Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic distribution of least squares estimators for linear models with dependent errors : regular designs
378
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. We show that for a large class of designs, the asymptotic covariance matrix is as simple as the independent and identically distributed case. We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions.
rate research
Read More
We study the performance of the Least Squares Estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$-th moment ($pgeq 1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard `entropy condition with exponent $alpha in (0,2)$, then the $L_2$ loss of the LSE converges at a rate begin{align*} mathcal{O}_{mathbf{P}}big(n^{-frac{1}{2+alpha}} vee n^{-frac{1}{2}+frac{1}{2p}}big). end{align*} Such a rate cannot be improved under the entropy condition alone. This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $pgeq 1+2/alpha$ moments, the $L_2$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/alpha$, there are (many) hard models at any entropy level $alpha$ for which the $L_2$ loss of the LSE converges at a strictly slower rate than other robust estimators. The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_2$ loss of the LSE can converge arbitrarily slowly when the independence fails. The key technical ingredient is a new multiplier inequality that gives sharp bounds for the `multiplier empirical process associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality.
We study the asymptotic properties of the SCAD-penalized least squares estimator in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. We are particularly interested in the use of this estimator for simultaneous variable selection and estimation. We show that under appropriate conditions, the SCAD-penalized least squares estimator is consistent for variable selection and that the estimators of nonzero coefficients have the same asymptotic distribution as they would have if the zero coefficients were known in advance. Simulation studies indicate that this estimator performs well in terms of variable selection and estimation.
The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk, and study the a.o. of several cross validation (CV) based hyper-parameter estimators including the leave $k$-out CV method, generalized CV method, $r$-fold CV method and hold out CV method. We find the former three methods can be a.o. under mild assumptions, but not the last one, and we use Monte Carlo simulations to illustrate the efficacy of our findings.
The paper continues the authors work on the adaptive Wynn algorithm in a nonlinear regression model. In the present paper it is shown that if the mean response function satisfies a condition of `saturated identifiability, which was introduced by Pronzato cite{Pronzato}, then the adaptive least squares estimators are strongly consistent. The condition states that the regression parameter is identifiable under any saturated design, i.e., the values of the mean response function at any $p$ distinct design points determine the parameter point uniquely where, typically, $p$ is the dimension of the regression parameter vector. Further essential assumptions are compactness of the experimental region and of the parameter space together with some natural continuity assumptions. If the true parameter point is an interior point of the parameter space then under some smoothness assumptions and asymptotic homoscedasticity of random errors the asymptotic normality of adaptive least squares estimators is obtained.
For the class of Gauss-Markov processes we study the problem of asymptotic equivalence of the nonparametric regression model with errors given by the increments of the process and the continuous time model, where a whole path of a sum of a deterministic signal and the Gauss-Markov process can be observed. In particular we provide sufficient conditions such that asymptotic equivalence of the two models holds for functions from a given class, and we verify these for the special cases of Sobolev ellipsoids and Holder classes with smoothness index $> 1/2$ under mild assumptions on the Gauss-Markov process at hand. To derive these results, we develop an explicit characterization of the reproducing kernel Hilbert space associated with the Gauss-Markov process, that hinges on a characterization of such processes by a property of the corresponding covariance kernel introduced by Doob. In order to demonstrate that the given assumptions on the Gauss-Markov process are in some sense sharp we also show that asymptotic equivalence fails to hold for the special case of Brownian bridge. Our results demonstrate that the well-known asymptotic equivalence of the Gaussian white noise model and the nonparametric regression model with independent standard normal distributed errors can be extended to a broad class of models with dependent data.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
