Bayesian analysis of restricted penalized empirical likelihood
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Bayesian analysis of restricted penalized empirical likelihood Mahdieh Bayati1 · Seyed Kamran Ghoreishi2
· Jingjing Wu3
Received: 15 November 2019 / Accepted: 5 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce restricted empirical likelihood and restricted penalized empirical likelihood estimators. These estimators are obtained under both unbiasedness and minimum variance criteria for estimating equations. These scopes produce estimators which have appealing properties and particularly are more robust against outliers than some currently existing estimators. Assuming some prior densities, we develop the Bayesian analysis of the restricted empirical likelihood and the restricted penalized empirical likelihood. Moreover, we provide an EM algorithm to approximate hyper-parameters. Finally, we carry out a simulation study and illustrate the theoretical results for a real data set. Keywords Empirical likelihood · Restricted empirical likelihood · Bayesian optimization · Gibbs sampling · EM algorithm · Estimating equations
1 Introduction Empirical likelihood (EL) is a statistical tool that provides robust inferences for given data under some general conditions. The EL approach was initially used by Thomas et al. (1957) and then further developed by Owen (1990). Nowadays, many statisticians use this method to analyze various real-world data.
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Seyed Kamran Ghoreishi [email protected] Mahdieh Bayati [email protected] Jingjing Wu [email protected]
1
Department of Statistics, Science and Research branch, Islamic Azad University, Tehran, Iran
2
Department of Statistics, University of Qom, Qom, Iran
3
Department of Mathematics and Statistics, University of Calgary, Calgary, Canada
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M. Bayati et al.
Owen (1988 and 1990) developed and applied the EL ratio statistics to some nonparametric problems. The author proved that these statistics have asymptotically a chi-squared distribution. Furthermore, the author developed confidence intervals and hypothesis tests for model parameters based on likelihood ratio statistics under a parametric model. The asymptotic properties and some essential corrections of these statistics were presented in DiCiccio and Romano (1989) and Hall and La Scala (1990). Qin and Lawless (1994) showed that EL, along with some appropriate estimating equations, provides an acceptable non-parametric fit to data. Practically, in the EL approach, estimates are obtained by maximizing the EL function based on estimating equations along with some additional restrictions. Of these constraints, we would point out the zero expectation value of estimating equations under the probability model { p1 , p2 , . . . , pn } of observations. Newey and Smith (2004) and Chen and Cui (2006 and 2007) showed that the obtained estimators have some appealing statistical properties; for more details, see Owen (2001) and Qin and Lawless (1994). In addition to the studies by Chen et al. (2009), Hjort et al. (2009), Tang and Leng (2010), Leng and T
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