Art Owen's contribution to the Discussion of “Estimating means of bounded random variables by betting” by Ian Waudby-Smith and Aaditya Ramdas

Abstract

I congratulate the authors on some very interesting work making connections between likelihood, betting and martingales. What caught my eye was the connection to the empirical likelihood (EL) (Owen, 2001) and the dual likelihood (Mykland, 1995). The hindsight optimal λ solves 0 = (cid:80) ti =1 ( x i − m ) / (1+ λ T ( x i − m )) = 0 corresponding to the observation weights w i ∝ 1 / (1 + λ T ( x i − m )) used in empirical likelihood. Two common alternatives use weights w i ∝ (cid:0) 1 − λ T ( x i − m ) (cid:1) and w i ∝ exp( − λ T ( x i − m )) arising from L 2 and entropy criteria, respectively (for different vectors λ ). The entropy weights connect to exponential tilting and logistic regression; see for instance Hainmueller (2012). The EL weights perform a kind of reciprocal tilting that gives them some special power properties. Kitamura (2003) shows that empirical likelihood tests cannot be dominated by other regular tests for moment restrictions, in a large deviations sense. His result is a nonparametric counterpart to the likelihood test optimality result of Hoeffding (1965) for multinomial distributions. Lazar and Mykland (1998) find that for true parametric models, empirical likelihood matches their power to second order and at third order either the empirical or the parametric tests could have greater power. In some overspecified moment models, empirical likelihood has such high power for detecting lack of fit that, under lack of fit, there cannot exist any pseudo-true value of the parameter for which the maximum EL estimate is root-n consistent (Schennach, 2007). We can now add that authors’ hindsight optimality to this list of power properties.