## Institute of Mathematical Statistics Lecture Notes - Monograph Series

### Student’s t-test for scale mixture errors

Gábor J. Székely

#### Abstract

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 $4\sqrt3/5 = 1.385\dots > \Phi^{-1}(0.9)=1.282\dots$.

#### Chapter information

Source
Javier Rojo, ed., Optimality: The Second Erich L. Lehmann Symposium (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 9-15

Dates
First available in Project Euclid: 28 November 2007

http://projecteuclid.org/euclid.lnms/1196283952

Digital Object Identifier
doi:10.1214/074921706000000365

Subjects
Primary: 62F03: Hypothesis testing 62E99: None of the above, but in this section
Secondary: 62F04

Rights

#### Citation

Székely, Gábor J. Student’s t-test for scale mixture errors. Optimality, 9--15, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000365. http://projecteuclid.org/euclid.lnms/1196283952.

#### Subjects

Primary 62F03 62E99

Secondary 62F04

#### Keywords

generalized $t$-tests; symmetric errors; Gaussian scale mixture errors