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Added by
Xiaoying Tian Harris
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
Authors
Xiaoying Tian Harris
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Estimation of the prediction error of a linear estimation rule is difficult if the data analyst also use data to select a set of variables and construct the estimation rule using only the selected variables. In this work, we propose an asymptotically unbiased estimator for the prediction error after model search. Under some additional mild assumptions, we show that our estimator converges to the true prediction error in $L^2$ at the rate of $O(n^{-1/2})$, with $n$ being the number of data points. Our estimator applies to general selection procedures, not requiring analytical forms for the selection. The number of variables to select from can grow as an exponential factor of $n$, allowing applications in high-dimensional data. It also allows model misspecifications, not requiring linear underlying models. One application of our method is that it provides an estimator for the degrees of freedom for many discontinuous estimation rules like best subset selection or relaxed Lasso. Connection to Steins Unbiased Risk Estimator is discussed. We consider in-sample prediction errors in this work, with some extension to out-of-sample errors in low dimensional, linear models. Examples such as best subset selection and relaxed Lasso are considered in simulations, where our estimator outperforms both $C_p$ and cross validation in various settings.
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We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
It is known that there is a dichotomy in the performance of model selectors. Those that are consistent (having the oracle property) do not achieve the asymptotic minimax rate for prediction error. We look at this phenomenon closely, and argue that the set of parameters on which this dichotomy occurs is extreme, even pathological, and should not be considered when evaluating model selectors. We characterize this set, and show that, when such parameters are dismissed from consideration, consistency and asymptotic minimaxity can be attained simultaneously.
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