Example DRMST Objective Functions
Bayes Risk Function (“Savage Framework” Optimal):dn?(?) = ??(?,?*)p(?*|x1,...,xn)d?*
Special cases of this difficult integral typically estimated by DRMST-type risk functions!dn?(?) = n-1? c(?;xt) + kn(?), with c(?;xt) = -log(p(xt| ?)) and
- MLE, QMLE (e.g., White, 1982) : kn(?) = 0
- MAP (e.g., Golden, 1996) : kn(?) = -(1/n) ln (p(?))
- AIC (e.g., Akaike, 1973): kn(?) = q/n [q=parameter vector dimension]
- BIC/SIC (e.g., Schwarz, 1978) : kn(?) = (q/2n) ln(n)
- GBIC (e.g., Djuric, 1998) : kn(?) = (q/2n) ln(n ?t det(?2 ct))
- MDL/SC (e.g., Rissanen, 1996) : kn(?) = (q/2n) ln(n/2?) +(ln ?[det(?2 c)]1/2d?)/n