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.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

Likelihood ratio test for simultaneous testing of the mean and the variance of a normal distribution

In this paper, we consider the simultaneous testing of the mean and the variance of a normal distribution. The exact distribution of the likelihood ratio test statistic is obtained, which is not available in the literature. The critical points of the exact test are reported. We also consider some of the other exact and asymptotic tests. The powers of these tests are compared using the Monte Carlo simulations.     

Weighted combinations of rank tests for location-scale alternatives

All existing location-scale rank tests use equal weights for the components. We advocate the use of weighted combinations of statistics. This approach can partly be substantiated by the theory of locally most powerful tests. We specifically investi= gate a Wilcoxon-Mood combination. We give exact critical values for a range of weights. The asymptotic normality of the test statistic is proved under a general hypothesis and Chernoff-Savage conditions. The asymptotic relative efficiency of this test with respect to unweighted combinations shows that a careful choice of weights results in a gain in efficiency.     

Computing the Exact Fisher Information Matrix of Periodic State-Space Models

The purpose of this article is to develop algorithms for computing the exact Fisher information matrix of periodic time-varying state-space models. We first present a relatively simple recursive algorithm which computes the elements of the exact information matrix without involving numerical differentiation, since all required derivatives are analytically evaluated. The proposed algorithm extends the procedure due to Cavanaugh and Shumway (1996 Cavanaugh , J. E. , Shumway , R. H. ( 1996 ). On computing the expected Fisher information matrix for state-space model parameters . Statist. Probab. Lett. 26 : 347 – 355 .[Crossref], [Web of Science ®] , [Google Scholar]) to the periodic state-space framework. Exploiting the approach used in Klein et al. (2000 Klein , A. , Mélard , G. , Zahaf , T. ( 2000 ). Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules . Linear Alg. Applic. 321 : 209 – 232 .[Crossref], [Web of Science ®] , [Google Scholar]), a second algorithm is proposed in order to obtain the exact information matrix as a whole instead of element by element. The algorithms are first developed in a general framework and then specialized to the case of a periodic Gaussian vector autoregressive moving-average (PVARMA) model.     

Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Exact Slopes of Test Statistics for the Multivariate Exponential Family

The objective of this paper is to investigate exact slopes of test statistics { T_n } when the random vectors X ₁, ..., X_n are distributed according to an unknown member of an exponential family { P _θ; θ∈Ω. Here Ω is a parameter set. We will be concerned with the hypothesis testing problem of H ₀θ∈Ω₀ vs H ₁: θ∉Ω₀ where Ω₀ is a subset of Ω. It will be shown that for an important class of problems and test statistics the exact slope of { T_n } at η in Ω−Ω₀ is determined by the shortest Kullback–Leibler distance from {θ: T_n (λ(θ)) = T_n (λ(π))} to Ω₀, λ_θ = E _θ)( X ).     

Power comparisons of single-sample winsorized T,trimmed T and student's T

The objective of this study is to evaluate the performance of Winsorized t which was proposed to be used as one alternative to the student's t statistic when the parent population is non-normal. The criterion we use is the power of the test. Comparative performance of the Winsorized t, trimmed t, and Studen$tCs t for normal populations and for Studen$tCs t distribution with 8, 4, and 2 degrees of freedom was investigated.     

Exact and approximate distributions of the chi-square statistic for equiprobability

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.     

A note on the most powerful invariant test of Rayleigh against exponential distribution

Abstract

The problem of testing Rayleigh distribution against exponentiality, based on a random sample of observations is considered. This problem arises in survival analysis, when testing a linearly increasing hazard function against a constant hazard function. It is shown that for this problem the most powerful invariant test is equivalent to the “ratio of maximized likelihoods” (RML) test. However, since the two families are separate, the RML test statistic does not have the usual asymptotic chi-square distribution. Normal and saddlepoint approximations to the distribution of the RML test statistic are derived. Simulations show that saddlepoint approximation is more accurate than the normal approximation, especially for tail probabilities that are the main values of interest in hypothesis testing.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants,Moments, and Applications

This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921 Fisher , R. A. ( 1921 ). On the “probable error” of a coefficient of correlation deduced from a small sample . Metron 1 : 1 – 32 . [Google Scholar]). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953 Hotelling , H. ( 1953 ). New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Ser. B 15 : 192 – 232 . [Google Scholar]) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.     

Approximation of the Fisher price index by using Lowe,Young, and AG Mean indices

The Consumer Price Index (CPI) approximates changes in the costs of household consumption assuming the constant utility (COLI, Cost of Living Index). In practice, the Laspeyres price index is used to measure the CPI despite the fact that many economists consider the superlative indices to be the best approximation of COLI. The Fisher index is one of the superlative indices and additionally it satisfies most of tests from the axiomatic price index theory. Nevertheless, the Fisher price index makes use of current-period expenditure data and its usefulness in CPI measurement is limited. In this article, we verify the utility of using the Lowe, Young, and AG Mean indices for Fisher price index approximation. We confirm this utility in a simulation study and we provide an empirical proof.     

Discrete approximations to the Csiszár,Renyi, and Fisher measures of information

The problem of loss of information due to the discretization of data and its estimate is studied for various measures of information. The results of Ghurye and Johnson (1981) are generalized and supplemented for the Csiszár and Renyi measures of information as well as for Fisher's information matrix.     

Some rememarks on locally most powerful unbiased tests for the detection of mixtures of poisson or binomial distributions

In t h i s note mixture models are used to represent overdispersion relative to Poisson or binomial distributions. We flnd a sufflclent condition on the mixing distribution underich the detection of mixture departures from the Poisson or binomial adrnits a locally most powerful unbiased test. The conditions specify plynoria: relations between the variance and mean of Le glxing distribution.     

Shrinkage testimators for the shape parameter of weibull distributin under type ii censoring

In this paper we propose some shrinkage testimators for the shape parameter of the Weibull distribution when censored samples are available and study their properties. Comparison of the testimators with Singh and Bhatkulikar (1978) and with the usual estimator, interms of mean squared error are made. It is shown that the proposed testimators     

Power comparisons between the F-test,the studentised range test,and an optimal test of the equality of several normal means

The power assessment of tests of the equality of k normal means such as the k treatment means in a one-way fixed effects analysis of variance model is addressed. Power assessment is considered in terms of a constraint on the range of the treatment means. The power properties of the standard F-test and Studentised range test are compared with those of an optimal (minimax) test procedure, which is known to maximise power levels under this constraint. It is shown that the standard test procedures compare well with the optimal test procedure, and in particular, the Studentised range test is shown to be practically as good as optimal in this setting.     

A Simple Analysis of the Exact Probability Matching Prior in the Location-Scale Model

It has long been asserted that in univariate location-scale models, when concerned with inference for either the location or scale parameter, the use of the inverse of the scale parameter as a Bayesian prior yields posterior credible sets that have exactly the correct frequentist confidence set interpretation. This claim dates to at least Peers, and has subsequently been noted by various authors, with varying degrees of justification. We present a simple, direct demonstration of the exact matching property of the posterior credible sets derived under use of this prior in the univariate location-scale model. This is done by establishing an equivalence between the conditional frequentist and posterior densities of the pivotal quantities on which conditional frequentist inferences are based.     

A Confidence Interval and Test for the Mean of an Asymmetric Distribution

ABSTRACT

A confidence interval and test are obtained for the mean of an asymmetric distribution using a random sample of size n. The method is based on N. J. Johnson's (1978 Johnson , N. J. ( 1978 ). Modified, t tests and confidence intervals for asymmetrical populations. J. Amer. Statist. Assoc. 73 ( 363 ): 536 – 544 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) modified t-test, where terms of Cornish–Fisher expansions involving the third moment are used to adjust the conventional statistic to have more closely a Student's t-distribution with n ? 1 degrees of freedom. Johnson's (1978 Johnson , N. J. ( 1978 ). Modified, t tests and confidence intervals for asymmetrical populations. J. Amer. Statist. Assoc. 73 ( 363 ): 536 – 544 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) test cannot be inverted uniquely, so a corresponding confidence interval for the mean may be disjointed. However, an artificial term of small order can be added to make inversion of the test a uniquely defined operation, which prevents such disjointedness. The resulting one-sided and two-sided intervals perform better than others in the literature with skewed distributions, and have good performance with a normal distribution. The two-sided interval may be recommended for general use if the sample size is 10 or more and the nominal confidence coefficient is 95% or less, or if the sample size is 30 or more and the confidence coefficient is 99% or less.     

A Test for the Equality of Means of Two Groups with Different Variances When the Sample Size and the Dimension are Large

We consider a class of test statistics including the Dempster trace criterion in the case of two groups without assuming equal covariance matrices. The test statistics in the class are valid when the dimension is larger than the sample size. We obtain asymptotic distributions of the test statistics in the class and use these distributions to derive the limiting power in each case. We obtain the most powerful test in the class with respect to this limiting power.