Student t-statistic distribution for non-Gaussian populations
João Paulo Martins · Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) · 2010
The exact distribution of t(n-1) = √n X n[-μ/Sn is easily derived when the parent population is Gau (μ, σ), since the sample mean Xn and sample standard deviationSn are independent. However this is an exceptional situation, since, the. independence of Xn and S2 n is a characterization of the Gaussian populations. When Y isn't Gaussian, the exact distribution of Tn-1 = √n Y n-μ/Sn 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 Tn-1 = √n Y-μ/Sn 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 Tn-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 Xn and S2n suggests a strong relation between the population skewness and the. attraction or repulsion behaviour between Xn and S2n. To support this statement some, simulation work is done.