Empirical Information Criterion (EIC Sample Clauses

Empirical Information Criterion (EIC. Kolaczyk(1995) [42] proposed an alternate extension of Akaike’s information xxxxx- xxxx (AIC) for longitudinal data that is based on empirical likelihood. Using the same notation as in the previous discussion of empirical likelihood, let X1, . . . , Xn be independent and identically distributed d-variate random variables with a common distribution function F. Recall that, by definition, the empirical likelihood function is n n L(F ) = ∏ dF (xi) = ∏ pi, i=1 i=1 where pi = dF (xi) = Pr(X = xi). Further, the empirical log-likelihood ratio can be expressed as ∑ [ ] n lEL (θ) = log 1 + tT (θ)g(xi; θ) . Kolaczyk [42] notes that, by construction, the set {pi (θ) ; i = 1 . . . , n} is a proper probability distribution on the sample {x1, . . . , xn}. Hence, an empirical likelihood analogue of the Kullback-Leibler (KL) information can be obtained by replacing the log-likelihood ratio statistic with the empirical likelihood alternative ∑ log {pi (θ∗)} p i (θ∗) , i=1 pi (θ) where θ∗ denotes the true value of the parameter vector of interest. In words, this means that the loss entailed in modeling θ∗ by θ is defined to be the Kullback- Leibler distance between two discrete empirical distributions corresponding to these two parameter values. It follows that, under the regularity conditions outlined in Kolaczyk[42], the empirical information criterion can be expressed as EIC(k) = −2lEL (θ˜k) + d, where θ˜k denotes the estimated parameter of the fitted submodel. The EIC(k) statistic has many appealing properties. First, it does not depend on any parametric assumptions regarding the distribution of the data and requires only that the estimating function being considered is unbiased. Further, EIC(k) is an asymptotically unbiased estimate of the risk of modeling θ∗ by the fitted submodel θ˜k. Unfortunately, the fact that EIC(k) is an asymptotically unbiased estimate of risk does not ensure that it will be a successful criterion for selecting the optimal model from among a set of candidate models and the effectiveness of EIC for model selection has not yet been investigated.