Finite-sample distributions of self-normalised sums

(Logan, Mallows, Rice & Shepp 1973) analyse the limit probability distribution of the statistic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${S_n}\left( p \right)=\sum\nolimits_{i=1} {{X_i}/{{\left( {{{\sum\nolimits_{i=1} {|{X_j}|} }^p}} \right)}^{1/p}}}$</EquationSource> </InlineEquation> as n → ∞, when X{ini} is in the domain of attraction of a stable law with stability index α. By simulations, we provide quantiles of the usual critical levels of the finite-sample distributions of the Student\rss t-statistic as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$_\alpha {t_\xi }\left( n \right)={S_n}\left( p \right){\left[ {\left( {n - 1} \right)/\left( {n - S_n^2\left( p \right)} \right)} \right]^{1/2}}$</EquationSource> </InlineEquation> with p=2. The response surface method is used to provide approximate quantiles of the finite-sample distributions of the Student’s t-statistic. Copyright Physica-Verlag 2003