Finite sample distributions of self-normalized sums (Q1887224)

Let \(X_i,i=1,\dots,n\), be independent, \(\alpha\)-stable random variables with \(1\leq \alpha\leq 2\) and \(\beta=0\). The Student-t statistics \[ {}_\alpha t(n)={}_\alpha S(n) \left( (n-1)/ (n-{}_\alpha S^2(n) \right)^{1/2} \] is considered with \({}_\alpha S(n)=\sum_{i=1}^n X_i/ \sqrt{\sum_{i=1}^n X_i^2}\). The author investigates quantiles of \({}_\alpha t(n)\) via simulations. Tables of such quantiles are given for \(n=1,\dots,30,\infty\), \(\alpha=1,1.1,\dots,2\), and levels \(0.9\), \(0.95\), \(0.99\). Approximate formulas for the differences between \({}_2 t(n)\) and \({}_\alpha t(n)\) quantiles are proposed.