A Comparison of the Estimators of the Scale Parameter of the Errors Distribution in the L1 Regression

The L1 regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To calculate the standard errors of the L1 estimators, construct confidence intervals and test hypotheses about the parameters of the model, or to calculate a robust coefficient of determination, it is necessary to have an estimate of a scale parameterτ. This parameter is such that τ2/n is the variance of the median of a sample of size n from the errors distribution. [1] proposed the use of , a consistent, and so, an asymptotically unbiased estimator of τ. However, this estimator is not stable in small samples, in the sense that it can increase with the introduction of new independent variables in the model. When the errors follow the Laplace distribution, the maximum likelihood estimator of τ, say , is the mean absolute error, that is, the mean of the absolute residuals. This estimator always decreases when new independent variables are added to the model. Our objective is to develop asymptotic properties of under several errors distributions analytically. We also performed a simulation study to compare the distributions of both estimators in small samples with the objective to establish conditions in which is a good alternative to for such situations.

GOAL PROGRAMMING ALGORITHM FOR A TYPE OF LEAST ABSOLUTE VALUE REGRESSION PROBLEM

This paper develops goal programming algorithm to solve a type of least absolute value (LAV) problem. Firstly, we simplify the simplex algorithm by proving the existence of solutions of the problem. Then, we present a goal programming algorithm on the basis of the original techniques. Theoretical analysis and numerical results indicate that the new method contains a lower number of deviation variables and consumes less computational time as compared to current LAV methods.

Incorporation of source contributions to improve the accuracy of soil heavy metal mapping using small sample sizes at a county scale

Estimating heavy metal(HM) distribution with high precision is the key to effectively preventing Chinese medicinal plants from being polluted by the native soil. A total of 44 surface soil samples were gathered to detect the concentrations of eight HMs(As, Hg, Cu, Cr, Ni, Zn, Pb, and Cd) in the herb growing area of Luanping County, northeastern Hebei Province, China. An absolute principal component score-multiple linear regression(APCS-MLR) model was used to quantify pollution source contributions to soil HMs. Furthermore, the source contribution rates and environmental data of each sampling point were simultaneously incorporated into a stepwise linear regression model to identify the crucial indicators for predicting soil HM spatial distributions. Results showed that 88% of Cu, 72% of Cr, and 72% of Ni came from natural sources;50% of Zn, 49% of Pb, and 59% of Cd were mainly caused by agricultural activities;and 44% of As and 56% of Hg originated from industrial activities. When three-type(natural, agricultural, and industrial) source contribution rates and environmental data were simultaneously incorporated into the stepwise linear regression model, the fitting accuracy was significantly improved and the model could explain 31%–86% of the total variance in soil HM concentrations. This study introduced three-type source contributions of each sampling point based on APCS-MLR analysis as new covariates to improve soil HM estimation precision, thus providing a new approach for predicting the spatial distribution of HMs using small sample sizes at the county scale.