On inference about rare events

Abstract

Despite the increasing volume of data in modern statistical applications, critical patterns and events have often little, if any, representation. This is not unreasonable, given that such variables are critical precisely because they are rare. We then have to raise the natural question: when can we infer something meaningful in such contexts? The focal point of this thesis is the archetypal problem of estimating the probability of symbols that have occurred very rarely, in samples drawn independently from an unknown discrete distribution. Our first contribution is to show that the classical Good-Turing estimator that is used in this problem has performance guarantees that are asymptotically non-trivial only in a heavy-tail setting. This explains the success of this method in natural language modeling, where one often has Zipf law behavior. We then study the strong consistency of estimators, in the sense of ratios converging to one. We first show that the Good-Turing estimator is not universally consistent. We then use Karamata's theory of regular variation to prove that regularly varying heavy tails are sufficient for consistency. At the core of this result is a multiplicative concentration that we establish both by extending the McAllester-Ortiz additive concentration for the missing mass to all rare probabilities and by exploiting regular variation. We also derive a family of estimators which, in addition to being strongly consistent, address some of the shortcomings of the Good-Turing estimator. For example, they perform smoothing implicitly. This framework is a close parallel to extreme value theory, and many of the techniques therein can be adopted into the model set forth in this thesis. Lastly, we consider a different model that captures situations of data scarcity and large alphabets, and which was recently suggested by Wagner, Viswanath and Kulkarni. In their rare-events regime, one scales the finite support of the distribution with the number of samples, in a manner akin to high-dimensional statistics. In that context, we propose an approach that allows us to easily establish consistent estimators for a large class of canonical estimation problems. These include estimating entropy, the size of the alphabet, and the range of the probabilities.