Testing the Equality of Location Parameters for Skewed Distributions Using S 1 with High Breakdown Robust Scale Estimators

Abstract

A simulation study had been carried out to compare the Type I error and power of S 1, a statistic recommended by Babu et al. (1999) for testing the equality of location parameters for skewed distributions. Othman et al. (in press) showed that this statistic is robust to the underlying populations and is also powerful. In our work, we modified this statistic by replacing the standard errors of the sample medians with four alternative robust scale estimators; the median absolute deviation (MAD) and three of the scale estimators proposed by Rousseeuw and Croux (1993); Q n , S n , and T n These estimators were chosen based on their high breakdown value and bounded influence function, and in addition, they are simple and easy to compute. Even though MAD is more appropriate for symmetric distributions (Rousseeuw and Croux, 1993), due to its popularity and for the purpose of comparison, we decided to include it in our study. The comparison of these methods was based on their Type I error and the power for J = 4 groups in an unbalanced design having heterogeneous variances. Data from the Chi-square distribution with 3 degrees of freedom were considered. Since the null distribution of S 1 is intractable, and its asymptotic null distribution may not be of much use for practical sample sizes, bootstrap methods were used to give a better approximation. The S 1 statistic combined with each of the scale estimators was shown to have good control of Type I errors.