Titirut Mekbunditkul , Pachitjanut Siripanich Tobit-Piecewise Estimator of the Regression Coefficient Titirut Mekbunditkul * , Pachitjanut Siripanich School of Applied Statistics, National Institute of Development Administration. * EMAIL: tple@hotmail.com Abstract: In this paper, an alternative estimator of the regression coefficient, called TP estimator, is proposed based on the idea of Tobit and piecewise regression, in order to fit a data with outliers. A suitable likelihood function is derived for desired conditions so that the TP estimator is the maximum likelihood estimator (MLE). It is found that the regression line obtained by the proposed method is preferable to the least square (LS) and others method as shown by various examples. Keywords: Outliers, Tobit regression model, Piecewise regression model, MLE I. Introduction An important problem usually found in regression analysis is that data are from two or more contaminated distributions and hence outliers are appeared. These cause a regression line away from most of the data points. To cope this problem, some may fit the model by deleting the outliers or “down- weight” the outliers. Many robust regression methods were introduced by various authors. This paper introduces an alternative method called TP, abbreviated from Tobit- piecewise. Consider the piecewise regression first proposed by Quandt [4], for instance, fit one data set, by two regression lines in stead of a single one so that two distributions of error are taken into account as they should be. In addition, based on Tobit regression introduced by Tobin [9], putting limited value for outliers is one procedure of “down-weighted” value (reduce effect) of outliers at the inner fences of the data (Hyndman [1]). Consider a scatter diagram of a data set in Figure 1, the points in a circle are obviously outliers. When this data is fitted by LS, the outliers have strong influential on regression line so that it does not represent the bulk of the data meanwhile we fit the data by either Tobit or piecewise regression that is better performance than the ordinary regression. In stead, if we combine the Tobit and piecewise modeling ideas, called the TP model described in the next section, we will get the TP regression that yields the best fitting among 4 regression lines for this data. Hence, the TP method is an alternative robust method for situation when error are from two distributions of difference various and/or outliers exist. A data set in Figure 1, for example, supports such “belief”. Figure 1. The comparison of 4 regressions fitting II. Fundamental Notions Basic Idea Related to the Tobit Regression Model Tobin [9] introduced the Tobit regression to fit a linear relationship by putting limit on values of some variables. As explained by Tobin, there are some phenomena where the dependent variable being the total durable goods expenditure has some observed data take value as zero during time of survey. This zero value does not mean that the observation has never bought the durable goods but only during the time of survey he has not. In another case, the dependent variable, the students’ monthly expenditure, has some observations of higher expenditure than the rest because they are rich persons. Consequently, this zero value or the higher expenditure is assumed to be the lower or upper limit, respectively, in Tobit regression analysis. In this paper, we assign an upper or lower value to the data point in order to limit value of outliers instead of ignoring (weighted by zero) them and then construct an alternative estimator of regression coefficient. In particularly, bounds are the locally inner fences. Consider a two-limit Tobit model (Tobin [9], Rosett [5] and Jöreskog [2]) which is a relationship of variables Y and as shown in the model (1). The observed variable satisfies Y * Y The 4 th International Conference on Operations and Supply Chain Management, Hong Kong & Guangzhou, Jul.25 to Jul.31, 2010 365