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

F. Angeletti, E. Bertin, P. Abry, Signal Processing, (2012), 2848-2865, ### Abstract

This contribution aims at studying the behaviour of the classical sample moment estimator,$S(n,q)=\sum_k=1^n X_k^q/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 $X=\exp Y$ with the tail of $Y$ behaving like $\exp(-y^\rho)$. 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 $q_c(n)$ is defined as the largest statistical order $q$ up to which the sample mean estimator $S(n,q)$ correctly accounts for the ensemble average $\E X^q$,for a given $n$.
A practical estimator for the critical moment $q_c(n)$ 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 amongst the observed samples.
Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.