M-estimators of location for functional data

Abstract

M-estimators of location are widely used robust estimators of the center of univariate or multivariate real-valued data. This paper aims to study M-estimates of location in the framework of functional data analysis. To this end, recent developments for robust nonparametric density estimation by means of M-estimators are considered. These results can also be applied in the context of functional data analysis and allow to state conditions for the existence and uniqueness of location M-estimates in this setting. Properties of these functional M-estimators are investigated. In particular, their consistency is shown and robustness is studied by means of their breakdown point and their influence function. The finite-sample performance of the M-estimators is explored by simulation. The M-estimators are also empirically compared to trimmed means for functional data.

Robust functional regression based on principal components

“…Neither functional principal components derived from the covariance operator nor least-squares regression methods are robust to anomalous observations and this remains true even if penalized estimators are used. To obtain a robust method we instead propose combining M-estimators of location for functional data, (Sinova et al, 2018), functional principal components based on projection pursuit, (Bali et al, 2011), and MM estimators for regression (Yohai, 1987). We briefly review these ideas and explain their place in our proposal.…”

Robust multivariate and functional archetypal analysis with application to financial time series analysis

“…M-estimators try to lower the large influence of outliers by changing the square loss function for a less rapidly increasing loss function. Eugster and Leisch (2011) defined a loss function from R m to R. However, Sinova et al (2018) established several conditions of the loss function ρ for functional M-estimators, the first of which is that the loss function is defined from R + to R and the loss is specified as ρ(||r i ||). Furthermore, ρ(0) should be zero, ρ(x)/x should tend towards zero, when x tends towards zero, and ρ should be differentiable and ρ and φ(x) = ρ (x)/x should be both continuous and bounded, where we assume that φ(0) := lim x→0 ρ (x)/x exists and is finite.…”

“…Details about properties of functional M-estimators, such as their consistency and robustness by means of their breakdown point and their influence function can be found in Sinova et al (2018). Sinova et al (2018) also analyzed the common families of loss functions. We follow the ideas of Sinova et al (2018) and the Tukey biweight or bisquare family of loss function (Beaton and Tukey (1974)) with tuning parameter c is considered, since this loss function copes with extreme outliers well.…”

“…Sinova et al (2018) also analyzed the common families of loss functions. We follow the ideas of Sinova et al (2018) and the Tukey biweight or bisquare family of loss function (Beaton and Tukey (1974)) with tuning parameter c is considered, since this loss function copes with extreme outliers well. Therefore, RSS in equations 1, 2 and 3 are replaced by n i=1 ρ c (||r i ||), where · denotes the Euclidean norm for vectors, the L 2 -norm for univariate functions and corresponding norm for P -variate functions, and ρ c (||r i ||) is given by…”

“…Eugster and Leisch (2011) considered a kind of M-estimators for multivariate real-valued data (m-variate), where the domain of their loss function is not R + , but R m . Recently, Sinova et al (2018) considered functional M-estimators for the first time, where the domain of their loss function is R + . We base our proposal to robustify archetypal solutions in the multivariate and functional case on this last kind of loss function, which is commonly used in robust analysis (Maronna et al (2006)).…”