Two-Sample Plug-in Empirical Likelihood Method with Applications to Structural Relationship Models
Janis Valeinis
Faculty of Physics and Mathematics, University of Latvia, Riga, Latvia

Since Owen (1988, 1990) has introduced the empirical likelihood method for statistical inference, it has obtained great interest among researchers. It is not only appealing as a nonparametric procedure, but also due to the data determined asymmetric confidence regions and Bartlett correctability (see, for example, Owen (2001)). Recently the empirical likelihood method has been explored for several two-sample problems. Valeinis (2007) showed that many two-sample problems can be generalized in one general framework (see also Valeinis et al. (2010), Valeinis and Cers (2011)). It covers the difference of the two-sample means, distribution functions, quantile functions, probability-probability and quantile-quantile plots, ROC curves and structural relationship models for a fixed structural parameter.

General plug-in empirical likelihood has been introduced by Hjort et al. (2009). Using the results of PhD thesis of Valeinis (2007) we extend their approach and introduce the two-sample plug-in empirical likelihood method. The asymptotic distribution of the test statistic is derived. For a practical application we check the goodness-of-fit of structural relationship models introduced by Freitag and Munk (2005). This model contains the simple two-sample location, the location-shift model and also the well known Lehmann's alternative model.

In order to check the goodness-of-fit of the structural relationship models we also introduce and derive asymptotic limiting distributions for the location, the location-scale and the Lehmann's alternative empirical processes. In order to apply these results for practical data problems we use the smoothed bootstrap method. Finally, we also show some simulation study with empirical coverage accuracies and real data problems.

References:

Freitag, G. and Munk, A. (2005). On Hadamard differentiability in k-sample semiparametric models with applications to the assessment of structural relationships. Journal of Multivariate Analysis, 94(1), 123-158.

Hjort, N. L., McKeague I. W. and van Keilegom I. (2009). Extending the scope of empirical likelihood. The Annals of Statistics, 37(3), 1079-1111.

Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2), 237-249.

Owen, A. (1990). Empirical likelihood ratio confidence regions. The Annals of Statistics, 18(1), 90-120.

Owen, A. (2001). Empirical likelihood. Chapman & Hall, Boca Raton, FL.

Valeinis, J. (2007). Confidence bands for structural relationship models, Ph.D. Dissertation, University of Goettingen, Goettingen.

Valeinis, J., Cers, E. and Cielens, J. (2010). Two-sample problems in statistical data modelling. Mathematical modelling and analysis, 15(1), 137-151.

Valeinis, J. and Cers, E. (2011). Extending the two-sample empirical likelihood function. Preprint (available at http://home.lu.lv/∼valeinis/en/index.html).

Keywords: empirical likelihood; structural relationship model; plug-in; empirical processes

Biography: At present I am working as an assistant professor at the University of Latvia. I obtained my master degree in Kaiserslautern, Germany in 2003 and my PhD degree in Göttingen, Germany in 2007. After that I returned back to my homeland Latvia and now I am teaching statistics, financial mathematics, probability theory and stochastic processes at different levels (bachelor and master levels). My research interests concern mostly nonparametric statistical procedures: the empirical likelihood method, bootstrap, nonparametric goodness-of-fit tests, nonparametric regression and density estimation both for the independent and dependent data.