, Volume 77, Issue 5, pp 695720
First online:
A new bounded loglinear regression model
 HaiYing WangAffiliated withDepartment of Mathematics and Statistics, University of New Hampshire
 , Nancy FlournoyAffiliated withDepartment of Statistics, University of Missouri Email author
 , Eloi KpameganAffiliated withClinical and Nonclinical Biostatistics, Novavax
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In this paper we introduce a new regression model in which the response variable is bounded by two unknown parameters. A special case is a bounded alternative to the four parameter logistic model. The four parameter model which has unbounded responses is widely used, for instance, in bioassays, nutrition, genetics, calibration and agriculture. In reality, the responses are often bounded although the bounds may be unknown, and in that situation, our model reflects the datagenerating mechanism better. Complications arise for the new model, however, because the likelihood function is unbounded, and the global maximizers are not consistent estimators of unknown parameters. Although the two sample extremes, the smallest and the largest observations, are consistent estimators for the two unknown boundaries, they have a slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes in the one sample case; but even these consistent estimators do not obtain the optimal convergence rate. To obtain efficient estimation, we suggest using the local maximizers of the likelihood function, i.e., the solution to the likelihood equations. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.
Keywords
Asymptotics Consistency Linear model Logistic model Maximum likelihood estimation Parameter dependent support Title
 A new bounded loglinear regression model
 Journal

Metrika
Volume 77, Issue 5 , pp 695720
 Cover Date
 201407
 DOI
 10.1007/s001840130460x
 Print ISSN
 00261335
 Online ISSN
 1435926X
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Asymptotics
 Consistency
 Linear model
 Logistic model
 Maximum likelihood estimation
 Parameter dependent support
 Industry Sectors
 Authors

 HaiYing Wang ^{(1)}
 Nancy Flournoy ^{(2)}
 Eloi Kpamegan ^{(3)}
 Author Affiliations

 1. Department of Mathematics and Statistics, University of New Hampshire, N315C Kingsbury Hall, 33 Academic Way, Durham, NH , 03824, USA
 2. Department of Statistics, University of Missouri, 146 Middlebush Hall, Columbia, MO , 65211, USA
 3. Clinical and Nonclinical Biostatistics, Novavax, 9920 Belward Campus Dr, Rockville, MD , 20850, USA