Resumen de Tail index estimation for discrete heavy-tailed distributions with application to statistical inference for regular markov chains
Patrice Bertail, Stéphan Clemençon, Carlos Fernandez
It is the purpose of this paper to investigate the issue of estimating the regularityindex β > 0 of a discrete heavy-tailed r.v. S, i.e. a r.v. S valued in N∗ such that P(S > n) = L(n) · n−β for all n ≥ 1, where L : R∗+ → R+ is a slowly varying function. Such discrete probability laws, referred to as generalized Zipf’s laws sometimes, are commonly used to model rank-size distributions after a preliminary rangesegmentation in a wide variety of areas such as e.g. quantitative linguistics, social sciences or information theory. As a first go, we consider the situation where inference isbased on independent copies S1, ..., Sn of the generic variable S. The estimator βwe propose can be derived by means of a suitable reformulation of the regularly varying condition, replacing S’s survivor function by its empirical counterpart. Under mildassumptions, a non-asymptotic bound for the deviation between βand β is established,as well as limit results (consistency and asymptotic normality). Beyond the i.i.d. case,the inference method proposed is extended to the estimation of the regularity indexof a regenerative β-null-recurrent Markov chain. Since the parameter β can be thenviewed as the tail index of the (regularly varying) distribution of the return time ofthe chain X to any (pseudo-) regenerative set, in this case, the estimator is constructedfrom thesuccessiveregeneration times. Because the durations between consecutive regeneration times are asymptotically independent, we can prove that the consistency of the estimator promoted is preserved. In addition to the theoretical analysis carried
