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A note on orthogonal series regression function estimators

100%
Popiński W.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 3
281-291
EN
The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_i,Y_i)$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^1$[a,b]. The constructed estimators are of the form $\widehat f_n(x) = \sum_{k=0}^{N(n)}\widehat c_ke_k(x)$, where the coefficients $\widehat c_0,\widehat c_1,...,\widehat c_N$ are determined by minimizing the empirical risk $n^{-1}\sum_{i=1}^n(Y_i - \sum_{k=0}^Nc_ke_k(X_i))^2$. Sufficient conditions for consistency of the estimators in the sense of the errors $E_X\vert f(X)-\widehat f_n(X)\vert^2$ and $n^{-1}\sum_{i=1}^nE(f(X_i)-\widehat f_n(X_i))^2$ are obtained.
2
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Orthogonal series regression estimators for an irregularly spaced design

100%
Popiński W.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 3
309-318
EN
Nonparametric orthogonal series regression function estimation is investigated in the case of a fixed point design where the observation points are irregularly spaced in a finite interval [a,b]i ⊂ ℝ. Convergence rates for the integrated mean-square error and pointwise mean-square error are obtained in the case of estimators constructed using the Legendre polynomials and Haar functions for regression functions satisfying the Lipschitz condition.
3
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Generalized kernel regression estimatefor the identification of Hammerstein systems

100%
Mzyk G.
International Journal of Applied Mathematics and Computer Science
|
2007
|
tom 17
|
nr 2
189-197
EN
A modified version of the classical kernel nonparametric identification algorithm for nonlinearity recovering in a Hammerstein system under the existence of random noise is proposed. The assumptions imposed on the unknown characteristic are weak. The generalized kernel method proposed in the paper provides more accurate results in comparison with the classical kernel nonparametric estimate, regardless of the number of measurements. The convergence in probability of the proposed estimate to the unknown characteristic is proved and the question of the convergence rate is discussed. Illustrative simulation examples are included.
4

On robust GMM estimation with applications in economics and finance

88%
Steland A.
Discussiones Mathematicae Probability and Statistics
|
2000
|
tom 20
|
nr 1
63-83
EN
Generalized Methods of Moments (GMM) estimators are a popular tool in econometrics since introduced by Hansen (1982), because this approach provides feasible solutions for many problems present in economic data where least squares or maximum likelihood methods fail when naively applied. These problems may arise in errors-in-variable regression, estimation of labor demand curves, and asset pricing in finance, which are discussed here. In this paper we study a GMM estimator for the rank modelingapproach (RMA), which analyzes the ordinal structure of a response variable. Assuming m-dependent data consistency and asymptotic normality of the proposed estimator are shown including the important case that the instruments depend on lagged regressors.Consistent estimators for the asymptotic covariance matrices are proposed. Further, the construction of minimum variance RMA-GMM estimators is discussed. Finite sample properties are studied by a small simulation study.
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