Bounds on iterated coerror functions and their ratios

Author:
D. E. Amos

Journal:
Math. Comp. **27** (1973), 413-427

MSC:
Primary 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1973-0331723-2

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.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**D. F. Barrow and A. C. Cohen Jr.,*On some functions involving Mill’s ratio*, Ann. Math. Statistics**25**(1954), 405–408. MR**0061319****[3 Z]**Z. W. Birnbaum,*An inequality for Mill’s ratio*, Ann. Math. Statistics**13**(1942), 245–246. MR**0006640****[4]**A. V. Boyd,*Inequalities for Mills’ ratio*, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs.**6**(1959), 44–46 (1959). MR**0118856****[5]**Walter Gautschi,*Computational aspects of three-term recurrence relations*, SIAM Rev.**9**(1967), 24–82. MR**0213062**, https://doi.org/10.1137/1009002**[6]**Walter Gautschi,*Recursive computation of the repeated integrals of the error function.*, Math. Comp.**15**(1961), 227–232. MR**0136074**, https://doi.org/10.1090/S0025-5718-1961-0136074-9**[7]**Robert 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. Statistics**12**(1941), 364–366. MR**0005558****[8]**Yûsaku Komatu,*Elementary inequalities for Mills’ ratio*, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs.**4**(1955), 69–70. MR**0079844****[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 Yûsaku Komatu*, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs.**4**(1956), 110. MR**0083529****[11]**W. D. Ray and A. E. N. T. Pitman,*Chebyshev polynomial and other new approximations to Mills’ ratio*, Ann. Math. Statist.**34**(1963), 892–902. MR**0153101**, https://doi.org/10.1214/aoms/1177704012**[12]**Harold Ruben,*A convergent asymptotic expansion for Mill’s ratio and the normal probability integral in terms of rational functions*, Math. Ann.**151**(1963), 355–364. MR**0155374**, https://doi.org/10.1007/BF01470826**[13]**Harold Ruben,*A new asymptotic expansion for the normal probability integral and Mill’s ratio*, J. Roy. Statist. Soc. Ser. B**24**(1962), 177–179. MR**0139226****[14]**Harold Ruben,*Irrational fraction approximations to Mills’ ratio*, Biometrika**51**(1964), 339–345. MR**0173306**, https://doi.org/10.1093/biomet/51.3-4.339**[15]**M. R. Sampford,*Some inequalities on Mill’s ratio and related functions*, Ann. Math. Statistics**24**(1953), 130–132. MR**0054890****[16]**L. R. Shenton,*Inequalities for the normal integral including a new continued fraction*, Biometrika**41**(1954), 177–189. MR**0061785**, https://doi.org/10.1093/biomet/41.1-2.177**[17]**Robert F. Tate,*On a double inequality of the normal distribution*, Ann. Math. Statistics**24**(1953), 132–134. MR**0054891****[18]**V. R. Rao Uppuluri,*A stronger version of Gautschi’s inequality satisfied by the gamma function*, Skand. Aktuarietidskr**1964**(1964), 51–52 (1965). MR**0180703**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0331723-2

Keywords:
Iterated coerror function,
error function,
coerror function,
Mill's ratio,
probability integral

Article copyright:
© Copyright 1973
American Mathematical Society