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Asymptotic inversion of the incomplete beta function. (English) Zbl 0755.33002
From the author’s summary: The normalized incomplete beta function $$I\sb x(a,b)={1\over B(a,b)}\int\sb 0\sp x t\sp{a-1}(1-t)\sp{b-1}dt$$ is inverted for large values of $a$ and $b$. The approximations are obtained by using uniform asymptotic expansions of the incomplete beta function, in which an error function or an incomplete gamma function is the dominant term. The inversion problem starts by inverting this dominant term and further terms in the expansion are obtained by using standard perturbation methods. Numerical results indicate that the asymptotic method can already be used for $a+b\geq 5$ for an accuracy of four correct digits.

MSC:
 33B20 Incomplete beta and gamma functions 30E10 Approximation in the complex domain 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text:
References:
 [1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions with formulas, graphs and mathematical tables. Nat. bur. Standards appl. Ser. 55 (1964) · Zbl 0171.38503 [2] Temme, N. M.: The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions. SIAM J. Math. anal. 13, 239-253 (1982) · Zbl 0489.41031 [3] Temme, N. M.: Incomplete Laplace integrals: uniform asymptotic expansions with application to the incomplete beta function. SIAM J. Math. anal. 18, 1638-1663 (1987) · Zbl 0641.33002 [4] N.M. Temme, Asymptotic inversion of incomplete gamma functions, Math. Comp., to appear. · Zbl 0759.33001