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| THE RATES OF CONVERGENCE OF M-ESTIMATORS FORPARTLY LINEAR MODELS IN DEPENDENT CASES |
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Citation： |
Shi Peide,Chen Xiru.THE RATES OF CONVERGENCE OF M-ESTIMATORS FORPARTLY LINEAR MODELS IN DEPENDENT CASES[J].Chinese Annals of Mathematics B,1996,17(3):301~316 |
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Net amount： 833 |
Authors： |
Shi Peide; Chen Xiru |
Foundation： |
Project supported by the postdoctoral fellowship and the National Natural Science Foundation of China |
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| Abstract： |
Consider the partly linear model $Y_i=X_i' \beta_0+g_0(T_i)+e_i$, where $\{(T_i,X_i)\}_1^{\infty}$ is a strictly stationary sequence of random variables, the $e_i$'s are i.i.d. random errors, the $Y_i$'s are real-valued responses, $\beta_0$ is a $d$-vector of parameters, $X_i$ is a $d$-vector of explanatory variables, $T_i$ is another explanatory variable ranging over a nondegenerate compact interval. Based on a segment of observations $(T_1,X_1' ,Y_1),\cdots,(T_n,X_n' ,Y_n)$, this article investigates the rates of convergence of the $M$-estimators for $\beta_0$ and $g_0$ obtained from the minimization problem $$ \sum_{i=1}^n\rho(Y_i-X_i' \beta-g_n(T_i)) =\min_{\beta\in R^d,\,g_n\in \Cal F_n}, $$where $\Cal F_n$ is a space of $B$-spline functions of order $m+1$ and $\rho(\cdot)$ is a function chosen suitably. Under some regularity conditions, it is shown that the estimator of $g_0$ achieves the optimal global rate of convergence of estimators for nonparametric regression, and the estimator of $\beta_0$ is asymptotically normal. The $M$-estimators here include regression quantile estimators, $L_1$-estimators, $L_P$-norm estimators, Huber's type $M$-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis $H_0:A'\beta_0=\tilde\beta$ are also discussed, where $\tilde\beta$ is a given vector and $A$ is a known $d\times d_0$ matrix with rank $d_0$. |
Keywords： |
Partly linear model, $M$-estimator, $L_1$-norm estimator, $B$-spline, Optimal rate of convergence, Strictly stationary sequence, $\beta$-mixing |
Classification： |
62G07, 62M10 |
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