Robust estimation of the population mean using quantile regression under systematic sampling

Abstract

ABSTRACT Regression ratio mean estimators of a study variable are defined as the coefficients provided by the ordinary least-squares regression of on a given auxiliary variable . They can be improved by using the coefficient of variation and the coefficient of kurtosis of . The influence of outliers on the estimates of the population mean of is neutralized by calculating robust regression coefficients, obtained by the method of either least absolute deviations, Huber-M, Huber-MM, Hampel-M, Tukey-M, or adjusted least squares. These robust coefficients are used to estimate the population mean of under simple random sampling. Extension to systematic sampling—which is a probability sampling in which every element of the population has equal probability of inclusion to be drawn—using the coefficients provided by quantile regression—whose coefficients result from the minimization of the sum of absolute deviations rather than from the square deviations from the regression line—requires ratio estimators of the population mean of . The mean square errors of these estimators are expressed analytically. If the quantile regression coefficient is greater than the ratio of the covariance between the study and the auxiliary variables to the variance of the auxiliary variable minus a function of the mean or the coefficient of variation, skewness, or kurtosis of and , then the proposed robust quantile regression mean estimator of is more efficient than the ratio estimators in the presence of outliers under systematic sampling. The reason is that these estimators only use regression coefficients and not the ratio between the population mean and sample means of the auxiliary variable . The aforementioned condition occurs with the values of the case study. For empirical data of 176 forest strips, the proposed estimate of the volume of timber is over 30% more efficient than the ratio estimates based on quantile regression coefficients.