On laplace continued fraction for the normal integral

✍ Scribed by Chu-In Charles Lee

• Publisher: Springer Japan
• Year: 1992
• Tongue: English
• Weight: 655 KB
• Volume: 44
• Category: Article
• ISSN: 0020-3157
• DOI: 10.1007/bf00048673

✦ Synopsis

The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability (~(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, (P(x)/¢(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 -a is required.