統計学輪講（第２５回）

日時    2004年 10月 12日(火)　15時〜16時40分
場所    経済学部新棟３階第３教室
講演者  藤澤 洋徳(統計数理研究所)
演題    Adaptive parameter estimation both for robustness and efficiency
概要：
 Robust and efficient parameter estimation using a density power weight 
has been studied (Windham, 1995; Basu et al., 1998; Jones et al., 2001). 
The non-negative power parameter $\beta$ controls the performance of 
the resulting estimator, which is more efficient if $\beta$ is near zero 
and more robust if $\beta$ is near one. The estimation method successfully 
works when an appropriate power parameter $\beta$ is selected a priori, 
but no universal selection method has been created. This paper suggests 
a method for adaptively selecting $\beta$ in accordance with the observations,
using robust cross-validation. Consequently, adaptive parameter estimation 
is constructed both for robustness and efficiency. We first prepare a loss 
of using the resulting estimator and next evaluate the risk given as the 
expectation of the loss. The optimal power parameter is selected as the 
minimizer of the risk function estimated by the cross-validation. An important 
issue is to choose a loss function suitable for robust inference. This paper 
presents a robustly estimable loss function even if the ratio of outliers to 
observations is substantial. The minimization of the risk in $\beta$ is shown 
to reflect a trade-off between bias and variance in the resulting estimator. 
Data analyses and numerical studies are illustrated. For synthetic data sets 
including the underlying data and outliers, the suggested method properly 
selected different power parameters in accordance with different outliers 
but gave similar estimates to the maximum likelihood estimates based only 
on the underlying data.

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Tokyo University