Extending the scope of empirical likelihood

Abstract

This article extends the scope of empirical likelihood methodology in three directions: to allow for plug-in estimates of nuisance parameters in estimating equations, slower than $\sqrt{n}$-rates of convergence, and settings in which there are a relatively large number of estimating equations compared to the sample size. Calibrating empirical likelihood confidence regions with plug-in is sometimes intractable due to the complexity of the asymptotics, so we introduce a bootstrap approximation that can be used in such situations. We provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results.Comment: Published in at http://dx.doi.org/10.1214/07-AOS555 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generalized empirical likelihood M testing for semiparametric models with time series data

“…The results of this paper generalize and extend results of Bertail (2006), Bravo (2009), Hjort, McKeague, andvan Keilegom (2009) and many others on EL inferences for semiparametric models with independent and identically distributed observations. The new results are the following: First we use the same kernel based smoothing used by Smith (1997) and Kitamura and Stutzer (1997) and propose two general types of test statistics, one based on an appropriately corrected GEL criterion function and one based on a Lagrange Multiplier (LM henceforth) approach.…”

“…In such case, it is not necessary to account for the presence of h in the asymptotic distribution of β, which greatly simplifies the calculation of the asymptotic variance. Condition 5(a) is directly assumed by Andrews (1994a) and is also considered by Hjort, McKeague, and van Keilegom (2009); condition 5(b) is assumed by Newey (1994). Note that for h = h (z 2t ) sufficient conditions for condition 5(a) are Assumptions 5(b) and 4(a).…”

Peter Hall's Contribution to Empirical Likelihood

“…Hjort, McKeague andChen, Peng andQin (2009) studied the properties of EL, allowing the number of parameters to diverge at some polynomial rate with the sample size. Their results show that asymptotically, the limiting distribution of the empirical log-likelihood ratio can still be characterized by the χ 2 distribution in the sense that with appropriate scaling and normalization, the empirical log-likelihood ratio converges in distribution to the standard normal distribution when the number of parameters is diverging.…”

Empirical Phi-discrepancies and quasi-empirical likelihood: exponential bounds

“…In Hjort et al [24], convergence of empirical likelihood is investigated when q is allowed to increase with n. They show that convergence to a χ 2 distribution still holds when q = O(n 1 3 ) as n tends to infinity.…”

“…Several approaches have been recently proposed in the literature to handle such problems. A first one is to discretize the problem and to retain a reasonable number of constraints (typically of order n 1/3 , see Hjort et al [24]) before applying the empirical likelihood procedure. Another way to discretize the problem is to consider what is called in the statisticallearning literature a skeleton class approximating F. One problem is of course the choice of the constraints and the loss of efficiency induced by this choice.…”