An Adjustment to the Bartlett's Test for Small Sample Size

The Bartlett's test (1937) for equality of variances is based on the χ² distribution approximation. This approximation deteriorates either when the sample size is small (particularly < 4) or when the population number is large. According to a simulation investigation, we find a similar varying trend for the mean differences between empirical distributions of Bartlett's statistics and their χ² approximations. By using the mean differences to represent the distribution departures, a simple adjustment approach on the Bartlett's statistic is proposed on the basis of equal mean principle. The performance before and after adjustment is extensively investigated under equal and unequal sample sizes, with number of populations varying from 3 to 100. Compared with the traditional Bartlett's statistic, the adjusted statistic is distributed more closely to χ² distribution, for homogeneity samples from normal populations. The type I error is well controlled and the power is a little higher after adjustment. In conclusion, the adjustment has good control on the type I error and higher power, and thus is recommended for small samples and large population number when underlying distribution is normal.     

Energy statistics: A class of statistics based on distances

Energy distance is a statistical distance between the distributions of random vectors, which characterizes equality of distributions. The name energy derives from Newton's gravitational potential energy, and there is an elegant relation to the notion of potential energy between statistical observations. Energy statistics are functions of distances between statistical observations in metric spaces. Thus even if the observations are complex objects, like functions, one can use their real valued nonnegative distances for inference. Theory and application of energy statistics are discussed and illustrated. Finally, we explore the notion of potential and kinetic energy of goodness-of-fit.     

Methods to test for equality of two normal distributions

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

ROBUST ESTIMATION AND HYPOTHESIS TESTS FOR FIRST-ORDER THRESHOLD AUTOREGRESSIVE MODELS

Results of Petrucelli & Woolford (1984) for a first-order threshold autoregressive model are considered from a robust point of view. Robust estimators of the threshold parameters of the model are obtained and their asymptotic normality is proved. Testing the equality of the threshold parameters is considered using the robust analogues of Wald and score test statistics. Limiting distributions of these statistics are given under both null and alternative hypotheses.     

NONPARAMETRIC TWO SAMPLE TESTS BASED ON AREA STATISTICS

ABSTRACT

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions against inequality or testing stochastic dominance of distributions in one or either direction against nondominance. Though three of the statistics considered have already been suggested in literature, two of them are new and deserve our interest. The finite sample distributions of the statistics (under F=G) can be calculated via recursion formulae. Two tables with critical values of the new statistics are included. The asymptotic distribution of the properly normalized versions of the area statistics are functionals of the Brownian bridge. The distribution functions and quantiles thereof are obtained by Monte Carlo simulation. Finally, the power functions of the two new tests based on area statistics are compared to the power functions of the tests based on the corresponding supremum statistics, i.e., statistics of the Kolmogorov–Smirnov type.     

Tests for the multivariate two-sample problem based on empirical probability measures

Nonparametric tests are proposed for the equality of two unknown p-variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution-free in the general p-variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techniques.     

EXACT DISTRIBUTION OF CERTAIN GENERAL TEST STATISTICS IN MULTIVARIATE ANALYSIS

In this paper, exact solution of Wilks' type-B integral equation has been obtained in its most general form as a series of weighted gamma distributions. This general result then gives the distributions of many test statistics in multivariate analysis. In particular the distributions of Wilks' Λ, the sphericity test criterion, and Bartlett's test statistic, are derived in easily computable form.     

Asymptotic expansions of the null distributions of some test statistics for profile analysis in general distributions

This paper discusses asymptotic expansions for the null distributions of some test statistics for profile analysis under non-normality. It is known that the null distributions of these statistics converge to chi-square distribution under normality [Siotani, M., 1956. On the distributions of the Hotelling's T²$T^2$-statistics. Ann. Inst. Statist. Math. Tokyo 8, 1–14; Siotani, M., 1971. An asymptotic expansion of the non-null distributions of Hotelling's generalized T²$T^2$-statistic. Ann. Math. Statist. 42, 560–571]. We extend this result by obtaining asymptotic expansions under general distributions. Moreover, the effect of non-normality is also considered. In order to obtain all the results, we make use of matrix manipulations such as direct products and symmetric tensor, rather than usual elementwise tensor notation.     

Divergence statistics: sampling properties and multinomial goodness of fit and divergence tests

φ-divergence .statistics are obtained by either replacing both distributions involved in the argument of the φ -divergence measure by their sample estimates or replacing one distribution and considering the other as given. The sampling properties of estimated divergence-type measures are investigated. Approximate means and variances are derived and asymptotic distributions are obtained. Tests of goodness of fit of observed frequencies to expected ones and tests of equality of divergences based on two or more multinomial samples are constructed.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

On the distribution of lrt for testing the equality of exponential distributions

In this paper, an asymptotic expansion of the distribution' of the likelihood ratio criterion for testing the equality of p one-parameter exponential distributions is obtained for unequal sample sizes. The expansion is obtained up to the order of n^-3 with the second term of the order of n^-2 so that the first term of this expansion alone should provide an excellent approximation to the distribution for moderately large values of n, where n is the combined sample size.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

Precedence tests for equality of two distributions based on early failures of ranked set samples

In this article, two different types of precedence tests, each with two different test statistics, based on ranked set samples for testing the equality of two distributions are discussed. The exact null distributions of proposed test statistics are derived, critical values are tabulated for both set size and number of cycles up to 8, and the exact power functions of these two types of precedence tests under the Lehmann alternative are derived. Then, the power values of these two test procedures and their competitors based on simple random samples and based on ranked set samples are compared under the Lehmann alternative exactly and also under a location-shift alternative by means of Monte Carlo simulations. Finally, the impact of imperfect ranking is discussed and some concluding remarks are presented.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Behavior of agreement measures in the presence of zero cells and biased marginal distributions

Kappa and B assess agreement between two observers independently classifying N units into k categories. We study their behavior under zero cells in the contingency table and unbalanced asymmetric marginal distributions. Zero cells arise when a cross-classification is never endorsed by both observers; biased marginal distributions occur when some categories are preferred differently between the observers. Simulations studied the distributions of the unweighted and weighted statistics for k=4, under fixed proportions of diagonal agreement and different patterns off-diagonal, with various sample sizes, and under various zero cell count scenarios. Marginal distributions were first uniform and homogeneous, and then unbalanced asymmetric distributions. Results for unweighted kappa and B statistics were comparable to work of Muñoz and Bangdiwala, even with zero cells. A slight increased variation was observed as the sample size decreased. Weighted statistics did show greater variation as the number of zero cells increased, with weighted kappa increasing substantially more than weighted B. Under biased marginal distributions, weighted kappa with Cicchetti weights were higher than with squared weights. Both statistics for observer agreement behaved well under zero cells. The weighted B was less variable than the weighted kappa under similar circumstances and different weights. In general, B's performance and graphical interpretation make it preferable to kappa under the studied scenarios.     

The asymptotic distributions of maximum likelihood ratio test and maximally selected χ2-test in binomial observations

We considered binomial distributed random variables whose parameters are unknown and some of those parameters need to be estimated. We studied the maximum likelihood ratio test and the maximally selected χ²-test to detect if there is a change in the distributions among the random variables. Their limit distributions under the null hypothesis and their asymptotic distributions under the alternative hypothesis were obtained when the number of the observations is fixed. We discussed the properties of the limit distribution and found an efficient way to calculate the probability of multivariate normal random variables. Finally, those results for both tests have been applied to examples of Lindisfarne's data, the Talipes Data. Our conclusions are consistent with other researchers' findings.     

Improved bonferroni procedures for testing overall and pairwise homogeneity hypotheses

Simes' (1986) improved Bonferroni test is verified by simulations ^?to control the α-level when testing the overall homogeneity hypothesis with all pairwise t statistics in a balanced parallel group design. Similarly, this result was found to hold (for practical purposes) in various underlying distributions other than the normal and in some unbalanced designs. To allow the use of step-up procedures based on pairwise t statistics, simulations were used to verify that Simes' test, when applied to testing multiple subset homogeneity hypotheses with pairwise t statistics also keeps the level ? α. Some robustness as above was found here too. Tables of the simulation results are provided and an example of a step-up Hommel-Shaffer type procedure with pairwise comparisons is given.     

A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.