ON LAPLACE CONTINUED-FRACTION FOR THE NORMAL INTEGRAL

The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability PHI-BAR(x), it is simple, it converges for x > 0, and it is by far the best approximation for x greater-than-or-equal-to 3. The Laplace continued fraction is rederived as an extreme case of admissible bounds of the Mills' ratio, PHI-BAR(x)/phi(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 greater-than-or-equal-to 1 and it is recommended for the entire range of x if a maximum absolute error of 10(-4) is required.