A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Robust tests for the equality of variances for clustered data

Tests for the equality of variances are often needed in applications. In genetic studies the assumption of equal variances of continuous traits, measured in identical and fraternal twins, is crucial for heritability analysis. To test the equality of variances of traits, which are non-normally distributed, Levene [H. Levene, Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed. Stanford University Press, Palo Alto, California, 1960, pp. 278–292] suggested a method that was surprisingly robust under non-normality, and the procedure was further improved by Brown and Forsythe [M.B. Brown and A.B. Forsythe, Robust tests for the equality of variances, J. Amer. Statis. Assoc. 69 (1974), pp. 364–367]. These tests assumed independence of observations. However, twin data are clustered – observations within a twin pair may be dependent due to shared genes and environmental factors. Uncritical application of the tests of Brown and Forsythe to clustered data may result in much higher than nominal Type I error probabilities. To deal with clustering we developed an extended version of Levene's test, where the ANOVA step is replaced with a regression analysis followed by a Wald-type test based on a clustered version of the robust Huber–White sandwich estimator of the covariance matrix. We studied the properties of our procedure using simulated non-normal clustered data and obtained Type I error rates close to nominal as well as reasonable powers. We also applied our method to oral glucose tolerance test data obtained from a twin study of the metabolic syndrome and related components and compared the results with those produced by the traditional approaches.     

Smooth tests for the gamma distribution

The gamma distribution is often used to model data with right skewness. Smooth tests of goodness of fit are proposed for this distribution. Their powers are compared with powers of the Anderson–Darling test and tests based on the empirical Laplace transform, the empirical moment generating function and the independence of the mean and coefficient of variation that characterizes the gamma distribution.     

Discriminating between distributions using feed-forward neural networks

A relevant problem in many applicatory contexts is to test whether some given observations follow one of two possible probability distributions. The vast literature produced over the years on this topic does not identify a tool which can be easily adopted to any situation but only finds solutions to specific comparisons. Recently, an easy to implement procedure for discrimination between two distributions based on feed-forward neural networks has been proposed giving interesting results. In this work this procedure is further investigated in terms of power, neural network architecture and expected statistical properties of the test statistic for small, moderate and large sample sizes, in a wide range of symmetric and skewed alternatives.     

Estimating the Pareto parameters under progressive censoring data for constant-partially accelerated life tests

Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this paper, assuming that the lifetime of items under use condition follow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based on progressively Type-II censored samples are considered. The likelihood equations of the model parameters and the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtain the maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIs are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credible interval of the involved parameters. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and to compare the performance of different corresponding CIs considered.     

Non-parametric weighted tests for independence based on empirical copula process

We propose a class of flexible non-parametric tests for the presence of dependence between components of a random vector based on weighted Cramér–von Mises functionals of the empirical copula process. The weights act as a tuning parameter and are shown to significantly influence the power of the test, making it more sensitive to different types of dependence. Asymptotic properties of the test are stated in the general case, for an arbitrary bounded and integrable weighting function, and computational formulas for a number of weighted statistics are provided. Several issues relating to the choice of the weights are discussed, and a simulation study is conducted to investigate the power of the test under a variety of dependence alternatives. The greatest gain in power is found to occur when weights are set proportional to true deviations from independence copula.     

Multivariate multi-sample tests for location based on data depth

A notion of data depth is used to measure centrality or outlyingness of a data point in a given data cloud. In the context of data depth, the point (or points) having maximum depth is called as deepest point (or points). In the present work, we propose three multi-sample tests for testing equality of location parameters of multivariate populations by using the deepest point (or points). These tests can be considered as extensions of two-sample tests based on the deepest point (or points). The proposed tests are implemented through the idea of Fisher's permutation test. Performance of earlier tests is studied by simulation. Illustration with two real datasets is also provided.     

Numerical testing of evolution theories †

Fisher's method of combining independent tests is used to construct tests of means of multivariate normal populations when the covariance matrix has intraclass correlation structure. Monte Carlo studies are reported which show that the tests are more powerful than Hotelling's T ²-test in both one and two sample situations.     

Multivariate outlier tests with structured co variance matrices

Outlier tests are developed for multivariate data where there is a structure to the covariance or correlation matrix. Particular structures considered are the block diagonal structure where there are reasons to assume that one set of variables is independent of another, and the equicorrelation structure where it may be assumed that all pairs of variables have the same correlation. Likelihood ratio tests for an outlier are derived for these situations and critical values, under the null hypothesis of no outliers present, are determined for selected sample sizes and dimensions, using Bonferroni bounds or simulation. The powers of the tests are compared with those of the Wilks′ statistic for a variety of situations. It is shown that the test procedures which incorporate knowledge of the correlation structure have considerably greater power than the usual tests particularly in relatively small samples with several dimensions.     

Modelling interval data with Normal and Skew-Normal distributions

A parametric modelling for interval data is proposed, assuming a multivariate Normal or Skew-Normal distribution for the midpoints and log-ranges of the interval variables. The intrinsic nature of the interval variables leads to special structures of the variance–covariance matrix, which is represented by five different possible configurations. Maximum likelihood estimation for both models under all considered configurations is studied. The proposed modelling is then considered in the context of analysis of variance and multivariate analysis of variance testing. To access the behaviour of the proposed methodology, a simulation study is performed. The results show that, for medium or large sample sizes, tests have good power and their true significance level approaches nominal levels when the constraints assumed for the model are respected; however, for small samples, sizes close to nominal levels cannot be guaranteed. Applications to Chinese meteorological data in three different regions and to credit card usage variables for different card designations, illustrate the proposed methodology.