Bounds on iterated coerror functions and their ratios
HTML articles powered by AMS MathViewer
- by D. E. Amos PDF
- Math. Comp. 27 (1973), 413-427 Request permission
Abstract:
Upper and lower bounds on ${y_n} = {i^n}\;{\operatorname {erfc}}(x)$ and ${r_n} = {y_n}/{y_{n - 1}}, n \geqq 1, - \infty < x < \infty$, are established in terms of elementary functions. Numerical procedures for refining these bounds are presented so that ${r_n}$ and ${y_k},k = 0,1, \ldots ,n$, can be computed to a specified accuracy. Some relations establishing bounds on $r’_{n}$ and $r”_{n}$ are also derived.References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- D. F. Barrow and A. C. Cohen Jr., On some functions involving Mill’s ratio, Ann. Math. Statistics 25 (1954), 405–408. MR 61319, DOI 10.1214/aoms/1177728801
- Z. W. Birnbaum, An inequality for Mill’s ratio, Ann. Math. Statistics 13 (1942), 245–246. MR 6640, DOI 10.1214/aoms/1177731611
- A. V. Boyd, Inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 6 (1959), 44–46 (1959). MR 118856
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
- Walter Gautschi, Recursive computation of the repeated integrals of the error function, Math. Comp. 15 (1961), 227–232. MR 136074, DOI 10.1090/S0025-5718-1961-0136074-9
- 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 5558, DOI 10.1214/aoms/1177731721
- Yûsaku Komatu, Elementary inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 4 (1955), 69–70. MR 79844 K. B. Oldham, "Approximations for the $x \exp {x^2} \operatorname {erfc} x$ function," Math. Comp., v. 22, 1968, p. 454.
- H. O. Pollak, A remark on “Elementary inequalities for Mills’ ratio” by Yûsaku Komatu, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 4 (1956), 110. MR 83529
- 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 153101, DOI 10.1214/aoms/1177704012
- 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 155374, DOI 10.1007/BF01470826
- 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 139226
- Harold Ruben, Irrational fraction approximations to Mills’ ratio, Biometrika 51 (1964), 339–345. MR 173306, DOI 10.1093/biomet/51.3-4.339
- M. R. Sampford, Some inequalities on Mill’s ratio and related functions, Ann. Math. Statistics 24 (1953), 130–132. MR 54890, DOI 10.1214/aoms/1177729093
- L. R. Shenton, Inequalities for the normal integral including a new continued fraction, Biometrika 41 (1954), 177–189. MR 61785, DOI 10.1093/biomet/41.1-2.177
- Robert F. Tate, On a double inequality of the normal distribution, Ann. Math. Statistics 24 (1953), 132–134. MR 54891, DOI 10.1214/aoms/1177729094
- V. R. Rao Uppuluri, A stronger version of Gautschi’s inequality satisfied by the gamma function, Skand. Aktuarietidskr. 1964 (1964), 51–52 (1965). MR 180703, DOI 10.1080/03461238.1964.10413253
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 413-427
- MSC: Primary 65D20
- DOI: https://doi.org/10.1090/S0025-5718-1973-0331723-2
- MathSciNet review: 0331723