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 converges to a positive constant when the working and true measures are non-equivalent, and is smooth in the prediction location. We then propose a new way to estimate the MSPE of the quasi-EBLUP that accounts for covariance function uncertainty. Our new estimator is compared to four other prediction variance estimators. The new prediction variance estimator generally performs better than all other competitors, and the larger the misspecification of the covariance smoothness, the wider the difference among MSPE estimators.

翻译：最佳线性无偏预测（BLUP）在广义线性混合模型、空间模型和高斯过程回归（GPR）中一直是主导方法。除了其最优性质外，BLUP过程还量化了预测不确定性。然而，BLUP的一般实现步骤如下：（i）假设概率分布和协方差函数已知，仅协方差参数值未知；（ii）将参数估计值代入BLUP方程，得到估计最佳线性无偏预测（EBLUP）及其方差。在实际应用中，真实过程的协方差函数未知，选择错误的协方差模型（尤其是其平滑度）来估计参数会产生准EBLUP，其预测方差存在向下偏差。本文聚焦于GPR场景，首先证明：当工作度量与真实度量非等价时，误设对拟EBLUP均方预测误差（MSPE）的影响收敛于一个正常数，且在预测位置上光滑。随后我们提出一种新的方法来估计考虑协方差函数不确定性的拟EBLUP的MSPE。将新估计量与另外四种预测方差估计量进行比较。新预测方差估计量通常优于所有其他竞争者，且协方差平滑度误设越大，各MSPE估计量之间的差异越显著。