Application of Saddlepoint Approximation to Functional Equations 
Sunghoon Chung , Ph.D. Candidate, Preliminary Examination
Department of Statistics, Colorado State University
Application of Saddlepoint Approximation to Functional Equations
December 7, 2007
Room 008 Statistics
ABSTRACT
The saddlepoint approximation was originally introduced to the
statistics literature as a numerical method to obtain density (mass)
functions or distribution functions of the sample mean of i.i.d. random
variables (r.v.) and it is known to have relative error of order O(n^1)
and generally works well for small n or even n=1 cases.
For n=1, we may view the approximation as a numerical inversion of the
moment generating function (mgf). This perspective can be useful for
stochastic processes where the r.v. of interest can only be recognized
using the mgfs of related known random variables.
To investigate the case where the mgf of interest takes an explicit
form, we examine the PollaczekKinchin formula of M/G/1 queues. Our
simulation study shows the method performs adequately. Asymptotic
results are obtained and the bootstrap method is applied to obtain
confidence intervals.
The potential use for implicitly defined mgfs is also discussed.
