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Vol 27, No 4 (2018)
- Year: 2018
- Articles: 4
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13903
Article
On Optimal Cardinal Interpolation
Abstract
For the Hardy classes of functions analytic in the strip around real axis of a size 2β, an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery [12]. Below this method, based on the Jacobi elliptic functions, is shown to be optimal according to the criteria of Nonparametric Regression and Optimal Design.
In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant’s bias and variance mutually cancel each other. In the limiting case β → ∞, the optimal interpolant converges to the well-knownNyquist–Shannon cardinal series.
245-267
Asymptotic Distribution of Least Squares Estimators for Linear Models with Dependent Errors: Regular Designs
Abstract
We consider the usual linear regression model in the case where the error process is assumed strictly stationary.We use a result of Hannan, who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and the error process.We show that for a large class of designs, the asymptotic covariance matrix is as simple as in the independent and identically distributed (i.i.d.) case.We then estimate the covariance matrix using an estimator of the spectral density whose consistency is proved under very mild conditions.
268-293
On the Empirical Distribution Function of Residuals in Autoregression with Outliers and Pearson’s Chi-Square Type Tests
Abstract
We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is γn−1/2 with an unknown γ, n is the sample size. The autoregression parameters are unknown, they are estimated by any estimator which is n1/2-consistent uniformly in γ ≤ Γ < ∞. Using the residuals from the estimated autoregression, we construct a kind of empirical distribution function (e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. We obtain a stochastic expansion of this e.d.f., which enables us to construct a test of Pearson’s chi-square type for testing hypotheses about the distribution of innovations. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting level with respect to γ in a neighborhood of γ = 0.
294-311
On the Asymptotic Behavior of the Contaminated Sample Mean
Abstract
An observation of a cumulative distribution function F with finite variance is said to be contaminated according to the inflated variance model if it has a large probability of coming from the original target distribution F, but a small probability of coming from a contaminating distribution that has the same mean and shape as F, though a larger variance. It is well known that in the presence of data contamination, the ordinary sample mean looses many of its good properties, making it preferable to use more robust estimators. It is insightful to see to what extent an intuitive estimator such as the sample mean becomes less favorable in a contaminated setting. In this paper, we investigate under which conditions the sample mean, based on a finite number of independent observations of F which are contaminated according to the inflated variance model, is a valid estimator for the mean of F. In particular, we examine to what extent this estimator is weakly consistent for the mean of F and asymptotically normal. As classical central limit theory is generally inaccurate to copewith the asymptotic normality in this setting, we invokemore general approximate central limit theory as developed in [3]. Our theoretical results are illustrated by a specific example and a simulation study.
312-323