Art Owen’s contribution to the Discussion of ‘Estimating means of bounded random variables by betting’ by Waudby-Smith and Ramdas

原文摘要

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=∑i=1t(xi−m)/(1+λT(xi−m))=0 corresponding to the observation weights wi∝1/(1+λT(xi−m)) used in EL. Two common alternatives use weights wi∝(1−λT(xi−m)) and wi∝exp(−λT(xi−m)) arising from L2 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 EL 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, EL 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, EL 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 the present authors’ hindsight optimality to this list of power properties. A similar power optimality is achieved by the confidence bands of Berk and Jones (1979). They use the most significant binomial likelihood ratio from all n order statistics to form confidence bands with greater power than any weighted Kolmogorov–Smirnov test. It would be very interesting to see if the authors’ methods could produce an always valid version of the Berk–Jones bands.

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