Mie Fujiki

In this paper, we consider a modified deepest regression estimator (DRE) in logistic regression model. We propose an estimator that takes the median of all candidate fits with maximal regression depth. We compare a modified DRE with the maximum likelihood estimator, Firth's method, and the original DRE in logistic regression model by the computer simulations. We show that a modified DRE is not affected by overlap or complete separation.

Rousseeuw and Hubert (1999) introduced the regression depth method for linear regression models.Regression depth is defined as the smallest number of residuals that need to change sign. DRE is defined as the fit which makes regression depth the maximum relative to the data. DRE is a robust regression estimator and it has high asymptotic efficiency. However, in simple regression, the performance of DRE is not high in case of small sample size with outliers. We showed that mean squared error of a modified DRE using median is smaller than that other estimators in small and large sample size with outliers (Fujiki and Shirahata, 2011).

Moreover, due to the monotone invariance property of regression depth, DRE is invariant to monotone transformations of the response, though this property does not hold for least squares or other estimators such as least trimmed squares or S-estimators. Thereby, it is possible to apply DRE to more general models. In general, the maximum likelihood method is used to estimate regression parameters in logistic regression model. However, a maximum likelihood estimator does not exist in case of complete separation or quasi-complete separation. Firth (1993) suggested the method to remove bias of a maximum likelihood estimator, but this method is not investigated under near separation. Though Ohkura and Kamakura (2007) discussed the method to approximate an estimator using Firth's method to an estimator using the exact logistic regression, DRE is not yet compared with Firth's method in logistic regression model. Therefore, we consider that we apply a modified DRE in logistic regression model, and we also investigate an estimator using Firth's method and our DRE under near separation or overlap by the computer simulations.

**Keywords:** Robust estimation; Data depth; Logistic regression; Simulation

**Biography:** Area of Research: Regression analysis, Robust estimation, Data Depth, Statistics Education.

I study the robust estimation and the statistical education. Especially I am interested in the robust regression estimator using data depth. I want to propose the new estimator using data depth that improved to become more useful. Moreover, I'd like to investigate Japanese university students about an image of “Statistic” in detail.