Skip to main content Accessibility help
×
Home

Asymptotic expansions and saddlepoint approximations using the analytic continuation of moment generating functions

  • Ronald W. Butler (a1)

Abstract

Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$ , have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks’ likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.

Copyright

Corresponding author

*Postal address: Department of Statistical Science, Southern Methodist University, Dallas, TX 75275, USA.

Footnotes

Hide All

The supplementary material for this article can be found at http://doi.org/10.1017/jpr.2019.19.

Footnotes

References

Hide All
Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems Theory Appl. 10, 587.
Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd edn. John Wiley, Hoboken, NJ.
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.
Blanchet, J. and Glynn, P. (2006). Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Prob. 16, 951983.
Bleistein, N. (1966). Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Commun. Pure Appl. Math. 19, 353370.
Butler, R. W. (2007). Saddlepoint Approximations with Applications. Cambridge University Press.
Butler, R. W. (2017). Asymptotic expansions and hazard rates for compound and first-passage distributions. Bernoulli 23, 35083536.
Butler, R. W. (2019a). Asymptotic expansions and saddlepoint approximations using the analytic continuation of moment generating functions. Supplementary material. Available at http://doi.org/jpr.2019.19.
Butler, R. W. (2019b). Asymptotic expansions and saddlepoint approximations using the analytic continuation of probability generating functions. Submitted.
Copson, E. T. (1965). Asymptotic Expansions. Cambridge University Press.
Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transform. Springer, Berlin.
Embrechts, P., Maejima, M. and Teugels, J. L (1985). Asymptotic behaviour of compound distributions. ASTIN Bulletin 15, 4548.
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 5572.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.
Lugannani, R. and Rice, S. (1980). Saddle point approximations for the distribution of the sum of independent random variables. Adv. Appl. Prob. 12, 475490.
Murray, J. D. (1984). Asymptotic Analysis. Springer, New York.
Nist, Dlmf. (2014). NIST Digital Library of Mathematical Functions. Release 1.0.8 of 2014-04-25. Available at https://dlmf.nist.gov.
Schatzoff, M. (1966). Exact distributions of Wilks’ likelihood ratio criterion. Biometrika 53, 347358.
Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems. Adv. Appl. Prob. 11, 701719.
Strawderman, R. L. (2004). Computing tail probabilities by numerical Fourier inversion: the absolutely continuous case. Statist. Sinica 14, 175201.
Sundt, B. (1982). Asymptotic behaviour of compound distributions and stop-loss premiums. ASTIN Bull. 15, 8998.
Tijms, H. C. (2003). A First Course in Stochastic Models. John Wiley, Chichester.
Widder, D. V. (1946). The Laplace Transform. Princeton University Press.

Keywords

MSC classification

Type Description Title
PDF
Supplementary material

Butler supplementary material
Supplementary material

 PDF (279 KB)
279 KB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed