On the Consistency of Least Squares Estimator in Models Sampled at Random Times Driven by Long Memory Noise: The Jittered Case

Abstract

In numerous applications data are observed at random times. Our main purpose is to study a model observed at random times incorporating a long memory noise process with a fractional Brownian Hurst exponent H. In this article, we propose a least squares (LS) 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 but 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 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.