Abstract
Let (X, Y) be a bivariate random vector and F(x) the marginal distribution function of X. The quantile regression (QR) function of Y on X is defined as r(u) = E[Y | F(X) = u] and the cumulative QR function (CQR) M(u) as its integral over [0, u]. The empirical counterpart based on a sample of size n is M n (u). In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on M in (u). A law of the iterated logarithm for the CQR process follows from our result.
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Tse, S.M. On the cumulative quantile regression process. Math. Meth. Stat. 18, 270–279 (2009). https://doi.org/10.3103/S1066530709030053
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DOI: https://doi.org/10.3103/S1066530709030053
Key words
- quantile regression function
- Lorenz curve
- strong Gaussian approximation
- induced order statistics
- partial sum process
- empirical process
- law of the iterated logarithm