Consistency Of A Class Of Estimators The Semi-parametric Regression Model For The Pairwise Negative Quadrant Dependent Samples

Engle talked about nonparametric estimates of the relation between weather and electricity, and established the semi-parametric regression model in 1986. y_i = x_iβ+ g (t_i ) + e_i .i = 1,2,…,n ..From then on more and more statisticians were attracted by it. Because the semi- parametric regression model has the parametric components and the nonparametric components, the model is more extensive and more elastic than the classical linear regression model and nonparametric regression model in the scientific research and the practical life.About the model , people have gained a lot of conclusions but the conclusions were finished on the bases of the hypothesis of errors independent . Actually the errors are independent in our lives. It is indispensable and important that studying the semi-parametric regression model for the errors are independent.Three kinds of the semi-parametric regression model are discussed in many articles:Modelâ… : y_i = x_iβ+ g (t_i) + e_i.i = 1,2,…,n .Modelâ…¡: y_i = x_iβ+ g (t_i) +σ_ie_i.i = 1,2,…,n .,Modelâ…¢: y^(j) ( x_in) = g (x_in) + e^(j)( x_in).1≤j≤m, 1≤i≤nUsually here the design points ( x_i , t_i , u_i )are known and nonrandom, g(.) and f(.) are unknown functions,βis an unknown parametric to be estimated, and the errors {e_i , i≥1} are random variables such as the independents ,the Negatively Associated random variables, the L^q-mixingale, pairwise NQD of random sequences. From 1986, people defined the least squares estimator (β|^)_n, the weighted least squares estimator (β|—) nofβ,the estimators (g|^)_n, (σ|^)_n² for g (t ),σ² in the articles [1～5], and people established their strong consistency under suitable conditions, see the articles [2,6～8] ,[9～11], [12～14].In this paper, we are going to considering the consistency of estimators in the modelâ… and modelâ…¡Considering the error {e_i , i≥1} is the pairwise NQD random samples in this paper, we must study their features. The concept of NQD was defined firstly by Lehmann [15] in 1966.Pairwise NQD are distinct from other independent variables, some famous variables are based on it such as NA. By virtue of the conclusions of WANG [16],WU[17],YANG[18],we establish the consistency of some estimators .Also we modify the conclusion of the article [11] and obtain the inequality the similar as YANG seeing the article [18] in the condition of limiting errors. Sum up, the main conclusions of this thesis are as follows:Lemma 6:Suppose the variable { X_i ; i≥1}is the Pair-wise NQD, and EX_i = 0, |X_i|≤M,a.s. and exist a real number t>0,Theorem 1Suppose the errors {e_j; j≥1} are the Pair-wise NQD, Ee_j = 0, Ee_j² =σ²<∞, E|e_j|^p<∞p > 2, then in the conditions of the①～⑨for the modelâ… , (β|^)_n→β.a.s.Theorem2Suppose the errors {e_j; j≥1} are the Pair-wise NQD, Ee_j = 0, Ee_j² =σ²<∞, E|e_j|^p<∞p > 2 then in the conditions of the①～⑨for the modelâ… , (g|^)_n(t )→g (t ).a.s.(?)t∈JTheorem3Suppose the errors {e_j; j≥1} are the Pair-wise NQD, Ee_j = 0, Ee_j² =σ²<∞, E|e_j|^p<∞p > 2 then in the conditions of the①～⑧for the modelâ… Theorem4:Suppose the errors {e_j; j≥1} are the Pair-wise NQD, Ee_j = 0, Ee_j² = 1,Ee_j⁴<∞, then in the conditions of the①～⑨for the modelâ…¡, (β|^)_n -β= O (n^-1/4).a.s.