L2 consistency of the kernel quantile estimator

É. Youndjé

Communications in Statistics - Theory and Methods, 2023, vol. 52, issue 17, 6111-6125

Abstract: Let F be a continuous distribution function and let Q be its associated quantile function. Let Fh be the kernel estimator of F and Qh that of Q. In this article the L2 right inversion distance between Qh and Q is introduced. It is shown that this distance can be represented in terms of Fh and F, more precisely it is established that the right inversion distance is equal to the conventional integrated squared error between Fh and F. This representation shows that any good bandwidth for Fh is a reasonable bandwidth for Qh and, this fact enables us to suggest methods to choose the smoothing parameter of Qh. Let Qĥcv be the kernel estimator of Q equipped with the global crossvalidation bandwidth ĥcv designed for Fh. Let Qĥpi be the linear kernel estimator of Q, ĥpi being the plug-in bandwidth function. A small scale simulation study presented in this paper contains some examples of distributions for which Qĥcv appears to be superior to Qĥpi. This paper also contains some properties of the classical L2 distance between Qh and Q.

Date: 2023
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DOI: 10.1080/03610926.2022.2026393

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