The Estimations On Semi-Varying Coefficient Models

Hastie and Tibshirani proposed the varying coefficient models, whichare defined as Y=sum from i=1 to pα_i(U)X_i+ε,where (U, X₁,X₂,…, X_p)^T is the vector of the given covariates and Y is theresponse variable,εis independent of (U, X₁, X₂,…, X_p)^T with E(ε)=0 andVar(ε)=σ². Since the models are more flexible and more adaptable thanthe linear models, they have been discussed by many people and appliedwidely in many fields.In practice, researchers often want to know whether the coefficients arereally varying or not. This amounts to test if every function coefficient isconstant, namely, test the null hypothesis H₀:α_i(U)=β_i, for some i. Themodels under H₀ for some i will be called semivarying coefficient models, which are defined as Y=sum from i=1 to pα_i(U)X_i+sum from j=1 to qβ_jZ_jε,where (U, X₁,X₂,...,X_p, Z₁,...,Z_q) is the vector of the given covariates and Y is the response variable,εis independent of(U, X₁, X₂,..., X_p, Z₁,..., Z_q) with E(ε)=0 and Var(ε)=σ². These mod-els consist of a nonparametric part that involves coefficient functions{α_i(·),i=1,2,...,p} and a linear part that involves constant coefficients{β_j,j=1,...,q}. Due to the curse of dimensionality, we assume U isunivariate.Obviously, ifα_i(·)≡0 (i=1,...,p), the semivarying coefficient mod-els become linear models, ifβ_j=0 (j=1,...,q), they become varyingcoefficient models. At the same time, if the constant coefficientβ_j is viewedas a function, the semivarying coefficient models can be regarded as a spe-cial case of the varying coefficient models. On the other hand, when p=1and X₁≡1, the models become partially linear models.Firstly, estimation on semivarying coefficient models with linear con-straints are studied in this paper. Suppose that the linear constraints Aβ=bis a consistent linear equation group, where A is a matrix of m×q withrank m, and b is a vector of m×1. Under the linear constraints, the con- strained profile least-squares estimation on semivarying coefficient modelsis discussed, and the asymptotic normality of which is investigated.Secondly, estimation on semivarying coefficient models with censoreddata are studied.Suppose that response variable {Y_i, 1≤i≤n} are not be observedfully for random censored, One can only observerd {(T_i,δ_i),1≤i≤n},where{V_i, 1≤i≤n} i.i.d, and independent of {Y_i, 1≤i≤n}.By transform data, the two step estimator of parametric component onsemivarying coefficient models are developed by local linear method andaverage method. The estimator of nonparametric component are developedby local linear method and backfitting technical. Asymptotic normalitiesof the estimators are investigated.Finally, the quasi-likelihood estimation on generalized semivarying co-efficient models are studied. Suppose that the conditional mean and theconditional variance gived byfor a given function V and unknown scale parameterσ², where U, Y∈ R, X∈R^p, Z∈R^q. The quasi-likelihood function Q(μ, y) is defined viaGeneralized semivarying coefficient models are defined viawhere g(·) is a given link function andα(u)=(α₁(u),…,α_p(u)) is aunknown vector of coeffifient functions,β=(β₁,…,β_q) is a unknownvector of constant coefficients.the procedures for estimation of the parametric component and thenonparametric component on generalized semi-varying coefficient modelsare developed as fellows: The coefficient functions in the nonparametriccomponent of generalized semi-varying coefficient models are estimatedvia local quasi-likelihood method with a small bandwidth. After replacingthe coefficient functions in the nonparametric component by the obtainedestimators, quasi-likelihood method is employed to produce the estimatorsof the constant coefficients in parametric part, then the final estimator ofnonparametric component are developed by local quasi-likelihood methodand backfitting technical. For this procedure, the asymptotic normalitiesof the estimation on parametric component and nonparametric componentare investigated.