Rare tail approximation using asymptotics and L^1 polar coordinates

by   Thomas Taimre, et al.

In this work, we propose a class of importance sampling (IS) estimators for estimating the right tail probability of a sum of continuous random variables based on a change of variables to L^1 polar coordinates in which the radial and angular components of the IS distribution are considered separately. When the asymptotic behaviour of the sum is known we exploit it for the radial change of measure, and the resulting estimator has the appealing form of the (known) asymptotic multiplied by a random multiplicative correction factor. Given we assume knowledge of the asymptotic behaviour of the sum in this framework, traditional notions of efficiency that appear in the rare-event literature hold little practical meaning here. Instead, we focus on the practical behaviour of the proposed estimator in the pre-asymptotic regime for right tail probabilities between roughly 10^-3 and 10^-7. The proposed estimator and procedure are applicable in both the heavy- and light-tailed settings, as well as for independent and dependent summands. In the case of independent summands, we find that our estimator compares favourably with exponential tilting (iid light-tailed summands) and the Asmussen--Kroese method (independent subexponential summands). However, for dependent subexponential summands using the same simple angular distribution as for the independent case, the performance of our estimator rapidly degenerates with increasing dimension, suggesting an open avenue for further research.


page 1

page 2

page 3

page 4


Efficient Importance Sampling for Large Sums of Independent and Identically Distributed Random Variables

We aim to estimate the probability that the sum of nonnegative independe...

Asymptotic distributions for weighted power sums of extreme values

Let X_1,n≤⋯≤ X_n,n be the order statistics of n independent random varia...

On the Tail Behaviour of Aggregated Random Variables

In many areas of interest, modern risk assessment requires estimation of...

Limit laws for the norms of extremal samples

Let denote S_n(p) = k_n^-1∑_i=1^k_n( log (X_n+1-i,n / X_n-k_n, n) )^p, w...

Limit theorems for the Multiplicative Binomial Distribution (MBD)

The sum of n non-independent Bernoulli random variables could be modeled...

Practical strategies for GEV-based regression models for extremes

The generalised extreme value (GEV) distribution is a three parameter fa...

A Universal Splitting Estimator for the Performance Evaluation of Wireless Communications Systems

We propose a unified rare-event estimator for the performance evaluation...

Please sign up or login with your details

Forgot password? Click here to reset