An Assumption for the Development of Bootstrap Variants of the Akaike Information Criterion in Mixed Models

This note provides a proof of a fundamental assumption in the verification of bootstrap AIC variants in mixed models. The assumption links the bootstrap data and the original sample data via the log-likelihood function, and is the key condition used in the validation of the criterion penalty terms. (See Assumption 3 of both Shibata, 1997, and Shang and Cavanaugh, 2007.) To state the assumption, let Y and Y ∗ represent the response vector and the corresponding bootstrap sample, respectively. Let θ represent the set of parameters for a candidate mixed model, and let θ̂ denote the corresponding maximum likelihood estimator based on maximizing the likelihood L(θ | Y ). With E∗ denoting the expectation with respect to the bootstrap distribution of Y , the assumption asserts that E∗ logL(θ̂ | Y ) = logL(θ̂ | Y ). We prove the assumption holds under parametric, semiparametric, and nonparametric bootstrapping.