  The optimal fingerprinting method for detection and attribution of climate
change is based on a multiple regression where each covariate has measurement
error whose covariance matrix is the same as that of the regression error up to
a known scale. Inferences about the regression coefficients are critical not
only for making statements about detection and attribution but also for
quantifying the uncertainty in important outcomes derived from detection and
attribution analyses. When there is no errors-in-variables (EIV), the optimal
weight matrix in estimating the regression coefficients is the precision matrix
of the regression error which, in practice, is never known and has to be
estimated from climate model simulations. We construct a weight matrix by
inverting a nonlinear shrinkage estimate of the error covariance matrix that
minimizes loss functions directly targeting the uncertainty of the resulting
regression coefficient estimator. The resulting estimator of the regression
coefficients is asymptotically optimal as the sample size of the climate model
simulations and the matrix dimension go to infinity together with a limiting
ratio. When EIVs are present, the estimator of the regression coefficients
based on the proposed weight matrix is asymptotically more efficient than that
based on the inverse of the existing linear shrinkage estimator of the error
covariance matrix. The performance of the method is confirmed in finite sample
simulation studies mimicking realistic situations in terms of the length of the
confidence intervals and empirical coverage rates for the regression
coefficients. An application to detection and attribution analyses of the mean
temperature at different spatial scales illustrates the utility of the method.
