A note on empirical likelihoods derived from pairwise score functions

Abstract

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large datasets with relative small sample sizes. Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.

Minimum Scoring Rule Inference

“…Other approaches, invoking the theory of unbiased estimating functions, have been suggested to produce likelihood ratio‐type statistics that allow reference to the usual asymptotic chi‐squared distribution (Lunardon et al, ; Lunardon & Ronchetti, ). However, in contrast to the proposed likelihood ratio‐type statistic, which directly adjusts the scoring rule itself, these tests are derived from an unbiased estimating function for which a convex function to be used in the likelihood ratio‐type statistics is usually unavailable.…”

Combining Euclidean and composite likelihood for binary spatial data estimation

“…Finally, we outline that the proposed method can be extended to other types of data and to the space time setting using spatio-temporal blocks (Kleiber & Porcu (2014); Martinez-Ruiz et al (2010); Ruiz-Medina et al (2008); Yu et al (2007)). In addition to that, differently from standard likelihood methods, it is well-known that the PL likelihood ratio test does not have a standard limiting distribution (Lunardon et al (2012); Pace et al (2011)). On the other hand, it could be shown that EU-based tests are chi square distributed.…”

“…See also Caragea & Smith (2006), Fuentes (2007), Lindsay (1988), Stein (2008), Stein et al (2004) and Vecchia (1988). Lunardon et al (2012), in the IID case, use empirical likelihood (EL) with moment conditions coming from the score function of the PL in order to overcome the problem of non standard asymptotic distribution of the likelihood ratio statistic of the PL. See also Qin & Lawless (1994) for a general reference on EL estimation for moment condition models.…”

“…Our proposal is in the same spirit of Lunardon et al (2012), but with three main differences. First, we consider an Euclidean likelihood (EU) estimator.…”

Model selection and model averaging in MACML-estimated MNP models

“…As an alternative we introduce another test, which has the appeal that there is no need to compute the Godambe information matrix. This test is based on the empirical likelihood framework, which in the context of CML estimation was first used by Lunardon et al (2013) to derive confidence regions for CML estimates. Using general results from Qin and Lawless (1994) it is straightforward to also introduce a likelihood-ratio-like test for CML.…”