Student's tt-test for scale mixture errors

Generalized t-tests are constructed under weaker than normal conditions. In the first part of this paper we assume only the symmetry (around zero) of the error distribution (i). In the second part we assume that the error distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical values can be computed from generalizations of Student's cumulative distribution function (cdf), $t_n(x)$. The cdf's of the generalized $t$-test statistics are denoted by (i) $t_n^S(x)$ and (ii) $t_n^G(x)$, resp. As the sample size $n\to \infty$ we get the counterparts of the standard normal cdf $\Phi(x)$: (i) $\Phi^S(x):=\operatorname {lim}_{n\to \infty}t_n^S(x)$, and (ii) $\Phi^G(x):=\operatorname {lim}_{n\to \infty}t_n^G(x)$. Explicit formulae are given for the underlying new cdf's. For example $\Phi^G(x)=\Phi(x)$ iff $|x|\ge \sqrt{3}$. Thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of $\Phi^G$ is $4\sqrt{3}/5=1.385... >\Phi^{-1}(0.9)=1.282...$.Comment: Published at http://dx.doi.org/10.1214/074921706000000365 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

DISCO analysis: A nonparametric extension of analysis of variance

In classical analysis of variance, dispersion is measured by considering squared distances of sample elements from the sample mean. We consider a measure of dispersion for univariate or multivariate response based on all pairwise distances between-sample elements, and derive an analogous distance components (DISCO) decomposition for powers of distance in $(0,2]$. The ANOVA F statistic is obtained when the index (exponent) is 2. For each index in $(0,2)$, this decomposition determines a nonparametric test for the multi-sample hypothesis of equal distributions that is statistically consistent against general alternatives.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS245 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org