No Arabic abstract
We study the parameter estimation problem of Vasicek Model driven by sub-fractional Brownian processes from discrete observations, and let {S_t^H,t>=0} denote a sub-fractional Brownian motion whose Hurst parameter 1/2<H<1 . The studies are as follows: firstly, two unknown parameters in the model are estimated by the least squares method. Secondly, the strong consistency and the asymptotic distribution of the estimators are studied respectively. Finally, our estimators are validated by numerical simulation.
We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal.
We study the problem of exact support recovery based on noisy observations and present Refined Least Squares (RLS). Given a set of noisy measurement $$ myvec{y} = myvec{X}myvec{theta}^* + myvec{omega},$$ and $myvec{X} in mathbb{R}^{N times D}$ which is a (known) Gaussian matrix and $myvec{omega} in mathbb{R}^N$ is an (unknown) Gaussian noise vector, our goal is to recover the support of the (unknown) sparse vector $myvec{theta}^* in left{-1,0,1right}^D$. To recover the support of the $myvec{theta}^*$ we use an average of multiple least squares solutions, each computed based on a subset of the full set of equations. The support is estimated by identifying the most significant coefficients of the average least squares solution. We demonstrate that in a wide variety of settings our method outperforms state-of-the-art support recovery algorithms.
We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.
In a regression setting with response vector $mathbf{y} in mathbb{R}^n$ and given regressor vectors $mathbf{x}_1,ldots,mathbf{x}_p in mathbb{R}^n$, a typical question is to what extent $mathbf{y}$ is related to these regressor vectors, specifically, how well can $mathbf{y}$ be approximated by a linear combination of them. Classical methods for this question are based on statistical models for the conditional distribution of $mathbf{y}$, given the regressor vectors $mathbf{x}_j$. Davies and Duembgen (2020) proposed a model-free approach in which all observation vectors $mathbf{y}$ and $mathbf{x}_j$ are viewed as fixed, and the quality of the least squares fit of $mathbf{y}$ is quantified by comparing it with the least squares fit resulting from $p$ independent white noise regressor vectors. The purpose of the present note is to explain in a general context why the model-based and model-free approach yield the same p-values, although the interpretation of the latter is different under the two paradigms.
In this paper, we consider an inference problem for the first order autoregressive process driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $abs{k}^{2H-2}$ times a function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA model and some others Gaussian noise are special examples that satisfy this assumption. We propose a second moment estimator and prove the strong consistency and give the asymptotic distribution. Moreover, when the limit distribution is Gaussian, we give the upper Berry-Esseen bound by means of Fourth moment theorem.