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Optimum percentile estimating equations for nonlinear random coefficient models
Institution:1. University of Texas Health Science Center at Houston (UTHealth), School of Public Health, Austin Campus, 1616 Guadalupe Street, Suite 6.300, Austin, TX 78701, USA;2. The University of Oklahoma Health Sciences Center, Oklahoma Tobacco Research Center, 655 Research Parkway, Suite 400, Oklahoma City, OK 73104, USA;3. University of Texas at Austin, Population Research Center, 305 E. 23rd Street, Stop G1800, Austin, TX 78712, USA;1. Laboratory on Engineering and Management Intelligence, Research Group of Operations Research and Decision Systems, Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest 1518, P.O. Box 63, Hungary;2. Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Hungary;3. Institute of Applied Mathematics, John von Neumann Faculty of Informatics, Óbuda University, Hungary
Abstract:In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.
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