Abstract
The density ratio model specifies that the log-likelihood ratio of two unknown densities is of known linear form which depends on some finite dimensional parameters. The model can be broadened to allow for m-samples in a quite natural way. Estimation of both parametric and nonparametric part of the model is carried out by the method of empirical likelihood. However the assumed linear form has an impact on the estimation and testing for the parametric part. The goal of this study is to quantify the effect of choosing an incorrect linear form and its impact to inference. The issue of misspecification is addressed by embedding the unknown linear form to some parametric transformation family which yields ultimately to its identification. Simulated examples and data analysis integrate the presentation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Box G.E.P., Cox D.R. (1964). An analysis of transformations. Journal of the Royal Statistical Society, Series B 26, 211–252 with discussion
Cox D.R., Snell E.J. (1989). The analysis of binary data (2nd ed). Chapman & Hall, London
Fokianos, K. (2003). Foundations of statistical inference In: Y. Haitovsky, H. R. Lerche and Y. Ritov (eds.), Haidelberg:Physica–Verlag, pp 131–140
Fokianos K., Kedem B., Qin J., Haferman J., Short D.A. (1998). On combining instruments. Journal of Applied Meteorology 37, 220–226
Fokianos K., Kedem B., Qin J., Short D.A. (2001). A semiparametric approach to the one-way layout. Technometrics 43, 56–64
Fokianos K., Peng A., Qin J. (1999). A generalized-moments specification test for the logistic link. The Canadian Journal of Statistics 27, 735–750
Gilbert P.B. (2000). Large sample theory of maximum likelihood estimates in semiparametric biased sampling models. The Annals of Statistics 28, 151–194
Gilbert P.B., Lele S.R., Vardi Y. (1999). Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika 86, 27–43
Gill R.D., Vardi Y., Wellner J.A. (1988). Large sample theory of empirical distributions in biased sampling models. The Annals of Statistics 16, 1069–1112
Kedem B., Wolff D.B., Fokianos K. (2004). Statistical comparisons of algorithms. IEEE Transactions on Instrumantetion and Measurement 53, 770–776
Owen A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249
Owen A.B. (2001). Empirical likleihood. Chapman and Hall/CRC, Boca Raton, Florida
Prentice R.L., Pyke R. (1979). Logistic disease incidence models and case–control studies. Biometrika 66, 403–411
Qin J. (1998). Inferences for case-control data and semiparametric two–sample density ratio models. Biometrika 85, 619–630
Qin J., Barwick M., Ashbolt R., Dwyer T. (2002). Quantifying the change of melanoma incidence by Breslow thickness. Biometrics 58, 665–670
Qin J., Lawless J.F. (1994). Empirical likelihood and general estimating functions. Annals of Statistics 22, 300–325
Qin J., Zhang B. (1997). A goodness of fit test for the logistic regression model based on case–control data. Biometrika 84, 609–618
Vardi Y. (1982). Nonparametric estimation in the presence of length bias. The Annals of Statistics 10, 616–620
Vardi Y. (1985). Empirical distribution in selection bias models. The Annals of Statistics 13, 178–203
White I.R., Thompson S.G. (2003). Choice of test for comparing two groups, with particular application to skewed outcomes. Statistics in Medicine 22, 1205–1215
Zhang B. (2001). An information matrix test for logistic regression models based on case-control data. Biometrika 88, 921–932
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Fokianos, K., Kaimi, I. On the Effect of Misspecifying the Density Ratio Model. Ann Inst Stat Math 58, 475–497 (2006). https://doi.org/10.1007/s10463-005-0022-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-005-0022-8