1. Slides prepared by Timothy I. Matis for SpringSim’06, April 4, 2006 Estimating Rare Event Probabilities Using Truncated Saddlepoint Approximations Timothy I. Matis, Ph.D Ivan G. Guardiola Department of Industrial Engineering Texas Tech University

2. Overview of Presentation (ordering is not strict ) How I got here What are saddlepoint approximations Are truncated approximations any good Why we should care Numerical demonstrations What is next

3. Texas Tech Trivia TTU is a comprehensive university TTU has Bobby Knight, and Mike Leach Cotton, cotton, and oil everywhere

4. Evolution of Research Topic Preliminary investigations into stochastic shortest path problems –The path that is shortest in expectation is not necessarily the shortest in probability –Convolution of random path lengths Preliminary investigations into Cross- Entropy Methods – R.Y. Rubinstein –Used to find the change of measure of the importance sampling density

5. Analytical Approximation CE methods are efficient, yet considerable computational effort is still required Can I quickly approximate the (rare) probabilities to at least the correct order? Truncated saddlepoint approximations are relatively simple and robustly accurate

6. Motivating Example Example in “Simulation” (2002) by S.M. Ross Consider the sum of of independent random variables where For Num large, find the rare event probability 

7. Motivating Example In this example, the change of measure for the IS density may be calculated exactly, yielding a Monte Carlo IS point estimate of  =3.17x10 -4 when Num=16 By contrast, a 3 rd order truncated saddlepoint approximation yields the estimate  =2.378x10 -4 when Num=16

8. Mathematica Code Note that K is the CGF of the convolution, h is a list of solutions, and f is the truncated saddlepoint approximated density, which is subsequently integrated numerically

9. Saddlepoint Approximations Daniels (seminal), Wang (bivariate), Renshaw (truncated) Saddlepoint approximations are accurate in the tails of the distribution (as opposed to Edgeworth or Guassian approximations)

10. Mathematical Development Let K i (  ) be the cumulant generating function (CGF) of X i The CGF of S=X 1 +…+X n is K(  )= K 1 (  )+…+ K n (  ) It follows that a saddlepoint approximation of the density function of S is given by where  o is the positive real solution of

11. Why Truncate ? It is likely that the solution to for  o will be messy, if even attainable Second order truncation reduces to Gaussian approximation, third brings in skewness, fourth brings in kurtosis, etc. Truncated saddlepoint approximations may be complex over some of the support, yet are often not in the tails

12. Truncation Development Let be the j th order truncated CGF The individual cumulants of the truncated CGF are found through differentiation of the CGF of the convolution

13. Truncation Development It follows that the truncated CGF is given by where  o is the positive real solution of the polynomial

14. Optimal Truncation Level? Truncated saddlepoint approximations converge to full saddlepoint approximations in the limit In a finite sense, however, increasing the truncation level does not monotonically decrease the error In practice, evaluate the truncated saddlepoint at multiple levels! (if possible)

15. Truncation Level Example For X 1 ~Normal(2,.5) and X 2 ~Exponential(1), estimate the rare event probability using truncated saddlepoint approximations. The solution to of the full saddlepoint is not reportable, thereby motivating the truncated approach

16. Truncation Level Example The table on the right gives j th order truncated saddlepoint approximations of  An IS based estimate of this probability is  =5.5x10 -4 j  35.18x10 -5 42.17x10 -4 53.82x10 -4 65.04x10 -4 75.71x10 -4 86.71x10 -4 96.41x10 -4 106.55x10 -4

17. What is Next? Truncated saddlepoint approximations of bivariate distributions Accuracy of truncated saddlepoint approximations when only moment closure estimates of the cumulants of a distribution are known

18. Questions Contact Information: Timothy I. Matis timothy.matis@ttu.edu (806) 742-3543