Robust pairwise multiple comparisons under short-tailed symmetric distributions

Abstract

In one-way ANOVA, most of the pairwise multiple comparison procedures depend on normality assumption of errors. In practice, errors have non-normal distributions so frequently. Therefore, it is very important to develop robust estimators of location and the associated variance under non-normality. In this paper, we consider the estimation of one-way ANOVA model parameters to make pairwise multiple comparisons under short-tailed symmetric (STS) distribution. The classical least squares method is neither efficient nor robust and maximum likelihood estimation technique is problematic in this situation. Modified maximum likelihood (MML) estimation technique gives the opportunity to estimate model parameters in closed forms under non-normal distributions. Hence, the use of MML estimators in the test statistic is proposed for pairwise multiple comparisons under STS distribution. The efficiency and power comparisons of the test statistic based on sample mean, trimmed mean, wave and MML estimators are given and the robustness of the test obtained using these estimators under plausible alternatives and inlier model are examined. It is demonstrated that the test statistic based on MML estimators is efficient and robust and the corresponding test is more powerful and having smallest Type I error.

Robust Post-Hoc Multiple Comparisons: Skew t Distributed Error Terms

“…If max | tij | > A * ij,α,a the null hypothesis given in equation ( 2) is rejected at the level of signicance α. A * ij,α,a represents a constant which is to be chosen so that the desired joint condence coecient 1−α is achieved as closely as possible (Balci & Akkaya, 2015). In this study the value of A * ij,α,a is simulated in order to achieve true α value.…”

“…Ringland (1983) use M estimators, Dunnett (1982) compared many robust estimators like Huber's wave (W24), bisqure (BS82), Hampel (H22), trimmed mean with winsorized standard deviation using 10% trimming (TM10), modied maximum likelihood (MML) estimator with 10% censoring (MML10) and sample mean in a modied T-method for pairwise multiple comparisons. Balci & Akkaya (2015) used MML estimators with using short tailed symmetric distributions and resulted that to avoid W24 and TM10 methodology under short tailed symmetric distributions. In this study we extend the comparisons with using Azzalini's skew t (St) distribution (Azzalini & Capitanio, 2003).…”

Commentary on S. Kumar and P. Chhaparwal, 2016. A robust unbiased dual to product estimator for population mean through Modified Maximum Likelihood in simple random sampling, Cogent Mathematics, 3:1168070

“…Modified maximum likelihood (MML) methodology was first introduced by Tiku (1967) and has been used by many scholars, such as Islam and Tiku (2010), Oral (2012), Balci and Akkaya (2015), Oral (2017), Oral and Danos (2018), Celik and Senoglu (2018), and so on, in various research areas. Kumar and Chhaparwal (2016) proposed a robust unbiased dual to product estimator for estimating population mean using the MML methodology.…”