An asymptotic expansion of a beta-type integral and its application to probabilities of large deviations

Authors:
J. C. Fu and R. Wong

Journal:
Proc. Amer. Math. Soc. **79** (1980), 410-414

MSC:
Primary 60F10; Secondary 41A60, 62E20

DOI:
https://doi.org/10.1090/S0002-9939-1980-0567982-6

MathSciNet review:
567982

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An asymptotic expansion is obtained for an incomplete beta-type integral, which arises in the study of probabilities of large deviations. The expansion obtained yields large deviation results for binomial, quantile, and related probabilities. Our approach is based on a generalized version of Laplace's method.

**[1]**R. R. Bahadur,*Some approximations to the binomial distribution function*, Ann. Math. Statist.**31**(1960), 43-54. MR**0120675 (22:11424)****[2]**-,*Some limit theorems in statistics*, Regional Conf. Ser. Appl. Math., Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1970.**[3]**S. A. Book,*Large deviation probabilities for order statistics*, Naval. Res. Logist. Quart.**18**(1971), 521-523. MR**0312640 (47:1196)****[4]**M. Fisz,*Probability theory and mathematical statistics*, 3rd ed., Wiley, New York, 1963. MR**0164358 (29:1655)****[5]**J. C. Fu,*The rate of convergence of consistent point estimators*, Ann. Statist.**3**(1975), 234-240. MR**0359155 (50:11610)****[6]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press, New York, 1974. MR**0435697 (55:8655)****[7]**J. Steinebach,*Exponential convergence properties of linear estimators under exponential and uniform distribution*, Metrika**24**(1977), 137-161. MR**0488436 (58:7975)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60F10,
41A60,
62E20

Retrieve articles in all journals with MSC: 60F10, 41A60, 62E20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0567982-6

Keywords:
Asymptotic expansion,
Laplace's method,
large deviation,
sample quantile

Article copyright:
© Copyright 1980
American Mathematical Society