期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2016
卷号：78
期号：2
页码：155-179
DOI：10.1007/s13171-016-0085-z
语种：English
出版社：Indian Statistical Institute
摘要：This paper discusses the asymptotic behavior of regression models under general conditions, especially if the dimensionality of the set of true parameters is larger than zero and the true model is not identifiable. Firstly, we give a general inequality for the difference of the sum of square errors (SSE) of the estimated regression model and the SSE of the theoretical true regression function in our model. A set of generalized derivative functions is a key tool in deriving such inequality. Under suitable Donsker condition for this set, we provide the asymptotic distribution for the difference of SSE. We show how to get this Donsker property for parametric models even though the parameters characterizing the best regression function are not unique. This result is applied to neural networks regression models with redundant hidden units when loss of identifiability occurs and gives some hints on how penalizing such models to avoid over-fitting.
关键词：Regression models ; Donsker class ; Loss of identifiability ; Multilayer neural networks
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