Abstract: In numerous applications, data are observed at random times. Our main purpose is to study a model observed at random times that incorporates a longmemory noise process with a fractional Brownian Hurst exponent H. We propose a least squares estimator in a linear regression model with long-memory noise and a random sampling time called “jittered sampling”. Specifically, there is a fixed sampling rate 1/N, contaminated by an additive noise (the jitter) and governed by a probability density function supported in [0, 1/N]. The strong consistency of the estimator is established, with a convergence rate depending on N and the Hurst exponent. A Monte Carlo analysis supports the relevance of the theory and produces additional insights, with several levels of long-range dependence (varying the Hurst index) and two different jitter densities.

Abstract: We study the estimation and prediction of Gaussian processes with spacetime covariance models belonging to the dynamical generalized Wendland (DGW) family, under fixed-domain asymptotics. Such a class is nonseparable, has dynamical compact supports, and parameterizes differentiability at the origin similarly to the space-time Matern class.
Our results are presented in two parts. First, we establish the strong consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameter associated with the DGW covariance model, under fixed-domain asymptotics. The second part focuses on optimal kriging prediction under the DGW model and an asymptotically correct estimation of the mean squared error using a misspecified model. Our theoretical results are, in turn, based on the equivalence of Gaussian measures under some given families of space-time covariance functions, where both space or time are compact. The technical results are provided in the online Supplementary material.