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Oracle inequality for conditional density estimation and an actuarial example. (English) Zbl 1440.62118
Summary: Conditional density estimation in a parametric regression setting, where the problem is to estimate a parametric density of the response given the predictor, is a classical and prominent topic in regression analysis. This article explores this problem in a nonparametric setting where no assumption about shape of an underlying conditional density is made. For the first time in the literature, it is proved that there exists a nonparametric data-driven estimator that matches performance of an oracle which: (i) knows the underlying conditional density, (ii) adapts to an unknown design of predictors, (iii) performs a dimension reduction if the response does not depend on the predictor, (iv) is minimax over a vast set of anisotropic bivariate function classes. All these results are established via an oracle inequality which is on par with ones known in the univariate density estimation literature. Further, the asymptotically optimal estimator is tested on an interesting actuarial example which explores a relationship between credit scoring and premium for basic auto-insurance for 54 undergraduate college students.

##### MSC:
 62G07 Density estimation 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62P05 Applications of statistics to actuarial sciences and financial mathematics
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