Symmetry, Asymmetry and Studentized Statistics

Inferences on the location parameter $\lambda$ in location-scale families can be carried out using Studentized statistics, i.e., considering estimators $\tilde{\lambda }$ of $\lambda$ and $\tilde{\delta }$ of the nuisance scale parameter $\delta$, in a statistic $T=g(\tilde{\lambda },\tilde{\delta })$ witha sampling distribution that does not depend on $(\lambda ,\delta )$. If both estimators are independent, then T is an externally Studentized statistic; otherwise, it is an internally Studentized statistic. For the Gaussian and for the exponential location-scale families, there are externally Studentized statistics with sampling distributions that are easy to obtain: in the Gaussian case, Student’s classic t statistic, since the sample mean $\tilde{\lambda }=\overline{X}$ and the sample standard deviation $\tilde{\delta }=S$ are independent; in the exponential case, the sample minimum $\tilde{\lambda }=X_{1:n}$ and the sample range $\tilde{\delta }=X_{n:n}-X_{1:n}$, where the latter is a dispersion estimator, which are independent due to the independence of spacings. However, obtaining the exact distribution of Student’s statistic in non-Gaussian populations is hard, but the consequences of assuming symmetry for the parent distribution to obtain approximations allow us to determine if Student’s statistic is conservative or liberal. Moreover, examples of external and internal Studentizations in the asymmetric exponential population are given, and an ANalysis Of Spacings (ANOSp) similar to an ANOVA in Gaussian populations is also presented.