On the computation and inversion of the cumulative noncentral beta distribution function

Abstract

The computation and inversion of the noncentral beta distribution B p,q (x, y) (or the noncentral F -distribution, a particular case of B p,q (x, y)) play an important role in different applications. In this paper we study the stability of recursions satisfied by B p,q (x, y) and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding x or y when a value of B p,q (x, y) is given. We provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed.

A new asymptotic representation and inversion method for the Student's t distribution

“…We use the representation given in (4) and first try to find 𝜉, then 𝑥 follows from the relation in (3). We assume that 𝑛 is large and use the asymptotic method as described in [12, §42.1] and in our papers [4], [5], [6].…”

Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters

“…It features in the normalizing constant of the generalized inverse normal (see [20]) and the Kummer beta generalized (see [15] and references therein) distributions. In a recent paper [5], where the computation and inversion of the cumulative noncentral beta distribution function is considered, the M -functions play a role in recurrence relations, series expansions and integral representations. At last, they appear in the theory of estimation and testing of the multiple correlation coefficient ([4, p. 671], [6]), of the t test ( [3]), of the simultaneous equations model ( [19], [17, p. 470]), as well as in other more special cases ( [7]).…”

New asymptotic representations of the noncentral t‐distribution

“…Remark 2. The function 𝑃 𝑛 (𝑥; 𝛿) introduced in (1) is a special case of the noncentral beta distribution that we considered in a recent paper 11 in the form…”

“…The integral of 𝑃 𝑛 (𝑥; 𝛿) in ( 11) also corresponds with a more general integral of the noncentral beta distribution; see [ 11 ,Equation (1.6)]. The Kummer functions in these integrals are related with parabolic cylinder functions, see [ 12 , Section 12.7(iv)].…”

“…For inverting the noncentral beta distribution and the Student's 𝑡-distribution, we refer to Refs. [ 11 ] and [6]. In [ 5 , Section 16], several approximations of percentage points in terms of elementary expressions are discussed in with comparisons from the literature factors in tables.…”