*J. Japan Statist. Soc.,* Vol. 44 (No. 1), pp. 25-41, 2014

## ESTIMATING SEMIPARAMETRIC VARYING COEFFICIENTS FOR GEOGRAPHICAL DATA IN A MIXED EFFECTS MODEL

Kenichi Satoh and Tetsuji Tonda

**Abstract. **
A geographical weighted regression model can be used for visualizing or interpreting the covariate effects that vary with location. This model is usually estimated by a locally weighted regression or a kernel smoothing method, but we can regard the regression coefficients as varying linear coefficients that can be obtained from a global linear regression. There are two types of design vectors, one of which expresses linearity and the other is prepared for nonlinearity, i.e., it assumes a semiparametric surface with varying coefficients. Ridge estimators can then be used to suppress overfitting of the nonlinear part. With a mixed effects model, optimization of the ridge parameters and estimation of the regression parameters can be simultaneously executed. The linear structure of the varying coefficients then provides an asymptotic confidence interval as a function of location, but it is wider than a common pointwise confidence interval. We derive some tests for the varying coefficients and offer two examples using real data to illustrate our methodology. The results of the applied tests are summarized as the uniformity and the linearity of the varying coefficients.

*Key words and phrases*:
Geographical weighted regression, mixed effects model, ridge parameter, semiparametric varying coefficient, simultaneous confidence interval.

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