ON LAPLACE CONTINUED FRACTION FOR THE NORMAL INTEGRAL

CHU-IN CHARLES LEE

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7

(Received August 10, 1989; revised November 26, 1990)

Abstract. The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability Φ̄(x), it is simple, it converges for x > 0, and it is by far the best approximation for x ≥ 3. The Laplace continued fraction is rederived as an extreme case of admissible bounds of the Mills’ ratio, Φ̄(x)/ϕ(x), in the family of ratios of two polynomials subject to a monotone decreasing absolute error. However, it is not optimal at any finite x. Convergence at the origin and local optimality of a subclass of admissible bounds are investigated. A modified continued fraction is proposed. It is the sharpest tail bound of the Mills’ ratio, it has a satisfactory convergence rate for x ≥ 1 and it is recommended for the entire range of x if a maximum absolute error of 10^{−4} is required.

Key words and phrases: Admissibility, approximation, convergence, Mills’ ratio, optimality, rational bound.