A new simple straightforward and practical asymptotic expansion for the incomplete beta function

Abstract

It is shown that by using elementary transformations, the Incomplete Beta Function can be turned into a form suggesting that a close approximation by a normal integral can be obtained. By using standard methods this approximation is then developed in the form of an asymptotic series consisting of four exhibited correction terms to the main normal integral. An alternative means of deriving the third of these terms is shown confirming the correctness of the calculations (paradoxically the fourth correction term is easily derived and needs no such check). From this asymptotic series an asymptotic series for a corresponding standard normal variable is derived. The excellence of these approximations is exhibited by studying two limiting cases: that of equal indices (equivalent to a Student's distribution) and that of one index becoming infinite so that we have in place of an Incomplete Beta Function an Incomplete Gamma Function. Some similar recent results in the literature are commented on. Tables to show how excellent the expansion is, even down to parameter values as low as unity, are provided for a selection of parameter values and percentage points.