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A convergent asymptotic expansion for Mill’s ratio and the normal probability integral in terms of rational functions. (English) Zbl 0116.11503


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statistics
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References:

[1] Barrow, D. F., andA. C. Cohen, jr.: On some functions involving Mill’s ratio. Ann. Math. Stat.25, 405-408 (1954). · Zbl 0055.37401 · doi:10.1214/aoms/1177728801
[2] Birnbaum, Z. W.: An inequality for Mill’s ratio. Ann. Math. Stat.13, 245-246 (1942). · Zbl 0060.29607 · doi:10.1214/aoms/1177731611
[3] ?? Effect of linear truncation on a multinormal population. Ann. Math. Stat.21, 272-279 (1950). · Zbl 0038.09201 · doi:10.1214/aoms/1177729844
[4] Boyd, A. V.: Inequalities for Mill’s ratio. Rep. Stat. App. Res. Union of Japanese Scientists and Engineers.6, 44-46 (1959).
[5] Bromwich, T. J.: Theory of infinite series. London: Macmillan 1926. · JFM 52.0208.05
[6] Carleman, T.: Les fonctions quasi-analytiques. Paris: Gauthier-Villars 1926. · JFM 52.0255.02
[7] Franklin, Joel, andBernard Friedman: A convergent asymptotic representation for integrals. Proc. Camb. Phil. Soc.53, 612-619 (1957). · Zbl 0081.29103 · doi:10.1017/S0305004100032667
[8] Gordon, Robert T.: Values of Mill’s ratio of area to bounding ordinate of the normal probability integral for large values of the argument. Ann. Math. Stat.12, 364-366 (1941). · Zbl 0026.33201 · doi:10.1214/aoms/1177731721
[9] Haldane, J. B. S.: Simple approximations to the probability integral andP(x 2, 1) when both are small. Sankhya, A,23, 9-10 (1961). · Zbl 0095.33801
[10] Kendall, M. G., andA. Stuart: The advanced theory of statistics. Vol. 1, London: Charles Griffin and Co. 1958. · Zbl 0416.62001
[11] Laplace, P. S.: Traité de mécanique céleste. t.3, livre 10, Paris 1802.
[12] – Théorie analytique des probabilités. Paris 1812.
[13] Murty, V. N.: On a result of Birnbaum regarding the skewness ofX in a bivariate normal population. J. Indian Soc. Agric. Stat.4, 85-87 (1952).
[14] Pólya, G.: Remarks on computing the probability integral in one and two dimensions. Proc. Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: Univ. of California Press 1949. · Zbl 0040.07203
[15] Ruben, Harold: Probability content of regions under spherical normal distributions, III: The bivariate normal integral. Ann. Math. Stat.32, 171-186 (1961). · doi:10.1214/aoms/1177705149
[16] ?? A new asymptotic expansion for the normal probability integral and Mill’s ratio. J. Roy. Stat. Soc., Ser. B,24, 177-179 (1962). · Zbl 0109.13301
[17] Sampford, M. R.: Some inequalities on Mill’s ratio and related functions. Ann. Math. Stat.24, 130-132 (1953). · Zbl 0050.13503 · doi:10.1214/aoms/1177729093
[18] Shenton, L. R.: Inequalities for the normal integral including a new continued fraction. Biometrika41, 177-189 (1954). · Zbl 0056.12002
[19] Sheppard, W. F.: The probability integral. British Assn. Math. Tables 7. Cambridge University Press, 1939. · Zbl 0021.33705
[20] Stieltjes, S.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse.8, J, 1-122;9, A, 1-47; Oeuvres,2, 402-566 (1894). [See also Mémoires Présentes Par Divers Savantes à l’Académie des Sciences de l’Institut National de France33, 1-196 (1894).]
[21] Tate, Robert F.: On a double inequality of the normal distribution. Ann. Math. Stat.24, 132-135 (1953). · Zbl 0050.13504 · doi:10.1214/aoms/1177729094
[22] Wall, H. S.: Analytic theory of continued fractions. New York: D. van Nostrand Co. 1948. · Zbl 0035.03601
[23] Watson, G. N.: Bessel functions. Cambridge University Press 1922. · JFM 48.0412.02
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