Comment on basic structure of the asymptotic theory in dynamic nonlinear econometric models

In this paper we compare the performance of the exogeneity tests of Revankar, Revankar and Hartley and Wu-Hausman for the cases of two and three included endogenous variables. The distribution and power functions are evaluated using the conditional distributions given in Kariya and Hodoshima. Our results indicate that the Revankar's test is the most powerful for large values of the concentration parameter and the Revankar and Hartley test is the most powerful for small values of the concentration parameter.     

On optimal testing for the equality of equicorrelation: An example of loss in power

Sen Gupta (1988) considered a locally most powerful (LMP) test for testing nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. This paper constructs analogous tests for the symmetric multivariate normal distribution. It shows that the new test is uniformly most powerful invariant even in the presence of a nuisance parameter, σ². Further applications of LMP invariant tests to several equicorrelated populations have been considered and an extension to panel data modeling has been suggested.     

Some observations in testing for nonstandard mixtures

We consider optimal testing procedures for specific models of early and instantaneous failures in reliability studies. These models are typically used to accommodate lifetime data that have a higher concentration of failures near time zero. We show that it is possible to derive uniformly most powerful tests, for testing the mixing parameter in the instantaneous failure model, for general lifetime distributions. A novel procedure to test for early failures, which uses an invariance property of the maximum likelihood estimate of the nuisance parameter, is also presented.     

A note on a result for two SUR models

Revankar (1974, p. 190, equation (4.4)) obtains a result for the covariance matrices of the “Aitken” estimators of the regression coefficients parameter matrices of two SUR models. The present note supplies a simpler derivation of this result. It is obtained by using a known result in multivariate statistical analysis, see e.g., Sarkar (1981, p. 560, Theorem 3.1).     

Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

Unbiased and upper critical values of mean trace of multivariate beta for testing difference of two covariance matrices or several mean vectors

For the invariant unbiased level-α test of equality of two co-variance matrices, the quantities b and B satisfying the equations P(b≤T≤B) = 1-α, E(T|b≤T≤B) = E(T), where T is the mean trace of a multivariate beta, are required. Five and one per cent values of B are tabulated for m = 2,3(2)11,16; b can be obtained from B. Upper five and one per cent values of T are also included, as these are required for the locally most powerful invariant test of nullity of any source of difference in several mean vectors and the locally most powerful invariant one-sided test of equality of two covariance matrices. Lower critical values may be obtained from upper critical values.     

Testing homogeneity of control and treatment populations — local optimality and related issues

Durairajan and Raman (1996 a, b) studied the robustness of Locally most powerful invariant (LMPI) tests for compound normal model in control and treatment populations. In the present paper, the Locally most powerful (LMP) tests are constructed for no contamination in normal mixture model through testing the parameter of mixture of distributions and the mixing proportion. The expected performance of LMP tests are compared using Efron's Statistical Curvature on the lines of Sen Gupta and Pal (1991). The Locally most powerful similar (LMPS) tests for the equality of control and treatment populations in the presence of nuisance parameters are also constructed. Further, the null and non-null distributions of the test statistics are derived and some power computations are made. Received: September 1, 1999; revised version: August 31, 2000     

Partial Round-Robin Comparisons with Perfect Rankings

In teaching the development of uniformly most powerful unbiased (UMPU) tests, one rarely discusses the performance of alternative biased tests. It is shown, through the comparison of two independent Bernoulli proportions, that a biased test (the Z test) can be more powerful than the UMPU test (Fisher's exact test—randomized) in a large region of the alternative parameter space. A more general example is also given.     

On optimal tests for isotropy against the symmetric wrapped stable-circular uniform mixture family

The family of Symmetric Wrapped Stable (SWS) distributions can be widely used for modelling circular data. Mixtures of Circular Uniform (CU) with the former also have applications as a larger family of circular distributions to incorporate possible outliers. Restricting ourselves to such a mixture, we derive the locally most powerful invariant (LMPI) test for the hypothesis of isotropy or randomness of directions-expressed in terms of the null value of the mixing proportion, p, in the model. Global monotonicity of the power function of the test is established. The test is also consistent. Power values of the test for some selected parameter combinations, obtained through simulation reveal quite encouraging performances even for moderate sample sizes. The P 3 approach (SenGupta, 1991; Pal & SenGupta, 2000) for unknown p and rho and the non-regular case of unknown a, the index parameter, are also discussed. A real-life example is presented to illustrate the inadequacy of the circular normal distribution as a circular model. This example is also used to demonstrate the applications of the LMPI test, optimal P 3 test and a Daviesmotivated test (Davies, 1977, 1987). Finally, a goodness-of-fit test performed on the data establishes the plausibility of the above SWS-CU mixture model for real-life problems encountered in practical situations.     

The uniformly most powerful unbiased tests involving the f-distribution

The uniformly most powerful unbiased test in exponential family situations involving the F-distrlbution is illustrated with examples. Its performance is compared with that of the more conventional equal-tails test. A FORTRAN program for computing the critical values and tail sizes of this test is also presented.     

Comparison of testing procedures utilizing P-values and Bayes factors in some common situations

Bayesian alternatives to classical tests for several testing problems are considered. One-sided and two-sided sets of hypotheses are tested concerning an exponential parameter, a Binomial proportion, and a normal mean. Hierarchical Bayes and noninformative Bayes procedures are compared with the appropriate classical procedure, either the uniformly most powerful test or the likelihood ratio test, in the different situations. The hierarchical prior employed is the conjugate prior at the first stage with the mean being the test parameter and a noninformative prior at the second stage for the hyper parameter(s) of the first stage prior. Fair comparisons are attempted in which fair means the likelihood of making a type I error is approximately the same for the different testing procedures; once this condition is satisfied, the power of the different tests are compared, the larger the power, the better the test. This comparison is difficult in the two-sided case due to the unsurprising discrepancy between Bayesian and classical measures of evidence that have been discussed for years. The hierarchical Bayes tests appear to compete well with the typical classical test in the one-sided cases.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

Modified p-Value of Two-Sided Test for Normal Distribution with Restricted Parameter Space

This article proposes a modified p-value for the two-sided test of the location of the normal distribution when the parameter space is restricted. A commonly used test for the two-sided test of the normal distribution is the uniformly most powerful unbiased (UMPU) test, which is also the likelihood ratio test. The p-value of the test is used as evidence against the null hypothesis. Note that the usual p-value does not depend on the parameter space but only on the observation and the assumption of the null hypothesis. When the parameter space is known to be restricted, the usual p-value cannot sufficiently utilize this information to make a more accurate decision. In this paper, a modified p-value (also called the rp-value) dependent on the parameter space is proposed, and the test derived from the modified p-value is also shown to be the UMPU test.     

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.     

Table of percentage points of ratios of linear combinations of order statistics of samples from exponential distributions

A form of the distribution function of ratios of linear combinations of order statistics of samples from an exponential distribution is given. From the distribution, tables of percentage points of the statistic for α = .05, .95, and n = 3(1)50, and for censoring up to five observations are presented. Use of the tables is made to find critical values of the most powerful scale and location invariant test of exponentiality against uniformity, and also to find critical values for a test of outliers in an exponential population.     

Consistent GMM Residuals-Based Tests of Functional Form

This paper presents a consistent Generalized Method of Moments (GMM) residuals-based test of functional form for time series models. By relating two moments we deliver a vector moment condition in which at least one element must be nonzero if the model is misspecified. The test will never fail to detect misspecification of any form for large samples, and is asymptotically chi-squared under the null, allowing for fast and simple inference. A simulation study reveals randomly selecting the nuisance parameter leads to more power than supremum-tests, and can obtain empirical power nearly equivalent to the most powerful test for even relatively small n.     

Power and robustness of sib-pair linkage tests and extension to larger sibships

We investigate the power and robustness of Haseman and Elstonfs sib-pair test for genetic linkage between a marker locus and a locus affecting a quantitative trait, and compare the test to that of Penrose. The Haseman-Elston test is more powerful than Penrrose's test; its power is acceptable for cases of tight linkage and high heritability due to the hypothesized quantitative trait locus, but is quite low in other situations. Computer simulations indicate that both tests are valid for normally distributed trait values, and that the Haseman-Elston test is robust for a variety of continuous distributions of the trait values. Several linkage tests are developed for sib trios that are much more powerful , for the same total number of sibs, than the test on independent sib pairs. The Haseman-Elston test on all possible sib pairs is suggested for sibships of size larger than three and for samples including sibships of various sizes.     

Optimal confidence intervals for the geometric parameter

Abstract

This article discusses optimal confidence estimation for the geometric parameter and shows how different criteria can be used for evaluating confidence sets within the framework of tail functions theory. The confidence interval obtained using a particular tail function is studied and shown to outperform others, in the sense of having smaller width or expected width under a specified weight function. It is also shown that it may not be possible to find the most powerful test regarding the parameter using the Neyman-Pearson lemma. The theory is illustrated by application to a fecundability study.     

Testing for multiple upper and lower outliers in an exponential sample

Due to wide applicability and simplicity, the exponential distribution is the most commonly used distribution in reliability engineering and other life testing experiments. In this paper a test statistic for testing upper and lower outliers simultaneously in an exponential sample is proposed. However, the distribution of test statistic under the alternative is rather intricate, the null distribution is derived and critical values are obtained. A simulation study is also carried out to compare the performance of test and is found that the test based on this statistic is more powerful than the other two selected tests.     

An entropy-based test for goodness of fit of the von mises distribution

The maximum entropy characterization of the von Mises distribution on the circle is exploited to derive a consistent goodness of fit test for the von Mises distribution. Monte Carlo simulation results are tabulated giving critical values of the test statistic for various sample sizes and values of the concentration parameter. A power analysis is presented for various alternative hypotheses, comparing this entropy statistic to two other competing goodness of fit statistics. The entropy statistic is shown to compare favorably and may be more attractive, especially considering its ease of computation.