SELECTING SMOOTHING PARAMETERS WITH RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION : A PROCEDURE FOR EFFICIENT COMPUTATION AND ITS APPLICATION

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  • Sakamoto Wataru
    Division of Mathematical Science, Graduate School of Engineering Science, Osaka University

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  • 制限付き最尤推定法による平滑化パラメータの選定 : 効率的な計算方式とその適用
  • セイゲン ツキ サイユウ スイテイホウ ニ ヨル ヘイカツカ パラメータ ノ センテイ コウリツテキ ナ ケイサン ホウシキ ト ソノ テキヨウ

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In fitting smoothing splines, obtained as optimum solutions to penalized approaches, it should be useful for diagnosing nonlinear structures if the values of smoothing parameters that control their smoothness can be selected appropriately based on given data. It is necessary to reconsider the procedure for selecting smoothing parameters with restricted maximum penalized likelihood estimation from the viewpoint of the objective of nonparametric regression, that is, to explore functional relationships. The procedure is based on the representation of smoothing splines as linear mixed effect models. However, it takes much time to compute estimates using existent programs. For easier application to various models and efficient calculation of estimates, the restricted log likelihood and its derivatives are derived in such forms that are suitable for fitting smoothing splines, using BLUP equations. In applications to some examples in literature, it is shown that the procedure can give valid suggestion to regression structures as well as error variance structures.

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