# Estimation of Linear Vectorial Semiparametric Models by Least Squares

Kun Zhang 1 and Songlin Zhang 2
1 Laboratory of Geographic Information Science, East China Normal University, Shanghai, 200062, China
2 Department of Surveying and Geomatics, Tongji University, Shanghai, 200092, China

Abstract. Semiparametric model is a statistical model  consisting of both parametric and nonparametric components, which can be looked on as a mixture model. The theoretical properties of this model have been studied extensively, such as large-sample property. However, most researches are based on scalar value, in which the dimension of the observation is one at each moment. In the fields of spatial data processing, such as econometrics, GPS, engineering surveying, engineering deformation monitoring, etc, the dimension of the observations is always more than one at each moment, they are vectorial models other than scalar ones. This paper focuses on the estimating theory of vectorial semiparametric models under the least-square principle. We deduced the formulas of weighted function estimator and spline estimator. Kernel and nearest-neighbor are most common weighted functions. In kernel estimation, the weights of observations are determined by kernel functions which are always probability density function, such as Nadaraya-Watson kernel. Nearest- neighbor means only the nearest neighbors have effect on a certain observation point. Spline estimation considers the penalized least-squares problem, the criterion function trades off fidelity to the data against function smoothness. The difference between scalar and vectorial semiparametric model was compared. As to weighted function estimator, the difference is a single weight parameter and a weight matrix; as to spline estimator, the difference is the operation of multiplication and Kronecker product.

Keywords: linear vectorial semiparametric model; scalar semiparametric model; least-square estimator

In: Wan, Y. et al. (eds) Proceeding of the 8th international symposium on spatial accuracy assessment in natural resources and environmental sciences, World Academic Union (Press).

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