ベキ重み付き平滑化スプラインによる分散均一性の診断

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  • Diagnosing Homoscedasticity with the Power-weighted Smoothing Spline
  • ベ キ オモミツキ ヘイカツカ スプライン ニ ヨル ブンサン キンイツセイ ノ シンダン

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In nonparametric regression models, the requirements of homoscedasticity and so on are implicitly assumed, which leads to poor estimation of regression functions. A power weighted smoothing spline (PWSS) model, whose objective is to diagnose homoscedasticity as well as to estimate unknown nonlinear regression structure, is assumed. The responses in an additive regression model are power-transformed, and then their variances after transformation are assumed to be constant. Smoothing splines are obtained as estimated functions by maximizing the penalized likelihood, and a reweighted version of a backfitting algorithm is constructed. A power-transformation parameter and smoothing parameters, which control smoothness of the functions, are estimated by maximizing the marginal likelihood, based on Bayesian approaches to smoothing splines. A form of the marginal likelihood, which yields comparatively easy computation, is derived using the property that smoothing splines are the best linear unbiased predictor of a linear mixed model. Examination of some data sets from the literature and a simulation experiment show that the power transformation estimated with the PWSS model attains homoscedasticity while taking nonlinear structure into account.

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