Bounded influence nonlinear signed‐rank regression

Abstract

In this paper we consider weighted generalized‐signed‐rank estimators of nonlinear regression coefficients. The generalization allows us to include popular estimators such as the least squares and least absolute deviations estimators but by itself does not give bounded influence estimators. Adding weights results in estimators with bounded influence function. We establish conditions needed for the consistency and asymptotic normality of the proposed estimator and discuss how weight functions can be chosen to achieve bounded influence function of the estimator. Real life examples and Monte Carlo simulation experiments demonstrate the robustness and efficiency of the proposed estimator. An example shows that the weighted signed‐rank estimator can be useful to detect outliers in nonlinear regression. The Canadian Journal of Statistics 40: 172–189; 2012 © 2012 Statistical Society of Canada

Rank-based estimating equation with non-ignorable missing responses via empirical likelihood

“…Overall, it is not surprising that for heavy-tailed model errors and contaminated data such as in the presence of outliers, the rank-based approach provides robust and more efficient estimators than its least-squares counterpart (Hettmansperger & McKean, 2011;Bindele & Abebe, 2012) for the complete case analysis. However, when it comes to direct statistical inference (confidence intervals/regions) about the true regression coefficients from model (1.1) with responses missing not at random, via the simulation study and the real data example in this paper, it is proven that the empirical likelihood based on the rank-based estimating equation provides a more appealing alternative compared to its normal approximation inference and its least squares counterpart.…”

“…Assumption (I 1 ) is a regular assumption in the rank-based framework; see Hettmansperger & McKean (2011) and Bindele & Abebe (2012). Assumptions (I 2 ) − (I 4 ) are necessary to ensure the result in Theorem 2; see Einmahl & Mason (2005), Rao (2009) and Wied & Weißbach (2012).…”

“…Under the MAR assumption however, as pointed out in Little & Rubin (2002), the complete case analysis, that is, ignoring observations with missing response, will lead to an efficient ML estimator. It is worth pointing out that even for the complete case analysis, the rank-based approach introduced by Jaeckel (1972) outperforms the aforementioned approaches in terms of robustness and efficiency when dealing with heavy-tailed model errors and/or in the presence of outliers; see Hettmansperger & McKean (2011) for linear models and Bindele & Abebe (2012) for nonlinear models. Recently, for missing responses under the MAR assumption, a rank-based approach has been proposed by ?…”

“…The motivation behind the use of the Jaeckel (1972) objective function is threefold: (i) it is known to result in a robust and more efficient estimator compared to many of the mentioned estimation methods such as the least-squares (LS), the maximum likelihood (ML), and many other methods of moments including the least absolute deviation (LAD), (ii) it does not require model errors' distribution specification, and (iii) similar to the LS approach, it has a simple geometric interpretability. For more on these facts, see Hettmansperger & McKean (2011) and Bindele & Abebe (2012).…”

Generalised signed-rank estimation for nonlinear models with multidimensional indices

“…This is not the case for Wu's and Jennrich's sufficient conditions (see Kundu, 1993). Bindele and Abebe (2012) were able to establish the asymptotic and robustness properties of the generalised signed-rank (GSR) estimator for nonlinear regression models with iid errors without imposing such Lipschitz-type sufficient conditions. Thus, the GSR estimator appears to be a good candidate for dealing with nonlinear models with multidimensional indices, especially those of the harmonic variety.…”