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Nearly all estimators in statistical prediction come with an associatedtuning parameter, in one way or another. Common practice, given data, is tochoose the tuning parameter value that minimizes a constructed estimate of theprediction error of the estimator; we focus on Stein's unbiased risk estimator,or SURE (Stein, 1981; Efron, 1986) which forms an unbiased estimate of theprediction error by augmenting the observed training error with an estimate ofthe degrees of freedom of the estimator. Parameter tuning via SURE minimizationhas been advocated by many authors, in a wide variety of problem settings, andin general, it is natural to ask: what is the prediction error of theSURE-tuned estimator? An obvious strategy would be simply use the apparenterror estimate as reported by SURE, i.e., the value of the SURE criterion atits minimum, to estimate the prediction error of the SURE-tuned estimator. Butthis is no longer unbiased; in fact, we would expect that the minimum of theSURE criterion is systematically biased downwards for the true predictionerror. In this paper, we formally describe and study this bias.

Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?