Bounds on iterated coerror functions and their ratios
Author:
D. E. Amos
Journal:
Math. Comp. 27 (1973), 413427
MSC:
Primary 65D20
MathSciNet review:
0331723
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Abstract: Upper and lower bounds on and , are established in terms of elementary functions. Numerical procedures for refining these bounds are presented so that and , can be computed to a specified accuracy. Some relations establishing bounds on and are also derived.
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 [4]
 A. V. Boyd, "Inequalities for Mills' ratio," Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., v. 6, 1959, pp. 4446. MR 22 #9625. MR 0118856 (22:9625)
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 W. Gautschi, "Computational aspects of threeterm recurrence relations," SIAM Rev., v. 9, 1967, pp. 2482. MR 35 #3927. MR 0213062 (35:3927)
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 W. Gautschi, "Recursive computation of the repeated integrals of the error function," Math. Comp., v. 15, 1961, pp. 227232. MR 24 #B2113. MR 0136074 (24:B2113)
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 R. D. Gordon, "Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument," Ann. Math. Statist., v. 12, 1941, pp. 364366. MR 3, 171. MR 0005558 (3:171e)
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 Y. Komatu, "Elementary inequalities for Mills' ratio," Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., v. 4, 1955, pp. 3334. MR 0079844 (18:155f)
 [9]
 K. B. Oldham, "Approximations for the function," Math. Comp., v. 22, 1968, p. 454.
 [10]
 H. O. Pollak, "A remark on "Elementary inequalities for Mills' ratio," by Y. Komatu," Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 4, 1956, p. 40. MR 18, 722. MR 0083529 (18:722a)
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 W. D. Ray & A. E. N. T. Pitman, "Chebyshev polynomial and other new approximations to Mills' ratio," Ann. Math. Statist., v. 34, 1963, pp. 892902. MR 27 #3070. MR 0153101 (27:3070)
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 H. Ruben, "A convergent asymptotic expansion for Mills' ratio and the normal probability integral in terms of rational functions," Math. Ann., v. 151, 1963, pp. 355364. MR 27 #5308. MR 0155374 (27:5308)
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 H. Ruben, "A new asymptotic expansion for the normal probability integral and Mills' ratio," J. Roy. Statist. Soc. Ser. B, v. 24, 1962, pp. 177179. MR 25 #2662. MR 0139226 (25:2662)
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 H. Ruben, "Irrational fraction approximations to Mills' ratio," Biometrika, v. 51, 1964, pp. 339345. MR 30 #3519. MR 0173306 (30:3519)
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 M. R. Sampford, "Some inequalities on Mills' ratio and related functions," Ann. Math. Statist., v. 24, 1953, pp. 130132. MR 14, 995. MR 0054890 (14:995g)
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 L. R. Shenton, "Inequalities for the normal integral including a new continued fraction," Biometrika, v. 41, 1954, pp. 177189. MR 15, 884. MR 0061785 (15:884e)
 [17]
 R. F. Tate, "On a double inequality of the normal distribution," Ann. Math. Statist., v. 24, 1953, pp. 132134. MR 14, 995. MR 0054891 (14:995h)
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 V. R. R. Uppuluri, "A stronger version of Gautschi's inequality satisfied by the gamma function," Skand. Aktuarietidskr. v. 12, 1964, pp. 5152. MR 31 #4934. MR 0180703 (31:4934)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197303317232
PII:
S 00255718(1973)03317232
Keywords:
Iterated coerror function,
error function,
coerror function,
Mill's ratio,
probability integral
Article copyright:
© Copyright 1973
American Mathematical Society
