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Non-parametric estimation of conditional quantiles. (English) Zbl 0678.62049

Summary: Let (X,Y) be a two-dimensional random variable with a joint density function f(x,y) and a joint distribution function \[ F(x,y)=\int^{x}_{-\infty}\int^{y}_{-\infty}f(u,v)dv du. \] Following E. Nadaraya [Teor. Veroyatn. Primen. 9, 550-554 (1964; Zbl 0152.176); English translation in Theor. Probab. Appl. 9, 497-500 (1965)] and M. Rosenblatt [Conditional probability density and regression estimators. In: P. R. Krishnaiah (ed.), Multivariate analysis. II. Proc. 2nd Int. Symp. Multivariate analysis, Wright State Univ., Dayton/Ohio 1968, 25-31 (1969; for a review of the entire collection see Zbl 0212.220)] a class of nonparametric estimators of conditional quantiles of Y for a given value of X, based on a random sample from the above distribution, is proposed. It is shown that under some regularity conditions the estimators are strongly consistent and asymptotically normally distributed.

MSC:

62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
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