Generalised signed-rank estimation for nonlinear models with multidimensional indices

Abstract

We consider a nonlinear regression model when the index variable is multidimensional. Such models are useful in signal processing, texture modelling, and spatio-temporal data analysis. A generalised form of the signed-rank estimator of the nonlinear regression coefficients is proposed. This general form of the signed-rank (SR) estimator includes L p estimators and hybrid variants. Sufficient conditions for strong consistency and asymptotic normality of the estimator are given. It is shown that the rate of convergence to normality can be different from √ n. The sufficient conditions are weak in the sense that they are satisfied by harmonic-type functions for which results in the current literature may not apply. A simulation study shows that certain generalised SR estimators (e.g. signed rank) perform better in recovering signals than others (e.g. least squares) when the error distribution is contaminated or is heavy-tailed.

Signed-rank analysis of a partial linear model with B-splines estimated monotone non parametric function

“…Note that the fact that ϕ is taken to be bounded, assumption (ii) on the weights a n (i) in Baklanov (2006) is systematically satisfied. Note that the above Lemma is not discussed in Baklanov (2006) but a proof of Lemma 3.4 can be found in Nguelifack et al (2014). A proof of Lemma 3.5 is provided in the Appendix.…”