A note on the consistency for the estimators of semiparametric regression model

Abstract

For the semiparametric regression model: $Y^{(j)}(x_{in},t_{in})=t_{in}\beta +g(x_{in})+e^{(j)}(x_{in}),~~1\le j\le m, 1\le i\le n,$ where $t_{in}\in \mathbb {R}$ and $x_{in}\in \mathbb {R}^{p}$ are known to be nonrandom, g is an unknown continuous function on a compact set A in $\mathbb {R}^{p}$, $e^{j}(x_{in})$ are $\rho ^{*}$-mixing random errors with mean zero, $Y^{(j)}(x_{in},t_{in})$ represent the j-th response variables which are observable at points $x_{in},t_{in}$. In this paper, we study the strong consistency and r-th ($1<r\le 2$) mean consistency for the estimators $\beta _{m,n}$ and $g_{m,n}$ of $\beta$ and g, respectively. The results obtained in this paper improve and extend the corresponding ones for $\rho ^{*}$-mixing random variables and other dependent sequences.