Critical moment definition and estimation, for finite size observation of log-exponential-power law random variables

This contribution aims at studying the behavior of the classical sample moment estimator, S(n,q)=∑k=1nXkq/n, as a function of the number of available samples n, in the case where the random variables X   are positive, have finite moments at all orders and are naturally of the form X=expY with the tail of Y   behaving like e−yρe−yρ. This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics, a critical moment qc(n) is defined as the largest statistical order q   up to which the sample mean estimator S(n,q)S(n,q) correctly accounts for the ensemble average EXqEXq, for a given n. A practical estimator for the critical moment qc(n) is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation among the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.

► Critical order definition for moment estimation from finite size data.
► Beyond critical moment, moment estimations diverge from the theoretical moments.
► Asymptotic properties based on insights from statistical physics.
► Construction and analysis of a practical estimator for the critical order.
► Customization of the estimation of the critical order for correlated data.