Empirical estimates for heteroscedastic hierarchical dynamic normal models
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
Empirical estimates for heteroscedastic hierarchical dynamic normal models S. K. Ghoreishi1 · Jingjing Wu2 Received: 8 June 2020 / Accepted: 31 October 2020 © Korean Statistical Society 2020
Abstract The available heteroscedastic hierarchical models perform well for a wide range of real-world data, but for data sets that exhibit a dynamic structure they seem fit poorly. In this work, we develop a two-level dynamic heteroscedastic hierarchical model and suggest some empirical estimators for the association hyper-parameters. Moreover, we derive the risk properties of the estimators. Our proposed model has the feature that the dependence structure among observations is produced from the hidden variables in the second level and not through the observations themselves. The comparison between various empirical estimators is illustrated through a simulation study. Finally, we apply our methods to a baseball data. Keywords Asymptotic optimality · Heteroscedasticity · Shrinkage estimators · Stein’s unbiased risk estimator (SURE) · Dynamic models Mathematics Subject Classification 62F15 · 62F30
1 Introduction Hierarchical modelling has become an increasingly popular statistical method in many disciplines such as biology, climatology, ecology, medicine and engineering. A hierarchical model is a multi-level model which integrates information from different sources to achieve coherent inferences of unknowns (e.g., Li et al. 2010; Barboza et al. 2014; Shand et al. 2018).
* S. K. Ghoreishi [email protected] Jingjing Wu [email protected] 1
Department of Statistics, Faculty of Sciences, University of Qom, Qom, Iran
2
Department of Mathematics and Statistics, University of Calgary, Calgary, Canada
13
Vol.:(0123456789)
Journal of the Korean Statistical Society
The earliest study of this model was for simultaneous estimation of several normal means; see James and Stein (1961) and Stein (1962). The work by James and Stein (1961) has promoted greatly the use of hierarchical models in recent decades. The shrinkage estimators in Stein (1962) with an interesting empirical Bayes interpretation have been the basis for developing shrinkage estimation in multilevel normal models. Efron and Morris (1973) has further shown the implications of Stein’s shrinkage estimators and has proposed several competing parametric empirical Bayes estimators. Practically, both parametric and non-parametric empirical Bayes properties of shrinkage estimators have been extensively studied under either a homoscedastic (equal sub-population variances) or heteroscedastic (unequal sub-population variances) assumption. For more details, see Berger and Strawderman (1996) and Brown and Greenshtein (2009). The parametric empirical Bayes models involve a way to estimate the parameters of the second-level model, so-called hyper-parameters. Generally, empirical Bayes maximum likelihood estimator (EBMLE) and empirical Bayes method of moments (EBMOM) have been the popular approaches in estim
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