Polar generation of random variates with the t -distribution

The "polar" method of Box and Muller uses two independent uniform variates in order to generate two independent normal variates. It can be adapted so that two variates from Student's t-distribution with parameter v are generated, though the two variates are now not independent. An algorithm based on the polar method is exact, inexpensive, and valid for all v > 0. Box and Muller's [1] polar method for generating random normal variates relies on two convenient properties of the normal distribution, which we may formulate as follows: (i) Let X N(O, 1). Then X can be regarded as the real part of a complex random variable Z which has a radial distribution (the contours of the density function of Z form circles centered at the origin); (ii) Write Z = X+ iY = Reie9. Then the distribution function FR (= 1GR) of R is a simple algebraic expression, so simple that it is invertible; that is, given G GR(r), we can write down a closed expression for r in term of G. The aim of this article is to show that properties (i) and (ii) are shared by the Student t-distribution with parameter v (the tv-distribution) defined by the density (1) fT(x) = B(v/2, 1/2)-i . v-1/2. (1 + x2/v)-(v+l)/2 Thus, we are asserting that the tv-distribution, like the normal, has a tractable radial parent. If T has the density (1), we shall write " T tV ". Many methods have been proposed for the generation of tv-variates. The most important ones are described in Devroye [2, pp. 445-450], whose masterly survey we shall not attempt to emulate. The faster algorithms may require either a comparatively great programming effort, or the expensive recalculation of certain quantities required by the algorithm, whenever v is changed. As Devroye notes, problems arise when v is small and the departure from normality is greatest, particularly in the region 0 0. We now show how the polar method may be applied to the tv-distribution to yield such an algorithm. Received by the editor December 23, 1992 and, in revised form, May 12, 1993. 1991 Mathematics Subject Classification. Primary 65C10; Secondary 62E15.