Two-sided Testing Problem Research Articles

Inference for subvectors and other functions of partially identified parameters in moment inequality models

This paper introduces a bootstrap-based inference method for functions of the parameter vector in a moment (in)equality model. These functions are restricted to be linear for two-sided testing problems, but may be non-linear for one-sided testing problems. In the most common case, this function selects a subvector of the parameter, such as a single component. The new inference method we propose controls asymptotic size uniformly over a large class of data distributions and improves upon the two existing methods that deliver uniform size control for this type of problem: projection-based and subsampling inference. Relative to projection-based procedures, our method presents three advantages: (i) it weakly dominates in terms of nite sample power, (ii) it strictly dominates in terms of asymptotic power, and (iii) it is typically less computationally demanding. Relative to subsampling, our method presents two advantages: (i) it strictly dominates in terms of asymptotic power (for reasonable choices of subsample size), and (ii) it appears to be less sensitive to the choice of its tuning parameter than subsampling is to the choice of subsample size.

On Variance Components in Semiparametric Mixed Models for Longitudinal Data

. First, to test the existence of random effects in semiparametric mixed models (SMMs) under only moment conditions on random effects and errors, we propose a very simple and easily implemented non-parametric test based on a difference between two estimators of the error variance. One test is consistent only under the null and the other can be so under both the null and alternatives. Instead of erroneously solving the non-standard two-sided testing problem, as in most papers in the literature, we solve it correctly and prove that the asymptotic distribution of our test statistic is standard normal. This avoids Monte Carlo approximations to obtain p-values, as is needed for many existing methods, and the test can detect local alternatives approaching the null at rates up to root n. Second, as the higher moments of the error are necessarily estimated because the standardizing constant involves these quantities, we propose a general method to conveniently estimate any moments of the error. Finally, a simulation study and a real data analysis are conducted to investigate the properties of our procedures.

Step-Up Tests for Comparing Several Exponential Populations with Control: Location Case

SYNOPTIC ABSTRACTLet π1…, πk be k treatments/ populations which are to be compared with a control treatment π0. We assume that an observation from population πi follows exponential distribution with probability density function (pdf) , where IA(.) is an indicator function of event A, θ is the common scale parameter and μi is the location parameter, (i = 0, 1, … k. In this paper, the problem of comparing several exponential populations/treatments with a control, in terms of their location parameters, is addressed by simultaneously testing k hypotheses on differences γi = πi, i = 1, …, k by using a step-up test procedure. A method for computing exact critical constants, which control the Type-I family-wise error rate at a preassigned level α, is discussed for one-sided and two-sided testing problems. The required constants are tabulated for specific values of the Type-I family-wise error rate and application of these constants to Pareto probability model is discussed.

Bayesian approach to estimation of accuracy in two-sided testing

For a two-sided hypothesis testing, it is known that there does not exist UMP tests. Uniformly most powerful unbiased (UMPU) tests and likelihood ratio test are the common approaches to two-sided testing. The p-values of these tests are usually used as evidence against null hypothesis. However, there are criticisms of p-values as a measure of evidence against null hypothesis for the two-sided testing problem in literature. Thus, in this paper, evidence measures derived from Bayesian approach are proposed to be the replacements of p-values and are investigated from both decision and testing perspectives. From decision theoretic framework, the proposed evidence measures can be demonstrated as admissible estimators, however, p-value of UMPU test are not admissible estimators; from testing aspect, the tests derived from p-values and the proposed evidence measures are shown to be UMPU tests. The Bayes estimator is better than p-value from theoretical aspect and it has the same merit as the p-value in testing point of view. Therefore, evidence measures derived from Bayesian approach are recommended in this paper.

Elements of Significance Testing with Equivalence Problems

The paper outlines an approach to the general methodological problem of equivalence assessment which is based on the classical theory of testing statistical hypotheses. Within this frame of reference it is natural to search for decision rules satisfying the same criteria of optimality which are customarily applied in deriving solutions to one- and two-sided testing problems. For three standard situations very frequently encountered in medical applications of statistics, a concise account of such an optimal test for equivalence is presented. It is pointed out that tests based on the well-known principle of confidence interval inclusion are valid in the sense of guaranteeing the prespecified level of significance, but tend to have an unnecessarily low efficiency.

A Note on Uniformly Most Powerful Two-Sided Tests

Abstract One of the standard examples (generally given in a first-year course on inference) of a testing situation in which there exists no uniformly most powerful test is that of the two-sided test of a normal mean. A fact, however, that is rarely mentioned about this example (also applicable to many other two-sided testing problems) is that the usual two-sided test is actually uniformly most powerful within the class of tests with critical regions that are symmetric about the null hypothesis (which is often the only class worth considering for the two-sided problem, in practice). Whereas this fact may be regarded as a straightforward application of the general theory of uniformly most powerful invariant tests; this particular point is not emphasized, nor is the fact generally mentioned in nonmathematical discussions. In this article, a simple, direct proof of this property of two-sided tests is given, thus offering the less theoretically oriented statistician a greater sense of security in the appropria...

On the second order local comparison between perturbed maximum likelihood estimators and Rao's statistic as test statistics

A bias-adjusted maximum likelihood estimator (mle), which has been shown to possess certain optimality criteria as an estimate of θ (see Ghosh, J. K., Sinha, B. K., and Joshi, S. N. (1982). In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, Eds.), Vol. 1, pp. 403–456. Academic Press, Orlando, FL) is compared with Rao's statistic as a test statistic for the standard two-sided testing problem. It is shown that Rao's statistic is locally superior to any bias-adjusted mle in the sense of Chandra and Joshi (1983, Sankhyā Ser. A 45 226–246). A second interpretation of a conjecture of C. R. Rao is proposed and Rao's statistic is shown to be superior as a test statistic according to the new interpretation as well. The last fact provides an interesting supplement to the results of Chandra and Joshi (1983, Sankhyā Ser. A 45 226–246). Furthermore, a partial reason for the inferiority of the likelihood ratio and Wald's statistic to Rao's statistic is supplied and certain regularity assumptions of the last paper are eliminated. Finally, the local powers of certain modified versions of Rao's and Wald's statistics (see Skovgaard, I. M. (1985). Ann. Statist. 13 534–551) are studied.

The Bahadur Efficiency of Tests of Some Joint Hypotheses

Abstract This article examines the relative efficiency of finite induced and Wald-like (or “infinite induced”) tests of some commonly encountered joint hypotheses. One two-sided testing problem and two types of two-sided testing problems concerning two parameters and normally distributed estimates are considered. The exact slopes of six test statistics are derived, and from these the Bahadur relative efficiency of tests based on any pair of the statistics can be easily computed. For the problems considered, Bahadur relative efficiency is equal to limiting Pitman efficiency, whereas Pitman relative efficiency coincides with exact finite sample relative efficiency. I present numerical results showing that Bahadur relative efficiency generally approximates Pitman efficiency rather poorly, although the patterns exhibited by these two efficiency criteria bear a reasonable similarity in most cases.

The Bahadur Efficiency of Tests of Some Joint Hypotheses

Abstract This article examines the relative efficiency of finite induced and Wald-like (or “infinite induced”) tests of some commonly encountered joint hypotheses. One two-sided testing problem and two types of two-sided testing problems concerning two parameters and normally distributed estimates are considered. The exact slopes of six test statistics are derived, and from these the Bahadur relative efficiency of tests based on any pair of the statistics can be easily computed. For the problems considered, Bahadur relative efficiency is equal to limiting Pitman efficiency, whereas Pitman relative efficiency coincides with exact finite sample relative efficiency. I present numerical results showing that Bahadur relative efficiency generally approximates Pitman efficiency rather poorly, although the patterns exhibited by these two efficiency criteria bear a reasonable similarity in most cases.

Asymptotic Behavior of Bayes Tests and Bayes Risk

In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.