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In recent years.the theory of dependent sequences such asφmixing sequence.Ψmix-ing sequence and NOD sequence and so on, has been sufficiently developed. Especially sev-eral important inequalities are obtained, for instance, Bernstein inequality, Rosenthal’type inequality, and so forth, which greatly improves the development of the applications in statistical fields. In this paper, the consistency of the estimators for semiparametric and nonparametric regressional models are estabilished under the dependent errors above.In Chapter 2, the semiparametric regression model Y(j)(xin,tin)=tinβ+g(xin)+ e(j)(xin),1≤j≤m,1≤i≤n is considered. Based on the methods of least squares and weight function, the estimatorsβm,n and gm,n(x) forβand g are defined, respectively. By using the truncating method, the moment inequalities forφmixing sequence andΨmixing sequence and the properties of the convex functions, we obtain the r-th moment consistency and the strong consistency for these estimators under some mild conditions, which generalize the correponding results of Hu(1997).In Chapter 3, the nonparametric regression model Yi=g(xi)+εi for i=1,2,…,n is discussed.The estimator gn(x) of the unknown function g(x) is defined. For NOD error sequence, under some mild conditions, we investigate the r-th moment consistency, the strong consistency and almostly complete convergence for the estimator. Moreover, under some uniform assumptions,the uniform moment consistency and the uniform strong con-sistency are also obtained. Since independent random variables and negatively associated random variables are the cases of NOD random variables, the results of independent and negatively associated sequences are generalized by the ones that we obtain.
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