Parameter estimation for random sampled Regression Model with Long Memory Noise

Abstract

In this article, we present the least squares estimator for the drift parameter in a linear regression model driven by the increment of a fractional Brownian motion sampled at random times. For two different random times, Jittered and renewal process sampling, consistency of the estimator is proven. A simulation study is provided to illustrate the performance of the estimator under different values of the Hurst parameter H.

Limit distribution of the least square estimator with observations sampled at random times driven by standard Brownian motion

“…Then D N converges almost surely, as N → ∞ to α 3 3 . Actually, the result of Araya et al (2019) has been obtained for α = 1, but after inspecting the proof, it is clear that the same arguments holds for every α ∈ (0, 1). Therefore, in order to obtain the asymptotic behavior of the LSE, we need to analyse the sequence A N in (7).…”

“…Our purpose is to analyze the asymptotic properties of the LSE (6), in particular its asymptotic normality in distribution. The denominator of the expression (7) has been already studied by Araya et al (2019). Let us recall their results (see Lemma 3.2 of Araya et al (2019)).…”

“…The denominator of the expression (7) has been already studied by Araya et al (2019). Let us recall their results (see Lemma 3.2 of Araya et al (2019)).…”

“…Let us now introduced the random times considered in our model (1). Our examples are inspired from Araya et al (2019) and Vilar (1995).…”

“…Our proofs are based on a sharp calculation of the mean square of the estimator and of its conditional distribution given the random times, which is Gaussian. We also use some results given by Araya et al (2019) where the asymptotic behavior of the denominator of the LSE estimator is obtained.…”