Abstract:
Best Linear Unbiased Prediction (BLUP) has been a dominant approach in Generalized Linear Mixed Models, spatial models, and Gaussian Process Regression (GPR). In addition to their optimal properties, BLUP procedures quantify prediction uncertainty. However, the general implementation of BLUP goes as follows:
(i) assume the probability distribution and covariance function are known and that only the covariance parameter values are unknown;
(ii) plug in parameter estimates into BLUP equations to get the Estimated Best Linear Unbiased Prediction (EBLUP) and its variance.
In applications, the reality is that the true covariance function for the process is unknown and choosing the wrong covariance model, particularly its smoothness, to estimate parameters yields a quasi-EBLUP whose prediction variance is biased downward. Focusing on a GPR context, in this paper we first demonstrate that the effect of misspecification on the mean squared prediction error (MSPE) of the quasi-EBLUP vanishes asymptotically when the working covariance family can match the high-frequency (smoothness) behavior of the true co-variance, remains bounded away from zero when the working predictor is asymptotically strictly inefficient at every working parameter value—the regime induced by covariance smoothness misspecification—and is smooth in the prediction location. We then attempt a new way to estimate the MSPE of the quasi-EBLUP that accounts for covariance function uncertainty: a calibration ratio on the observation scale, formed from cross-validated squared errors and model variances, is combined across observed inputs by a bounded kernel-weighted ratio of means and converted to the latent scale in a single step whose leading term contrasts two estimators of the measurement-error variance. Our new estimator is compared against five competing prediction-variance estimators for the latent process using coverage, interval length, and a proper interval scoring rule. Under correct specification the new estimator retains near-nominal coverage at a modest length premium; as covariance smoothness misspecification grows it attains coverage closest to nominal and, under severe misspecification, the best proper interval scores among all latent-target.
Place: Monzón 201
Hour: 10:45 a.m.