Simple approximations to the behrens—fisher distribution

Dayal and Dickey (1977) have published in this journal a rather efficient numerical integration procedure for the product of k Student t-densities, and point out the evaluation of Behrens-Fisher (BF) densities as an important special case. The present note adds to this three simple normal approximations to Behrens-Fisher tail probabilities, that will save computer time for someone using the Dayal-Dickey results, and even allow evaluation on a desk calculator for moderately large degrees of freedom. A direct normal approximation (method U) will be too coarse unless both degrees of freedom are large. A combination of the Peizer-Pratt (1968) approximation to the t-distribution and the Patil (1965) t-approximation to the BF distribution turns out to be very accurate. For very small degrees of freedom it may still be refined by an adhoc correction presented below. Other approximations and expansions turn out to be less satisfactory than the present trio. It facilitates a quick evaluation of BF probabilities and quantiles on a small computer or even a pocket calculator.