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.     

A multi sample test for the equality of coefficients of variation in normal populations

Two tests are derived for the hypothesis that the coefficients of variation of k normal populations are equal. The k samples may be of unequal size. The first test is the likelihood ratio test with the usual X²-approximation. A simulation study shows that the small sample behaviour under the null hypothesis is unsatisfactory. An alternative test, based on the sample coefficients of variation, appears to have somewhat better properties.     

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.     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

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.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

A permutation test approach to the choice of size k for the nearest neighbors classifier

The k nearest neighbors (k-NN) classifier is one of the most popular methods for statistical pattern recognition and machine learning. In practice, the size k, the number of neighbors used for classification, is usually arbitrarily set to one or some other small numbers, or based on the cross-validation procedure. In this study, we propose a novel alternative approach to decide the size k. Based on a k-NN-based multivariate multi-sample test, we assign each k a permutation test based Z-score. The number of NN is set to the k with the highest Z-score. This approach is computationally efficient since we have derived the formulas for the mean and variance of the test statistic under permutation distribution for multiple sample groups. Several simulation and real-world data sets are analyzed to investigate the performance of our approach. The usefulness of our approach is demonstrated through the evaluation of prediction accuracies using Z-score as a criterion to select the size k. We also compare our approach to the widely used cross-validation approaches. The results show that the size k selected by our approach yields high prediction accuracies when informative features are used for classification, whereas the cross-validation approach may fail in some cases.     

A test for independence in a multivariate exponential distribution with equal correlation coefficients

A rank statistic is considered which may be used for testing for total independence in a p-variate exponential distribution with equal correlation coefficients. Critical values for the statistic are provided for p = 3.4 and sample sizes less than or equal to 20. Finally, the small sample power performance of the rank test relative to that of the locally most powerful similar lest under the exponential alternative is evaluated.     

A new generalized p-value approach for testing equality of coefficients of variation in k normal populations

A new generalized p-value method is proposed for testing the equality of coefficients of variation in k normal populations. Simulation studies show that the type I error probabilities are close to the nominal level. The proposed test is also compared with likelihood ratio test, modified Bennett's test and score test through Monte Carlo simulation, the results demonstrate that the generalized p-value method has satisfactory performance in terms of sizes and powers.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

One-sided range test for testing against an ordered alternative under heteroscedasticity

In a one-way fixed effects analysis of variance model, when normal variances are unknown and possibly unequal, a one-sided range test for testing the null hypothesis H ₀ : μ ₁ = … = μ_k against an ordered alternative H_a : μ ₁ ≤ … ≤ μ_k by a single-stage and a two-stage procedure, respectively, is proposed. The critical values under H ₀ and the power under a specific alternative are calculated. Relation between the one-stage and the two-stage test procedures is discussed. A numerical example to illustrate these procedures is given.     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

Sur un test d'égalité des autocovariances de deux séries chronologiques

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite-sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by-product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L₁ norm.     

A comparative simulation study of the small sample powers of several goodness fit test

Eight goodness of fit tests are compared with respect to their simulated small sample power of detecting an inbreeding alternative to the Hardy-Weinberg null hypothesis. The Pearson's x ² test is found to be most powerful, and the small rample levels of this test are close to the nominal (x ²) significance levels. The use of conditional expectations, rather than expected frequencies based on ML estimates, increases the power and improves thc x ² fit to the true significance level. The small sample powers are also compared to the asymptotic (Pitman) pourer, based on the noncenlral x ² distribution.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Likelihood ratio test for independence with partial multivariate normal data

This article derives the likelihood ratio statistic to test the independence between (X ₁,…,X _r ) and (X _r+1,…,X _k ) under the assumption that (X ₁,…,X _k ) has a multivariate normal distribution and that a sample of size n is available, where for N observation vectors all components are available, while for M = (n + N) observation vectors, the data on the last q components, (X_k-q+1,…,X _k ) are missing (k+q≥r).     

Precedence tests and Lehmann alternatives

G = F ^k (k > 1); G = 1 − (1−F) ^k (k < 1); G = F ^k (k < 1); and G = 1 − (1−F) ^k (k > 1), where F and G are two continuous cumulative distribution functions. If an optimal precedence test (one with the maximal power) is determined for one of these four classes, the optimal tests for the other classes of alternatives can be derived. Application of this is given using the results of Lin and Sukhatme (1992) who derived the best precedence test for testing the null hypothesis that the lifetimes of two types of items on test have the same distibution. The test has maximum power for fixed κ in the class of alternatives G = 1 − (1−F) ^k , with k < 1. Best precedence tests for the other three classes of Lehmann-type alternatives are derived using their results. Finally, a comparison of precedence tests with Wilcoxon's two-sample test is presented. Received: February 22, 1999; revised version: June 7, 2000     

On Testing the Mean Equivalence of Treatments from Correlated Normal Populations

Let there be k equally correlated treatment populations under consideration, a Studentized range test is proposed to test the hypothesis of average mean equivalence against the alternative hypothesis of inequivalence. The maximum level and minimum power at some least favorable configurations of means are used to calculate the critical value and the required sample size simultaneously when testing a null against an alternative hypothesis. The range test is applied to a real world problem to find out if the stress levels among children at four time periods due to a newly built nearby airport are equivalent.