Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T18:19:29.907Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Approximation of Saddlepoint Expansions in Statistics

Published online by Cambridge University Press:  11 February 2009

Offer Lieberman
Offer Lieberman
Affiliation:
Monash University
Get access Rights & Permissions [Opens in a new window]

Abstract

The saddlepoint approximation as developed by Daniels [3] is an extremely accurate method for approximating probability distributions. Econometric and statistical applications of the technique to densities of statistics of interest are often hindered by the requirements of explicit knowledge of the c.g.f. and the need to obtain an analytical solution to the saddlepoint defining equation. In this paper, we show the conditions under which any approximation to the saddlepoint is justified and suggest a convenient solution. We illustrate with an approximate saddlepoint expansion of the Durbin-Watson test statistic.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 5 , December 1994 , pp. 900 - 916
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Barndorff-Nielsen, O.E.On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 (1983): 343–365.CrossRefGoogle Scholar
2Barndorff-Nielsen, O.E. & Cox, D.R.. Edgeworth and saddlepoint approximations with statistical applications (with discussion). Journal of the Royal Statistical Society B 41 (1979): 279–312.Google Scholar
3Daniels, H.E.Saddlepoint approximations in statistics. Annals of Mathematical Statistics 25 (1954): 631–650.CrossRefGoogle Scholar
4Daniels, H.E.Exact saddlepoint approximations. Biometrika 67 (1980): 59–63.CrossRefGoogle Scholar
5Daniels, H.E.Tail probability approximations. International Statistical Review 55 (1987): 37–48.CrossRefGoogle Scholar
6Daniels, H.E. & Young, G.A.. Saddlepoint approximation for the studentised mean with an application to the bootstrap. Biometrika 78 (1991): 169–179.CrossRefGoogle Scholar
7Durbin, J. & Watson, G.S.. Testing for serial correlation in least squares regression, I. Biometrika 37 (1950): 409–428.Google ScholarPubMed
8Durbin, J. & Watson, G.S.. Testing for serial correlation in least squares regression, II. Biometrika 38 (1951): 159–178.CrossRefGoogle ScholarPubMed
9Easton, G.S. & Ronchetti, E.. General saddlepoint approximations with applications to L statistics. Journal of the American Statistical Association 81 (1986): 420–430.Google Scholar
10Henshaw, R.C.Testing single-equation least squares regression models for autocorrelated disturbances. Econometrica 34 (1966): 646–660.CrossRefGoogle Scholar
11Imhof, P.J.Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961): 419–426.CrossRefGoogle Scholar
12Lieberman, O.Saddlepoint approximation for the distribution of a ratio of quadratic forms in normal variables. Journal of the American Statistical Association 89 (1994).CrossRefGoogle Scholar
13Lugannani, R. & Rice, S.. Saddlepoint approximation for the distribution of the sum of independent random variables. Advanced Applied Probability 12 (1980): 475–490.CrossRefGoogle Scholar
14McCullagh, P.Tensor Methods in Statistics. London: Chapman and Hall, 1987.Google Scholar
15Phillips, P.C.B.Edgeworth and saddlepoint approximations in first order noncircular autoregression. Biometrika 65 (1978): 91–98.CrossRefGoogle Scholar
16Phillips, P.C.B.Large deviation expansions in econometrics. Advances in Econometrics 5 (1986): 199–226.Google Scholar
17Reid, N.Saddlepoint methods and statistical inference. Statistical Science 3 (1988): 213–238.Google Scholar
18Rice, S.O.Distribution of quadratic forms in normal random variables-evaluation by numerical integration. Siam Journal of Statistics and Computing 1 (1980): 438–448.CrossRefGoogle Scholar
19Robinson, J.Saddlepoint approximations for permutation tests and confidence intervals. Journal of the Royal Statistical Society B 44 (1982): 91–101.Google Scholar
20Shively, T.S., Ansley, C.F. & Kohn, R.. Fast evaluation of the distribution of the Durbin-Watson and other invariant test statistics in time series regression. Journal of the American Statistical Association 85 (1990): 676–685.CrossRefGoogle Scholar
21Spady, R.H.Saddlepoint approximations for regression models. Biometrika 78 (1991): 879–889.CrossRefGoogle Scholar