A Critical Assessment of Simulated Critical Values

There is a wide variety of statistical problems (e.g., unit root and cointegration tests) where hypothesis testing involves the use of simulated rather than theoretical critical values. We argue that, in practice, the number of replications used to simulate critical values is often insufficient to provide the degree of precision that is implied. In particular, the number of replications needed is greatest for values in the tails of the distribution. We provide recommendations for approximating the number of replications needed to achieve a desired degree of precision.     

Divergence statistics: sampling properties and multinomial goodness of fit and divergence tests

φ-divergence .statistics are obtained by either replacing both distributions involved in the argument of the φ -divergence measure by their sample estimates or replacing one distribution and considering the other as given. The sampling properties of estimated divergence-type measures are investigated. Approximate means and variances are derived and asymptotic distributions are obtained. Tests of goodness of fit of observed frequencies to expected ones and tests of equality of divergences based on two or more multinomial samples are constructed.     

Locally optimal test for simultaneously testing the mean and variance of a normal distribution

A generalization of the locally most powerful unbiased (LMPU) test for the single parameter case to the k-parameter case was proposed by SenGupta and Vermeire (1986). In particular we defined a locally most mean power unbiased (LMMPU) test based on the mean curvature of the power hypersurface. Compared to the type C tests of Neyman and Pearson and the type D tests (Isaacson, 1951), LMMPU tests possess better theoretical properties and enjoy ease of construction of critical regions. In this paper we present an interesting example of a two-parameter univariate normal population for which Isaacson (1951, p. 233) was unsuccessful in finding a type D test. For the case of one observation, we prove that no Type D region exists but the LMMPU test is obtained - it is an example of a test with singular Hessian matrix for its power but is nevertheless a strictly locally unbiased (LU) test.     

COMPARING TESTS OF AUTOREGRESSIVE VERSUS MOVING AVERAGE ERRORS IN REGRESSION MODELS USING BAHADUR’S ASYMPTOTIC RELATIVE EFFICIENCY

ABSTRACT

The purpose of this paper is to use Bahadur's asymptotic relative efficiency measure to compare the performance of various tests of autoregressive (AR) versus moving average (MA) error processes in regression models. Tests to be examined include non-nested procedures of the models against each other, and classical procedures based upon testing both the AR and MA error processes against the more general autoregressive-moving average model.     

Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte Carlo experiments illustrate the finite sample performance of the modified portmanteau test.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

On goodness of fit tests based on spacings for type ii censored samples

A class of tests based on spacings is obtained for milticensored samples. Their asymptotic null as well as alternative distributions are obtained.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

中日服务型语文测试工作对比分析

如何评估个人或国民语文能力,将是语言学界热切关注的一个重要话题。对比中日两国的服务型语文测试工作,我们认为,中国的服务型语文测试工作应从当前的政府主导向民间主导转化,逐步实现测试目的公益化、测试对象大众化、行业管理严格化,从而形成良性的商业化模式,为更多国民提供更好的语言测试服务。     

The analysis of ratios and cross-product ratios of poisson variates with application to incidence rates

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.