Volume 31, Issue 2

This paper is devoted to the study of asymptotic properties of the regression function kernel estimate in the setting of continuous time stationary and ergodic data. More precisely, considering the Nadaraya–Watson type estimator, say m̂ T ( x ) , of the l -indexed regression function m ( x ) =𝔼 ( l ( Y )| X  =  x ) built upon continuous time stationary and ergodic data ( X t ,  Y t ) 0≤ t ≤ T , we establish its pointwise and uniform, over a dilative compact set, convergences with rates. Notice that the ergodic setting covers and completes various situations as compared to the mixing case and stands as more convenient to use in practice.

Nonparametric estimation of tail dependence can be based on a standardization of the marginals if their cumulative distribution functions are known. In this paper it is shown to be asymptotically more efficient if the additional knowledge of the marginals is ignored and estimators are based on ranks. The discrepancy between the two estimators is shown to be substantial for the popular Clayton and Gumbel–Hougaard models. A brief simulation study indicates that the asymptotic conclusions transfer to finite samples.

We show how regional prediction of car insurance risks can be improved for finer subregions by combining explanatory modeling with phenomenological models from industrial practice. Motivated by the control-variates technique, we propose a suitable combined predictor when claims data are available for regions but not for subregions. We provide explicit conditions which imply that the mean squared error of the combined predictor is smaller than the mean squared error of the standard predictor currently used in industry and smaller than predictors from explanatory modeling. We also discuss how a non-parametric random forest approach may be used to practically compute such predictors and consider an application to German car insurance data.

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).