A Note on Distribution and Density Estimation as a Regression Problem of Order Statistics

Résumé

We show how cumulative distribution function estimation (cdf) can be formulated as a regression problem, using order statistics. This results in a nonstandard nonparametric regression problem, with correlated but van- ishing errors. Nonparametric regression estimators can then be leveraged to obtain corresponding estimators of the cdf and of its derivatives. We illus- trate the approach, theoretically with Nadaraya-Watson regression estimator, and numerically with P-splines regression techniques. We discuss the issue of smoothing parameter selection for regression of correlated errors and show how to make use of known techniques from the regression framework to improve on the cdf and density estimation. Eventually, we draw some perspectives and show how the proposed framework could be extended to the multivariate setting.