Laplace record data

Basic properties of upper record values _XT(1),_XT(2),…,_XT(n)$X_{T(1)},X_{T(2)},\dots ,X_{T(n)}$ from a symmetric two-parameter Laplace distribution are established. In particular, unimodality of the density function and the exact expression of the mode are derived. Moreover, we obtain approximations of the first and second moment and the variance of _XT(k)$X_{T(k)}$ which provide close approximations even for moderate k. Additionally, limit laws and simulation of Laplace records are considered. Finally, we discuss maximum likelihood estimation in a location-scale family of Laplace distributions. We obtain nice representations of the estimators provided that the location parameter is unknown and present interesting properties of the established estimators. Some illustrative examples complete the presentation.     

Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n   consistent plug-in estimators of conditional expectations E(h(_Xn+1)|_X1,…,_Xn)$E(h(X_{n+1})|X_1,\dots ,X_n)$. More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F. For fixed h it suffices that h   is locally of bounded variation and locally Lipschitz in _L2(F)$L_2(F)$, and that the convolution of h and F   is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P(_Xn+1?·|_X1,…,_Xn)$P(X_{n+1}?·|X_1,\dots ,X_n)$ holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F-Donsker, assuming some smoothness of F. The proofs use empirical process arguments.