Tham khảo tài liệu 'handbook of empirical economics and finance _7', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Large Deviations Theory and Econometric Information Recovery 167 Empirical Parametric ED Problem and Empirical MaxMaxEnt The discussion of Subsection extends directly to the empirical parametric ED problem which CLLN implies should be solved by selecting p - 0 arg rnf J pG 0 II VN p - 0 en e with 0 0EMME where 0EMME arg inf I pG 0 IIVN . 0e The estimator 0EMME is known in Econometrics under various names such as maximum entropy empirical likelihood and exponential tilt. We call it the empirical MaxMaxEnt estimator EMME . Note that thanks to the convex duality the estimator 0EMME can equivalently be obtained as 0EMME arg sup inf log V VN Xi 0 exp -X u Xi e . R i i Example illustrates the extension of the parametric ED problem cf. Example to the empirical parametric ED problem. Example Let X 1 2 3 4 . Let a random sample of size N 100 from data-sampling distribution q induces N-type VN 7 42 24 27 100. Let in addition a random sample of size n 109 be drawn from q but it remains unavailable to us. We are told only that the sample mean is in the interval . Thus n 0 p - 0 J24 1 p xi 0 xi - 0 0 and 0 e . The objective is to select an n-empirical measure from n given the available information. CLLN dictates that we solve the problem by EMME. Since n is very large we can without much harm ignore rational nature ofn-types . vn - 0 e Qm and seek the solution among pmf s p - 0 e Rm. CLLN suggests the selection of p Q EMME . Since the average ffi y vNxi is outside of the interval convexity of the information divergence implies that 0 EMME . the lower bound of the interval. Kitamura and Stutzer 2002 were the first to recognize that LD theory through CLLN can provide justification for the use of the EMME estimator. The CLLNs demonstrate that selection of I -projection is a consistent method which in the case of a parametric possibly misspecified model n establishes consistency under misspecification of the EMME .
