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The consistency for the estimators of semiparametric regression model based on weakly dependent errors

Author

Listed:
  • Xuejun Wang

    (Anhui University)

  • Xin Deng

    (Anhui University)

  • Fengxi Xia

    (Anhui University)

  • Shuhe Hu

    (Anhui University)

Abstract

For the semiparametric regression model: $$Y^{(j)}(x_{in},~t_{in})=t_{in}\beta +g(x_{in})+e^{(j)}(x_{in}),~1\le j\le m,~1\le i \le n$$ Y ( j ) ( x i n , t i n ) = t i n β + g ( x i n ) + e ( j ) ( x i n ) , 1 ≤ j ≤ m , 1 ≤ i ≤ n , where $$x_{in}\in \mathbb {R}^p$$ x i n ∈ R p , $$t_{in}\in \mathbb {R}$$ t i n ∈ R are known to be nonrandom, g is an unknown continuous function on a compact set A in $$\mathbb {R}^p$$ R p , $$e^{(j)}(x_{in})$$ e ( j ) ( x i n ) are $$\tilde{\rho }$$ ρ ~ -mixing random variables with mean zero, $$Y^{(j)}(x_{in},t_{in})$$ Y ( j ) ( x i n , t i n ) are random variables which are observable at points $$x_{in}$$ x i n and $$t_{in}$$ t i n . In the paper, we establish the strong consistency, r-th ( $$r>2$$ r > 2 ) mean consistency and complete consistency for estimators $$\beta _{m,n}$$ β m , n and $$g_{m,n}(x)$$ g m , n ( x ) of $$\beta $$ β and g, respectively. The results obtained in the paper extend the corresponding ones for independent random variables and $$\varphi $$ φ -mixing random variables.

Suggested Citation

  • Xuejun Wang & Xin Deng & Fengxi Xia & Shuhe Hu, 2017. "The consistency for the estimators of semiparametric regression model based on weakly dependent errors," Statistical Papers, Springer, vol. 58(2), pages 303-318, June.
  • Handle: RePEc:spr:stpapr:v:58:y:2017:i:2:d:10.1007_s00362-015-0698-7
    DOI: 10.1007/s00362-015-0698-7
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    References listed on IDEAS

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    1. Soo Sung, 2011. "On the strong convergence for weighted sums of random variables," Statistical Papers, Springer, vol. 52(2), pages 447-454, May.
    2. Fraiman, Ricardo & Iribarren, Gonzalo Pérez, 1991. "Nonparametric regression estimation in models with weak error's structure," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 180-196, May.
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