A test for equality of variances with censored samples

A variance homogeneity test for type II right-censored samples is proposed. The test is based on Bartlett's statistic. The asymptotic distribution of the statistic is investigated. The limiting distribution is that of a linear combination of i.i.d. chi-square variables with 1 degree of freedom. By using simulation, the critical values of the null distribution of the modified Bartlett's statistic for testing the homogeneity of variances of two normal populations are obtained when the sample sizes and censoring levels are not equal. Also, we investigate the properties of the proposed test (size, power and robustness). Results show that the distribution of the test statistic depends on the censoring level. An example of the use of the new methodology in animal science involving reproduction in ewes is provided.     

Omnibus tests for the error distribution in the linear regression model

Test procedures are constructed for testing the goodness-of-fit of the error distribution in the regression context. The test statistic is based on an L ²-type distance between the characteristic function of the (assumed) error distribution and the empirical characteristic function of the residuals. The asymptotic null distribution as well as the behavior of the test statistic under contiguous alternatives is investigated, while the issue of the choice of suitable estimators has been particularly emphasized. Theoretical results are accompanied by a simulation study.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

Analysis of binary data for the balanced one-way classification

A test is proposed for testing the equality of proportions based on the data available from a one-way classification having t treatment conditions and n binary observations per treatment. The test statistic B is a constant multiple of the F-statistic which results when the analysis of variance procedure for the one-way classification is applied to the data and, hence, is computationally simple. The statistic B from this binary analysis of variance (BIANOVA) is distributed asymptotically as a chi-square random variable. The proposed test is uniformly more powerful than either the F-test indicated above or the Pearson chi-square test; however, the attained empirical level of significance is frequently higher than for either of these competitors and usually higher than the stated level of significance for smaller values of n (say n ≤ 20).     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

A method for constructing exact tests from test statistics that have unknown null distributions

A randomized procedure is described for constructing an exact test from a test statistic F for which the null distribution is unknown. The procedure is restricted to cases where F is a function of a random element U that has a known distribution under the null hypothesis. The power of the exact randomized test is shown to be greater in some cases than the power of the exact nonrandomized test that could be constructed if the null distribution of Fwere known.     

Test for Generalized Variance in Factor Analysis Model

We derive a likelihood ratio test for generalized variance a in factor analysis model. The asymptotic distribution of the test statistic follows chi-square distribution with one degree of freedom from a general theory of likelihood ratio test.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

An approximation of the distribution function of the LIML identifiability test statistic using the method of moments

This paper contains an application of the asymptotic expansion of a pFp() function to a problem encountered in econometrics. In particular we consider an approximation of the distribution function of the limited information maximum likelihood (LIML) identifiability test statistic using the method of moments. An expression for the Sth order asymptotic approximation of the moments of the LIML identifiability test statistic is derived and tabulated. The exact distribution function of the test statistic is approximated by a member of the class of F (variance ratio) distribution functions having the same first two integer moments. Some tabulations of the approximating distribution function are included.     

Generalized likelihood-ratio test of the number of components in finite mixture models

The number of components is an important feature in finite mixture models. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic does not have a chi-square limit distribution. It is very difficult to find a test with a specified significance level, and this is especially true for testing k — 1 versus k components. Most of the existing work has concentrated on finding a comparable approximation to the limit distribution of the log-likelihood-ratio statistic. In this paper, we use a statistic similar to the usual log likelihood ratio, but its null distribution is asymptotically normal. A simulation study indicates that the method has good power at detecting extra components. We also discuss how to improve the power of the test, and some simulations are performed.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

Improvement of the quality of the chi-square approximation for the ADF test on a covariance matrix with a linear structure

The asymptotically distribution-free (ADF) test statistic was proposed by Browne (1984). It is known that the null distribution of the ADF test statistic is asymptotically distributed according to the chi-square distribution. This asymptotic property is always satisfied, even under nonnormality, although the null distributions of other famous test statistics, e.g., the maximum likelihood test statistic and the generalized least square test statistic, do not converge to the chi-square distribution under nonnormality. However, many authors have reported numerical results which indicate that the quality of the chi-square approximation for the ADF test is very poor, even when the sample size is large and the population distribution is normal. In this paper, we try to improve the quality of the chi-square approximation to the ADF test for a covariance matrix with a linear structure by using the Bartlett correction applicable under the assumption of normality. By conducting numerical studies, we verify that the obtained Bartlett correction can perform well even when the assumption of normality is violated.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Order test for high-dimensional two-sample means

We propose a new method to test the order between two high-dimensional mean curves. The new statistic extends the approach of Follmann (1996) to high-dimensional data by adapting the strategy of Bai and Saranadasa (1996). The proposed procedure is an alternative to the non-negative basis matrix factorization (NBMF) based test of Lee et al. (2008) for the same hypothesis, but it is much easier to implement. We derive the asymptotic mean and variance of the proposed test statistic under the null hypothesis of equal mean curves. Based on theoretical results, we put forward a permutation procedure to approximate the null distribution of the new test statistic. We compare the power of the proposed test with that of the NBMF-based test via simulations. We illustrate the approach by an application to tidal volume traces.     

On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33–48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.     

A test for the variance residual life

In this paper we develop a test based on the empirical distribution function for the alternative representing 'decreasing variance residual life1 property. The test is consistent with asymptotically normal test statistic and is shown to perform well in the Pitman's asymptotic relative efficiency sense.     

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.     

Preliminary test of fit in a general class of conditionally heteroscedastic nonlinear time series

This article is concerned with a general class of conditionally heteroscedastic time series including possibly nonlinear and asymmetric autoregressive conditional heteroscedastic (ARCH) and generalized ARCH models. A problem of preliminary test of fit (PTF, hereafter) within the broad class under consideration is discussed. It is noted that contrary to usual tests in the literature of conditionally heteroscedastic time series, PTF does not require any specification of the conditional variance in advance. Based on the joint limit distributions of sample autocorrelations, a certain Portmanteau-type statistic for PTF is proposed, and its limit is shown to be a chi-square distribution. In addition, some simulation studies, under various innovations, are reported to support our theoretical results.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.