Student t-statistic distribution for non-Gaussian populations

The exact distribution of t_(n-1)=√n(X̅_n-μ)/(S_n) is easily derived when the parent population is Gau (μ, σ), since the sample mean X̅_n and sample standard deviationS_n are independent. However this is an exceptional situation, since the independence of X̅_n and S_n² is a characterization of the Gaussian populations. When Y isn´t Gaussian, the exact distribution of T_n-1=√n(Y̅_n-μ)/(S_n) is difficult to compute, due to the dependence structure tying the sample mean and variance. Our aim has been to investigate, for general parent Y with known skewness and kurtosis, whether there exists one type in the Pearson system of distributions which better approximates T_n-1 = √n(Y̅-μ)/(S_n), in the specific sense that it provides better approximations to the high quantiles of T_n-1 than the corresponding quantiles of t_(n-1). We show that the T_n-1 distribution for general parent can be approximated by a Pearson´s type IV distribution, an unexpected result since Student´s t distributions is not of Pearson´s type IV. We also show that this new approximation is better because skewness is taken into account. In fact, the covariance between X̅n and S_n² suggests a strong relation between the population skewness and the attraction or repulsion behaviour between X̅n and S_n². To support this statement some simulation work is done.