Jose H. Blanchet

Asymptotic robustness of estimators in rare-event simulation

The asymptotic robustness of estimators as a function of a rarity parameter, in the context of rare-event simulation, is often qualified by properties such as bounded relative error (BRE) and logarithmic efficiency (LE), also called asymptotic optimality. However, these properties do not suffice to ensure that moments of order higher than one are well estimated. For example, they do not guarantee that the variance of the empirical variance remains under control as a function of the rarity parameter. We study generalizations of the BRE and LE properties that take care of this limitation. They are named bounded relative moment of order k (BRM-k) and logarithmic efficiency of order k (LE-k), where k ≥ 1 is an arbitrary real number. We also introduce and examine a stronger notion called vanishing relative centered moment of order k, and exhibit examples where it holds. These properties are of interest for various estimators, including the empirical mean and the empirical variance. We develop (sufficient) Lyapunov-type conditions for these properties in a setting where state-dependent importance sampling (IS) is used to estimate first-passage time probabilities. We show how these conditions can guide us in the design of good IS schemes, that enjoy convenient asymptotic robustness properties, in the context of random walks with light-tailed and heavy-tailed increments. As another illustration, we study the hierarchy between these robustness properties (and a few others) for a model of highly reliable Markovian system (HRMS) where the goal is to estimate the failure probability of the system. In this setting, for a popular class of IS schemes, we show that BRM-k and LE-k are equivalent and that these properties become strictly stronger when k increases. We also obtain a necessary and sufficient condition for BRM-k in terms of quantities that can be readily computed from the parameters of the model.

Efficient rare-event simulation for the maximum of heavy-tailed random walks

Let

State-dependent importance sampling for regularly varying random walks

(X_n:n\geq 0)

A Markov Chain Approximation to Choice Modeling

be a sequence of i.i.d. r.v.s with negative mean. Set

Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue

S_0=0

Uniform renewal theory with applications to expansions of random geometric sums

and define

Quantifying Distributional Model Risk Via Optimal Transport

S_n=X_1+... +X_n

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case.

. We propose an importance sampling algorithm to estimate the tail of

Efficient Simulation of Light-Tailed Sums: an Old-Folk Song Sung to a Faster New Tune...

M=\max \{S_n:n\geq 0\}

Efficient rare event simulation for heavy-tailed compound sums

that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmunds (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.