Generalizing the Welch Test To Non-Zero Hypotheses on the Variance Component in the One-Way Random Effects Model Under Variance Heterogeneity

In an unbalanced and heteroscedastic one-way random effects model of analysis of variance we consider hypotheses concerning the between group variance where in a classical set-up the hypothesis whether this variance component is zero is tested. We consider the still open problem of testing also extended hypotheses concerning this parameter and generalize the prominent Welch test to deal with these hypotheses, too.     

On the sample breakdown robustness of some nonparametric tests

In this paper, relying on the sample breakdown points, we investigate the sample breakdown properties of some nonparametric tests. It is shown that the sample breakdown points of the sign test asymptotically dominate those of the Wilcoxon test for one–sided hypotheses, However, the different conclusion is derived in the case of testing some shrinking neighborhood hypotheses. The breakdown behaviors of the Kolmogorov test and X²–test are also explored. These studies unify or refine some existing breakdown analyses of tests.     

Two-stage tests of hypotheses in the general linear model

Two-stage (double sample) tests of hypotheses are presented for testing linear hypotheses in the general linear model. General and one-sided alternatives are considered. Computational techniques for computing critical points are discussed. Tables of critical points are presented. An example suggests that two-stage tests can achieve the same power as a fixed sample size test while reducing considerably the expected number of observations required for the test     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.     

Manipulating Statistical Hypotheses and Tests as Boolean Functions

Statistical hypotheses and test statistics are Boolean functions that can be manipulated using the tools of Boolean algebra. These tools are particularly useful for exploring multiple comparisons or simultaneous inference theory, in which multiparameter hypotheses or multiparameter test statistics may be decomposed into combinations of uniparameter hypotheses or uniparameter tests. These concepts are illustrated with both finite and infinite decompositions of familiar multiparameter hypotheses and tests. The corresponding decompositions of acceptance regions and rejection regions are also shown. Finally, the close relationship between hypothesis and test decompositions and Roy's union—intersection principle is demonstrated by a derivation of the union—intersection test of the univariate general linear hypothesis.     

Changepoint Analysis as a Method for Isotonic Inference

Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric inputoutput analysis. The slope change and inflection point models are introduced correspondingly as the corners of the polyhedral cones defined by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justified for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate non-linear transformation, leads to an exact and very efficient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.     

A Bayesian approach to sample size selection in a multiple hypotheses test for the Bernoulli model

The situation where a certain type of seed need to be classified into one of three different categories, according to its germination level, can be formulated as the simultaneous test of three statistical hypotheses. In this paper, the problem of assessing an optimal sample size for the simultaneous test of several hypotheses on a Bernoulli process is studied within the Bayesian framework, and a solution is obtained for a wide class of prior distributions and a logarithmic loss function.     

A Note on a Partition of the Likelihood Ratio Test for Autoregressive Covariance Structure

A sequence of nested hypotheses is presented for the examination of the assumption of autoregressive covariance structure in, for example, a repeated measures experiment. These hypotheses arise naturally by specifying the joint density of the underlying vector random variable as a product of conditional densities and the density of a subset of the vector random variable. The tests for all but one of the nested hypotheses are well known procedures, namely analysis of variance F-tests and Bartlett's test of equality of variances. While the procedure is based on tests of hypotheses, it may be viewed as an exploratory tool which can lead to model identification. An example is presented to illustrate the method.     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.     

A simple procedure for testing linear hypotheses about the parameters of a nonlinear model using weighted least squares

Suppose the same nonlinear function involving k parameters is fit to each of t populations. Suppose further it is of interest to compare a specific parameter of the models across the populations. Such comparisons can be expressed as linear hypotheses about the parameters of the nonlinear models. A weighted linear least squares (WLLS) procedure is proposed to test these linear hypotheses. The advantages and disadvantages of the WLLS procedure are discussed. This procedure is also compared to a nonlinear least squares procedure for testing these hypotheses in nonlinear models.     

On the poisson index of dispersion

A commonly used statistic for testing hypotheses concerning count data is the index of dispersion or variance test (c.f., Fisher 1950, Rao and Chakravarti 1956). The statistic is essentially the ratio of the sample variance to the sample mean,The hypothesis being tested is commonly referred to as the homogeneity hypothesis, often without further detailed specification. However, a closer study of the literature indicates that there are really two different hypotheses of interest.     

Sequential analysis applied to testing the scale parameter in the problem of the nile

The parameter O in the problem of the Nile is tested using the Sequential Probability Ratio Test (SPRT). It is observed that the test can be used to discriminate between two one–sided hypotheses.     

Testing hypotheses in truncated samples by means of divergence statistics

In this paper we consider the problem of testing hypotheses in parametric models, when only the first r (of n) ordered observations are known.Using divergence measures, a procedure to test statistical hypotheses is proposed, Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained.Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators for truncated samples are considered.Applications of these results in testing statistical hypotheses, on the basis of truncated data, are presented.The small sample behavior of the proposed test statistics is analyzed in particular cases.A comparative study of power values is carried out by computer simulation.     

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

Bayesian (mean) most powerful tests

A fundamental theorem in hypothesis testing is the Neyman‐Pearson (N‐P) lemma, which creates the most powerful test of simple hypotheses. In this article, we establish Bayesian framework of hypothesis testing, and extend the Neyman‐Pearson lemma to create the Bayesian most powerful test of general hypotheses, thus providing optimality theory to determine thresholds of Bayes factors. Unlike conventional Bayes tests, the proposed Bayesian test is able to control the type I error.     

Most mean powerful invariant test for testing two-dimensional parameter spaces

This paper investigates the application of the most mean powerful invariant test to the problem of testing for joint MA(1)–MA(4) disturbances against joint AR(1)–AR(4) disturbances in the linear regression model. The most mean powerful invariant test was introduced by Begum and King (Most mean powerful invariant test of a composite null against a composite alternative. Comp. Statist. Data Analysis, 2004, forthcoming) and is based on the generalized Neyman–Pearson lemma which provides an optimal test of certain composite hypotheses. The most mean powerful invariant test can be computationally intensive. Previous applications have only involved testing problems whose null hypotheses, after reduction through invariance arguments, are one dimensional. This is the first application involving null and alternative hypotheses which are two dimensional. A Monte Carlo experiment was conducted to assess the small sample performance of the test with encouraging results. The increase in dimension does increase significantly the computational effort required to apply the test.     

Multiple comparisons in two-factor testing of psychophysical thresholds by the delta test method

We propose an algorithm of multiple comparisons in two-factor testing of psychophysical thresholds. We use logistic regression combined with guessing rate and adopt the step-down procedure with Ryan–Einot–Gabriel–Welsch (REGW) significance levels for multiple test. We test hypotheses of main effects and interaction in two-factor problem by using the delta test statistics.     

The use of multi-state capture-recapture models to address questions in evolutionary ecology

Multi-state capture-recapture models can be used to estimate survival rates in populations that are stratified by location or by state variables associated with individual animals. In populations stratified by location, movement probabilities can be estimated and used to test hypotheses relevant to population genetics and evolutionary ecology. When the interest is in state variables, these models permit estimation and testing of hypotheses about state-specific survival probabilities. If the state variable of interest is reproductive activity or success, then the multi-state modeling approach can be used to test hypotheses about life history trade-offs and a possible cost of reproduction.     

Statistical test based on intuitionistic fuzzy hypotheses

In this paper, the classical statistical test based on intuitionistic fuzzy hypotheses in relation to the underlying population parametric is extended. In this approach, the type-I, type-II, power of test, and p-value are extended for intuitionistic fuzzy hypotheses. Throughout the paper, some applied examples are provided for both parametric and non parametric cases to clarify the discussions.     

A note on the manova model with double linear restrictions

In this note we show how one may construct goodness-of-fit tests to test hypotheses for the restricted MANOVA and GMANOVA models using the multivariate seemingly unrelated regression (MSUR) model.