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ON LAPLACE CONTINUED FRACTION FOR

THE NORMAL INTEGRAL

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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
\bar *Phi*(*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, \bar *Phi*(*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* __>__ 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.

**Source**
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