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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.
Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a rang
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 emp
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 th
In this paper, we built a new nonparametric regression estimator with the local linear method by using the mean squared relative error as a loss function when the data are subject to random right censoring. We establish the uniform almost sure consis
We consider a model where the failure hazard function, conditional on a covariate $Z$ is given by $R(t,theta^0|Z)=eta_{gamma^0}(t)f_{beta^0}(Z)$, with $theta^0=(beta^0,gamma^0)^topin mathbb{R}^{m+p}$. The baseline hazard function $eta_{gamma^0}$ and