulated. The usual testing procedure involves testing each endpoint separately at the same significance level using two-sample t-tests, and claiming drug efficacy only if each t-statistic is significant. In this paper we investigate properties of this procedure. We show that it is identical to both an intersection union test and the likelihood ratio test. A simple expression for the p-value is given. The level and power function are studied; it is shown that the test may be conservative and that it is biased. Computable bounds for the power function are established.     

Screening for Multiple Target Detection

We apply statistical selection theory to multiple target detection problems by analyzing the Mahalanobis distances between multivariate normal populations and a desired standard (a known characteristic of a target). We want to achieve the goal of selecting a subset that contains no non target (negative) sites, which entails screening out all non targets. Correct selection (CS) is defined according to this goal. We consider two cases: (1) that all covariance matrices are known; and (2) that all covariance matrices are unknown, including both heteroscedastic and homoscedastic cases. Optimal selection procedures are proposed in order to reach the selection goal. The least favorable configurations (LFC) are found. Tables and figures are presented to illustrate the properties of our proposed procedures. Simulation examples are given to show that our procedures work well. The log-concavity results of the operating characteristic functions are also given.     

A note on successive comparisons between several ordered variances

ABSTRACT

In this article, a procedure for comparisons between k (k ? 3) successive populations with respect to the variance is proposed when it is reasonable to assume that variances satisfy simple ordering. Critical constants required for the implementation of the proposed procedure are computed numerically and selected values of the computed critical constants are tabulated. The proposed procedure for normal distribution is extended for making comparisons between successive exponential populations with respect to scale parameter. A comparison between the proposed procedure and its existing competitor procedures is carried out, using Monte Carlo simulation. Finally, a numerical example is given to illustrate the proposed procedure.     

An essentially complete class of multiple decision procedures

Let π₁,…, π_k represent k(?2) independent populations. The quality of the ith population π_i is characterized by a real-valued parameter θ_i, usually unknown. We define the best population in terms of a measure of separation between θ_i's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.     

Empirical bayes rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

Combined Bayesian estimates for the equicorrelation coefficient

Combined Bayesian estimates for equicorrelation covariance matrices are considered. The case of a common equicorrelation p and possibly different standard deviations σ_l,σ_k among k experimental groups is examined first, and the Bayesian estimation of (σ, σ₁,σ_k) is discussed. Secondly, under the assumption of a common standard deviation and possibly different equicorrelations, the Bayesian estimation of (ρ₁,ρ_k,σ) is considered.     

On the estimation of restricted mean

In sampling from a continuous distribution with unknown mean μ and variance σ² the problem of estimation of μ, when it is known that μ∈(a, ∞) (or μ∈(-∞, b)), is considered. The estimators proposed here lie in the interval (a, ∞) (or (-∞, b)) almost surely. The performance of these estimators is compared to that of some known estimators in the case of sampling from a normal, exponential and a weighted difference of two independent chi-square distributions.     

Saturated designs for multivariate cubic regression

This paper deals with the problem of finding saturated designs for multivariate cubic regression on a cube which are nearly D-optimal. A finite class of designs is presented for the k dimensional cube having the property that the sequence of the best designs in this class for each k is asymptotically efficient as k increases. A method for constructing good designs in this class is discussed and the construction is carried out for 1?k?8. These numerical results are presented in the last section of the paper.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

A p-value for testing the equivalence of the variances of a bivariate normal distribution

A p-value is developed for testing the equivalence of the variances of a bivariate normal distribution. The unknown correlation coefficient is a nuisance parameter in the problem. If the correlation is known, the proposed p-value provides an exact test. For large samples, the p-value can be computed by replacing the unknown correlation by the sample correlation, and the resulting test is quite satisfactory. For small samples, it is proposed to compute the p-value by replacing the unknown correlation by a scalar multiple of the sample correlation. However, a single scalar is not satisfactory, and it is proposed to use different scalars depending on the magnitude of the sample correlation coefficient. In order to implement this approach, tables are obtained providing sub-intervals for the sample correlation coefficient, and the scalars to be used if the sample correlation coefficient belongs to a particular sub-interval. Once such tables are available, the proposed p-value is quite easy to compute since it has an explicit analytic expression. Numerical results on the type I error probability and power are reported on the performance of such a test, and the proposed p-value test is also compared to another test based on a rejection region. The results are illustrated with two examples: an example dealing with the comparability of two measuring devices, and an example dealing with the assessment of bioequivalence.