Distribution of Student’s Ratio Based on Half-Gaussian Population

Abstract

For a sample of size n drawn from a half-Gaussian population with location parameter μ and scale parameter σ, let $T\, = \,\sqrt n (\bar X - \mu )/S$ where $\bar X$ is the sample mean and S, the sample standard deviation given by $(n - 1)S^2 \, = \,\sum {(X_i - \bar X)^2 }$. The tail probability $G_n (t)\, = \,\{ T \geqslant t\}$ and the density $f_n (t)\, = \, - dG_n (t)/dt$ are obtained for values of $t\, \geqslant \,\left( {(n - 1)(n - 2)/2} \right)^{1/2}$. For the special cases n = 2,3,4 explicit expressions for G_n(t) and f_n(t) are obtained. For small n and large t, asymptotic tail probabilities for T with Gaussian, half-Gaussian and exponential parent distributions are compared.