For everyk≥1 consider the waiting time until each pattern of lengthk over a fixed alphabet of sizen appears at least once in an infinite sequence of independent, uniformly distributed random letters. Lettingn→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and
provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent.
Authors:Rolando Cavazos-Cadena and Graciela González-Farías
Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced,
and it is shown that, under each member of this class, the hypothesis H0: λ = 0 is invariant, where λ is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection
index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini’s centred
parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum
likelihood estimators is given under H0, and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.
Authors:Rafik Aguech, Sana Louhichi and Sofyen Louhichi
Newman, C. M., Asymptoticindependence and limit theorems for positively and negatively dependent random variables, in: Y. L. Tong, editor, Inequalities in Statistics and Probability , IMS Lecture Notes-Monograph Series 5 (1984), 127-140. MR 86i:60072