Abstract
The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](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, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. A. (eds.) (1972). Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C.
Birnbaum, Z. W. (1942). An inequality for Mills' ratio. Ann. Math. Statist., 13, 245–246.
Boyd, A. V. (1959). Inequalities for Mills' ratio, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 6, 44–46.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, New York.
Gordon, R. D. (1941). Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the arqument, Ann. Math. Statist., 12, 364–366.
Gray, H. L. and Schucany, W. R. (1968). On the evaluation of distribution functions, J. Amer. Statist. Assoc., 63, 715–720.
Gross, A. J. and Hosmer, D. W. (1978). Approximating tail areas of probability distributions Ann. Statist., 6, 1352–1359.
Hastings, C. (1955). Approximations for Digital Computers, Princeton University Press, New Jersey.
Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, Vol. 1, 4th ed., MacMillan, New York.
Kerridge, D. F. and Cook, G. W. (1976). Yet another series for the normal integral, Biometrika, 63, 401–403.
Komatu, Y. (1955). Elementary inequalities for Mills' ratio, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 4, 69–70.
Laplace, P. S. (1785). Memóire sur les approximations des formules qui sont fonction de trèsgrands nombres, 1–88, Histoire de l'Académie Royale des Sciences de Paris.
Laplace, P. S. (1805). Traité de Mécanique Celeste, Vol. 4, Courcier, Paris.
Laplace, P. S. (1812). Théorie Analytique de Probabilitiés, Vol. 2, Courcier, Paris.
Mitrinović, D. S. (1970). Analytic Inequalities, Springer, New York.
Patel, J. K. and Read, C. B. (1982). Handbook of the Normal Distribution, Dekker, New York.
Pollak, H. O. (1956). A remark on “Elementary inequalities for Mills' ratio” by Y. Komatu, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 4, p. 110.
Pólya, G. (1949). Remarks on computing the probability integral in one and two dimensions, Proc. First Berkeley Symp. on Math. Statist. Prob., 63–78, Univ. of California Press, Berkeley.
Ruben, H. (1963). A convergent asymptotic expansion for Mills' ratio and the normal probability integral in terms of rational functions, Math. Ann., 151, 355–364.
Ruben, H. (1964). Irrational fraction approximations to Mills' ratio, Biometrika, 51, 339–345.
Sampford, M. R. (1953). Some inequalities on Mills' ratio and related functions, Ann. Math. Statist., 24, 130–132.
Shenton, L. R. (1954). Inequalities for the normal integral including a new continued fraction, Biometrika, 41, 177–189.
Sheppard, W. F. (1939). The Probability Integral, British Association for the Advancement of Science, Mathematical Tables VII, Cambridge University Press, U.K.
Author information
Authors and Affiliations
Additional information
The efforts of the author were supported by the NSERC of Canada.
About this article
Cite this article
Lee, CI.C. On laplace continued fraction for the normal integral. Ann Inst Stat Math 44, 107–120 (1992). https://doi.org/10.1007/BF00048673
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00048673