The transform likelihood ratio method for rare event simulation with heavy tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation ( change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.

Disulfide bond mutagenesis and the structure and function of the head-to-tail macrocyclic trypsin inhibitor SFTI-1

SFTI-1 is a novel 14 amino acid peptide comprised of a circular backbone constrained by three proline residues, a hydrogen-bond network, and a single disulfide bond. It is the smallest and most potent known Bowman-Birk trypsin inhibitor and the only one with a cyclic peptidic backbone. The solution structure of [ABA(3,11)]SFTI-1, a disulfide-deficient analogue of SFTI-1, has been determined by H-1 NMR spectroscopy. The lowest energy structures of native SFTI-1 and [ABA(3,11)]SFTI-1 are similar and superimpose with a root-mean-square deviation over the backbone and heavy atoms of 0.26 +/- 0.09 and 1.10 +/- 0.22 Angstrom, respectively. The disulfide bridge in SFTI-1 was found to be a minor determinant for the overall structure, but its removal resulted in a slightly weakened hydrogen-bonding network. To further investigate the role of the disulfide bridge, NMR chemical shifts for the backbone H-alpha protons of two disulfide-deficient linear analogues of SFTI-1, [ABA(3,11)]SFTI-1[6,5] and [ABA(3,11)]SFTI-1[1,14] were measured. These correspond to analogues of the cleavage product of SFTI-1 and a putative biosynthetic precursor, respectively. In contrast with the cyclic peptide, it was found that the disulfide bridge is essential for maintaining the structure of these open-chain analogues. Overall, the hydrogen-bond network appears to be a crucial determinant of the structure of SFTI-1 analogues.

Improved algorithms for rare event simulation with heavy tails

The estimation of P(S-n > u) by simulation, where S, is the sum of independent. identically distributed random varibles Y-1,..., Y-n, is of importance in many applications. We propose two simulation estimators based upon the identity P(S-n > u) = nP(S, > u, M-n = Y-n), where M-n = max(Y-1,..., Y-n). One estimator uses importance sampling (for Y-n only), and the other uses conditional Monte Carlo conditioning upon Y1,..., Yn-1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.