Asymptotic test statistics for coefficients of variation

A one-sample asymptotically normal test statistic Is derived for testing the hypothesis that the coefficient of variation of a normal population is equal to a specified value. Based on this derivation, an asymptotically noraml two-sample test statistic and an asymptotically chi-square k-sample test statistic are derived for testing the hypothesis that the coefficients of variation of k ≥2 normal populations are equal. The two and k-sample test statistics allow for unequal sample sizes. Results of a simulation study which evaluate the size and power of the test statistics and compare the test statistics to earlier ones developed by McKay (1932) and Bennett (1976) are presented.     

A simulation study of the Bennett test and Miller test for coefficients of variation

In the article, properties of the Bennett test and Miller test are analyzed. Assuming that the sample size is the same for each sample and considering the null hypothesis that the coefficients of variation for k populations are equal against the hypothesis that k ? 1 coefficients of variation are the same but differ from the coefficient of variation for the kth population, the empirical significance level and the power of the test are studied. Moreover, the dependence of the test statistic and the power of the test on the ratio of coefficients of variation are considered. The analyses are performed on simulated data.     

Approximations of the critical region of the fbietkan statistic

The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x² and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.     

A network algorithm for the exact treatment of the 2×k contingency table

A common statistical problem encountered in biomedical research is to test the hypothesis that the parameters of k binomial populations are all equal. An exact test of significance of this hypothesis is possible in principle, the appropriate null distribution being a normalized product of k binomial coefficients. However, the problem of computing the tail area of this distribution can be formidable since it requires the enumeration of all sets of k binomial coefficients whose product is less than a given constant. Existing algorithms, all of which rely on explicit enumeration to generate feasible binomial coefficients     

Nonparametric Tests for Comparing Several Coefficients of Variation

We consider the problem of comparing (k + 1) coefficients of variation. We are interested in testing the null hypothesis that the coefficients of variation are equal against each of the alternatives: (a) some populations have different coefficients of variation and (b) the coefficients of variation are ordered. Three nonparametric test statistics are proposed and their asymptotic theory is developed. We compared the proposed tests together with another parametric test using two Monte Carlo studies to estimate their probabilities of Type I error and powers. An illustration of the proposed tests using a real data set is given.     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

Confidence Bounds and Hypothesis Tests for Normal Distribution Coefficients of Variation

For normally distributed populations, we obtain confidence bounds on a ratio of two coefficients of variation, provide a test for the equality of k coefficients of variation, and provide confidence bounds on a coefficient of variation shared by k populations.     

On Small Sample Properties of the Wald,LR and LM Tests in a Linear Model with AR(1) Errors

When the error terms are autocorrelated, the conventional t-tests for individual regression coefficients mislead us to over-rejection of the null hypothesis. We examine, by Monte Carlo experiments, the small sample properties of the unrestricted estimator of ρ and of the estimator of ρ restricted by the null hypothesis. We compare the small sample properties of the Wald, likelihood ratio and Lagrange multiplier test statistics for individual regression coefficients. It is shown that when the null hypothesis is true, the unrestricted estimator of ρ is biased. It is also shown that the Lagrange multiplier test using the maximum likelihood estimator of ρ performs better than the Wald and likelihood ratio tests.     

A test of simultaneous homogeneity against ordered alternatives in multifactorial design

A test of simultaneous homogeneity of main effects of several factors against an alternative hypothesis with simple order restrictions in main effects of more than one factor in a multifactorial design is considered. This can be regarded as an extension of Shorack's (1967) work where the alternative hypo t hqaLs involves simple order restriction in main effects of one faetor. We derive the likelihood ratio test statistic, E^-2 , and its null hypothesis distribution which involves the convolutions of PIrobabilities P(l,k) used in the statistical inference under order restriction , The powers of the E^-2 test and the usual F test are compar-ed by simulation.     

Bootstrap versus traditional hypothesis testing procedures for coefficients in least absolute value regression

Several approaches to hypothesis testing for coefficients in least absolute value regression are compared using a Monte Carlo simulation: likelihood ratio test, Lagrange multiplier test, and three versions of the bootstrap hypothesis test. Factors considered that might influence test performance include the disturbance distribution, the type of independent variable, and the sample size. Overall, the likelihood ratio and the bootstrap tests perform best, with the likelihood ratio test being marginally more powerful. Least absolute value tests are also compared to the standard t test and three versions of the bootstrapped t test for least squares regression.     

Simultaneous estimation of coefficients of variation

An improved asymptotic estimation theory for the coefficient of variation γ is developed under the homogeneity hypothesis that several coefficients of variation are the same. Assuming that homogeneity holds, it is advantageous to combine the data to estimate the common coefficient of variation. However, the combined estimator becomes inconsistent when the equality of the hypothesis does not hold. In this situation, estimators based on pretest and (James and Stein, 1961. Estimation with quadratic loss. Proceeding of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 361–379) principles are proposed. Asymptotic properties of the shrinkage estimator, positive-part and pretest estimators are discussed and compared with the standard and combined estimators. It is demonstrated that the positive part estimator utilizes the sample and nonsample information in a superior way relative to the ordinary shrinkage estimator.     

Small sample performance of large sample tests for tobit versus p -tobit

Large sample tests for the standard To bit model versus the ^p -Tobit model by Deaton and Irish (1984) are studied. The normalized one-tailed score test by Deaton and Irish (1984) is shown to be a version of Neyman's C(α) test that is valid for the non-standard problem of the null hypothesis lying on the boundary of the parameter space. Then, this paper reports the results of Monte Carlo experiments designed to study the small sample performance of large sample tests for the standard Tobit specification versus the ^p -Tobit specification.     

Use of the squared ranks test to test for the equality of the coefficients of variation

A distribution-free test for the equality of the coefficients of variation from k populations is obtained by using the squared ranks test for variances, as presented by Conover and Iman (1978) and Conover (1980), on the original observations divided by their respective expected values. Substitution of the sample mean in place of the expected value results in the test being only asymptotically distribution-free. Results of a simulation study evaluating the size of the test for various coefficient of variation values and probability distributions are presented.     

On Hotelling’s T2 test in a special paired sample case

In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.     

The sample breakdown points of tests

Several authors have taken the worst case breakdown measures in analyzing the robustness of a test. In general, these kinds of measures give only a rough picture of breakdown robustness of a test. To overcome this limitation, a new kind of breakdown measure of a test is defined as the smallest proportion of arbitrary outliers in the sample that can distort the test decision. It is called as the sample breakdown point of a test in this paper. A distinct advantage of this new measure is that it is directly concerned with the test decision based on the present sample and with the critical region of the test. The sample breakdown points of several commonly used tests of one-sided or two-sided hypotheses are calculated and their asymptotic properties are also established. By Monte Carlo simulations and asymptotic analysis, we show that the acceptance breakdown of the t-test and the Hotelling T²-test is slightly better than that of the sample mean test. Finally, we prove that, for a one-sided hypothesis testing of location, the sign test has the maximum sample breakdown points asymptotically within a class of M-tests and score-tests.     

Asymptotic inference for coefficients of variation

Asymptotic inference results for the coefficients of variation of normal populations are presented in this article. This includes formulas for test statistics, power, confidence intervals, and simultaneous inference. The results are based on the asymptotic normality of the sample coefficient of variation as derived by Miller (1991). An example which compares the homogeneity of bone test samples produced from two different methods is presented.     

A robust modification of the ordered-heterogeneity test

An ordered heterogeneity (OH) test is a test for a trend that combines a non-directional heterogeneity test with the rank-order information specified under the alternative. We propose two modifications of the OH test procedure: (1) to use the mean ranks of the groups rather than the sample means to determine the observed ordering of the groups, and (2) to use the maximum correlation out of the 2^k???1 – 1 possibilities under the alternative rather than the single ordering (1, 2, … , k), where k is the number of independent groups. A simulation study indicates that these two changes increase the power of the ordered heterogeneity test when, as common in practice, the underlying distribution may deviate from a normal distribution and the trend pattern is a priori unknown. In contrast to the original OH test, the modified OH test can detect all possible patterns under the alternative with a relatively high power.     

Repeated chi-square testing

A sequentialized version of the x²; goodness of fit test, called repeated x,²; test, is introduced. The form of the asymptotic distribution of the repeated x² test statistic is given under the null hypothesis as well as under local alternatives. For various numbers of cells Monte Carlo results are given for critical values, power and distribution of stopping time. Finally, the perfor-mance of the repeated and the fixed sample x² test are compared.     

A simple test for comparing regression curves versus one-sided alternatives

In this article we present a simple procedure to test for the null hypothesis of equality of two regression curves versus one-sided alternatives in a general nonparametric and heteroscedastic setup. The test is based on the comparison of the sample averages of the estimated residuals in each regression model under the null hypothesis. The test statistic has asymptotic normal distribution and can detect any local alternative of rate n^-1/2. Some simulations and an application to a data set are included.     

A parametric bootstrap approach for two-way error component regression models

In this article, the two-way error component regression model is considered. For the nonhomogenous linear hypothesis testing of regression coefficients, a parametric bootstrap (PB) approach is proposed. Simulation results indicate that the PB test, regardless of the sample sizes, maintains the Type I error rates very well and outperforms the existing generalized variable test, which may far exceed the intended significance level when the sample sizes are small or moderate. Real data examples illustrate the proposed approach work quite satisfactorily.