On optimal asymptotic tests of composite hypotheses with several constraints

Summary 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 = ( ₁, ₂, ..., _t) being under test, and the other = ( ₁, ₂, ..., _s) being the nuisance parameter.This investigation was supported (in part) by a research grant (No. GM-10525(2)) from the National Institutes of Health, Public Health Service.On leave from UniversitÄt Heidelberg, Germany, and supported by a NATO research scholarship.     

Power comparisons of tests of two multivariate hypotheses based on individual characteristic roots

Summary In this paper, power comparisons are made for tests of each of the following two hypotheses based on individual characteristic roots of a matrix arising in each case: (i) independence between ap-set and aq-set of variates in a (p+q)-variate normal population withp≦q and (ii) equality ofp-dimensional mean vectors ofl p-variate normal populations having a common covariance matrix. At first, a few lemmas are given which help to reduce the central distributions of the largest, smallest, second largest, and the second smallest roots in terms of incomplete beta functions or functions of them. Since the central distribution of the largest root has been discussed by Pillai earlier in several papers ([6], [8], [9], [11], [12], [13]) cdf’s of the three others in the central case are given. Further, the non-central distributions of the individual roots forp-3 are considered for the two hypotheses and that of the smaller root forp=2; that of the largest root forp=2 has been obtained by Pillai earlier, (Pillai [11], Pillai and Jayachandran [14]). The work of this author was supported by the National Science Foundation Grant No. GP-4600.     

Distribution-free tests for no effect of treatment in heteroscedastic functional data under both weak and long range dependence

In this paper, we present distribution-free tests to evaluate the effect of multiple treatments when there are a large number of repeated measurements from each subject nested in a treatment. We formulate new test statistics to account for heteroscedasticity and unbalanced designs. The asymptotic distributions for the test statistics are obtained when the repeated measurements from the same subject have long range dependence and weak dependence, respectively. The asymptotic results hold under the nonclassical setting in which the number of repeated measurements is large while the number of subjects per treatment may be small. A real application to compare cattle ear temperature profiles under different antibiotic treatments is given for illustration. Simulation studies are undertaken to compare the empirical performance of the proposed tests to commonly used methods.     

On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator

We consider the problem of the construction of the goodness-of-fit test in the case of continuous time observations of a diffusion process with small noise. The null hypothesis is parametric and we use a minimum distance estimator of the unknown parameter. We propose an asymptotically distribution free test for this model.     

A Monte Carlo comparison of studentized bootstrap and permutation tests for heteroscedastic two-sample problems

The present Monte Carlo study compares bootstrap and permutation tests for semiparametric heteroscedastic two-sample testing problems of Behrens-Fisher type. The underlying functionals to be tested are (a) the difference of the means and (b) the Wilcoxon functionalP(Y < X) which is invariant under strictly increasing transformations. The consideration leads to semiparametric modifications of Welch type tests for the Behrens-Fisher model and an extended two-sample Wilcoxon test which also works under some null hypothesis with non-exchangeable distributions. The present Monte Carlo study confirms the high quality of studentized permutation tests at finite sample size. They are typically better than tests with asymptotic critical values and for many situations and they are also better than two-sample bootstrap tests when their type I error probabilities are compared.     

On the asymptotic theory of rank order tests for experiments involving paired comparisons

Summary The problem of testing the hypothesis of no difference among several treatments for the case when the comparisons between the treatments is possible only in pairs has been considered by Durbin [3], Bradley and Terry [1], Elteren and Noether [4], and Mehra and Puri [6], among others. Following the lines of Sen and Puri [9], a new approach to the asymptotic theory of rank order tests for this problem is developed. This avoids the unnecessarily complicated and lengthy conditional approach of Mehra and Puri [6] and also simplifies the proofs considerably. Prepared under contract Nonr-285 (38) with the Office of Naval Research, and Research Grant GM-12868 with the National Institutes of Health, Public Health Service.     

Nearly minimax tests in the case of a finite number of hypotheses

A method of finding a minimax test in the class of tests belonging to the linear closure of a collection of hypothetical densities is considered.Translated from Staticheskie Metody, pp. 110–115, 1978.     

Subsampling tests of parameter hypotheses and overidentifying restrictions with possible failure of identification

We introduce a general testing procedure in models with possible identification failure that has exact asymptotic rejection probability under the null hypothesis. The procedure is widely applicable and in this paper we apply it to tests of arbitrary linear parameter hypotheses as well as to tests of overidentification in time series models given by unconditional moment conditions. The main idea is to subsample classical tests, like for example the Wald or the J test. More precisely, instead of using critical values based on asymptotic theory, we compute data-dependent critical values based on the subsampling technique.We show that under full identification the resulting tests are consistent against fixed alternatives and that they have exact asymptotic rejection probabilities under the null hypothesis independent of identification failure. Furthermore, the subsampling tests of parameter hypotheses are shown to have the same local power as the original tests under full identification.An algorithm is provided that automates the block size choice needed to implement the subsampling testing procedure. A Monte Carlo study shows that the tests have reasonable size properties and often outperform other robust tests in terms of power.     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

Large deviation theorems in the problem of testing two simple hypotheses

Limit theorems on large deviations of the logarithm of the likelihood ratio are proved for the problem of distinguishing two simple hypotheses in the general scheme of statistical experiments under the null hypothesis and under an alternative hypothesis. The theorems obtained are applied to the investigation of a decrease in the probability of errors of the first and second kind for the Neumann-Pearson criterion.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 227–235, February, 1995.     

Behaviour in small samples of some tests of non-nested hypotheses in non-stationary regressions and their bootstrap versions

Summary  In this paper, we use simulation methods to assess, in small samples, the performance of the Davidson and MacKinnon (1981) J-test when it is used to discriminate between two non-nested models with non-stationary regressors. We distinguish two cases: first, we assume that the sets of regressors of the two models are cointegrated; secondly, we consider the case where the regressors are not cointegrated. We also compare the behaviour of this test with that of the Fisher McAleer JA-type test and the bootstrap-adjusted J-tests, in order to assess its relative performance.     

On two conjectures of franz hering about convex surfaces

In this paper we show that from the point of view of the Baire categories for most convex bodies no shadow boundary is included in a hyperplane. A related, more quantitative question is also considered. It receives in general a negative answer and in theC ²-case a positive one, but remains open in theC ¹-case.     

On the generation of soft magneto-electric effects through Maxwell interactions

This contribution presents a finite-strain computational homogenization framework for soft magneto-electro-elastic solids. We use it in order to determine effective magneto-electric coupling coefficients of soft composites, which are driven by the interconnection of electrical and magnetical Maxwell stresses. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A characterization of spectral integral variation in two places for Laplacian matrices

We describe the graphs having the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing by 1 and the other Laplacian eigenvalues remaining fixed. For a certain subclass of graphs, we also characterize the Laplacian integral graphs with that property. Finally, we investigate a situation in which the algebraic connectivity is one of the eigenvalues that increases by 1.     

A characterization of a minimax test in the problem of testing two composite hypotheses

We obtain a characterization of a minimax test in the problem of testing two composite hypotheses via a solution of a dual problem. In comparison to previous results, our characterization is valid under much weaker assumptions, which became possible, in particular, due to a modification of a dual problem.     

Testing hypotheses about the equality of several risk measure values with applications in insurance

This paper explores statistical tests about the equality of risk measure values obtained using a distortion-based risk measure. We consider both the case in which the risk measure value is specified in the null hypothesis and the case in which it is not. In the former case, one- and two-sided alternatives are considered, and in the latter case, ordered and unordered alternatives are considered. Asymptotically most powerful tests are obtained, and asymptotic distributions of the test statistics are found using results about the asymptotic distributions of the risk measure values. Finally, we consider a numerical example and conclude the paper with notes on when the results of the paper could, or could not, be safely used.     

On a class of rank order tests for the identity of two multiple regression surfaces

Summary That the central theorem is valid for (forward) martingales is a result with a long history, beginning with Lévy [6], the most refined and recent results being due to Billingsley [3].Not altogether surprisingly, an analogous result holds for backwards martingales, and the proof, which parallels closely that of Billingsley, occupies Section 1. Examples are given in Section 2.