On the Distribution of the Weighted Difference of Two Independent Student Variables

Summary The distribution of the d-statistic, the weighted difference of two independent t-variables, is first obtained in integral form. This allows d to be interpreted as the ratio of two independent random variables, the numerator of which is a Student variable and the denominator a function of a Beta variable. It is suggested that the simplicity of the above integral form should enable extensive computation and tabulation of the d-statistic to be effected. Explicit forms for the distribution of d as finite or infinite series involving Incomplete Beta Function Ratios and Incomplete Gamma Function Ratios are derived for the special cases when (i) the degrees of freedom and weights of the component variables are equal and (ii) one of the components is normal.