Normalizing transformations of Student's t distribution

Abstract

SUMMARY The accuracies of several normalizing transformations of a t distribution with v degrees of freedom are examined for large and small values of v. Expansions of the inverse transformations in terms of powers of v-' are produced and the first few terms compared with Fisher's general expansion. For small values of v the transformations are used to determine approximate percentage points of t which are then compared with the exact percentage points. The problem of transforming a Student's t variable with v degrees of freedom to a standard normal variable has received attention for a variety of reasons. Quenouille (1953, p. 235) was concerned with the combination of results from a series of experiments and suggested analysing the treatment effects using an inverse sinh transformation. This transformation is related to Fisher's normalizing transformation of the product moment correlation coefficient and is fairly good provided v is not too small. Anscombe (1950) proposed a modified form of this transformation which is suitable for smaller values of v. Chu (1956) examined the proportional errors in using the normal cumulative distribution function as an approximation to the cumulative distribution function of t and considered a square root of a logarithmic transformation. Modified versions of this were proposed by Wallace (1959) who constructed bounds on the deviation from the exact normal deviates. Moran (1966) used an empirical approach to develop a simple transformation of a particular percentage point of the t distribution and showed that the approximation is very good even for quite small values of v. Scott & Smith (1970) used the general expansion of the percentage points of the t distribution in terms of the corresponding normal percentage points, given by Fisher (1926), to develop a simple transformation similar in form to Moran's but suitable for any percentage point of t. In the following sections the accuracies of these transformations are compared for a selection of percentage points for large and small values of v.