Robust CDF‐Filtering of a Location Parameter

ABSTRACT This paper introduces a novel framework for designing robust filters associated with signal plus noise models having symmetric observation density. The filters are obtained by a recursion where the innovation term is a transform of the cumulative distribution function of the residuals. The latter downweights extreme values by construction and allows the filters to be analytically tractable. The updating scheme naturally arises as the solution of an optimization problem, where the objective function is a continuous version of the quantile check function, formerly employed as a proper scoring function for quantiles and used to construct robust minimum contrast estimators. Stationarity, ergodicity and invertibility are derived under minimal assumptions and preserved under different parametric specifications. Estimation is carried out by the method of maximum likelihood and the asymptotic theory is developed under misspecification. As an illustration, the new filters are applied to brain scan data and compared across Gaussian, Student‐t, Cauchy and Logistic density specifications, with alternative methods. Additional results include a novel class of score‐driven models and a subgaussian density suitable for robust filtering and modelling, arising as the infinite sum of independent non‐identically distributed uniform random variables.