A Kiefer--Wolfowitz theorem for convex densities
| dc.creator | Balabdaoui, Fadoua | |
| dc.creator | Wellner, Jon A. | |
| dc.date | 2007-01-05 | |
| dc.date | 2007-09-07 | |
| dc.date.accessioned | 2026-07-07T08:35:13Z | |
| dc.date.available | 2026-07-07T08:35:13Z | |
| dc.description | Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\hat{F}_n$, which is, in fact, the least concave majorant of the empirical distribution function $\mathbb {F}_n$, differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1}\log n)^{2/3}$ almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions $F$ with convex decreasing densities $f$, but with the maximum likelihood estimator $\hat{F}_n$ of $F$ replaced by the least squares estimator $\widetilde{F}_n$: if $X_1,..., X_n$ are sampled from a distribution function $F$ with strictly convex density $f$, then the least squares estimator $\widetilde{F}_n$ of $F$ and the empirical distribution function $\mathbb {F}_n$ differ in the uniform norm by no more than a constant times $(n^{-1}\log n)^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall [J. Approximation Theory 1 (1968) 209--218], Hall and Meyer [J. Approximation Theory 16 (1976) 105--122], building on earlier work by Birkhoff and de Boor [J. Math. Mech. 13 (1964) 827--835]. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor [A Practical Guide to Splines (2001) Springer, New York]. | |
| dc.description | Published at http://dx.doi.org/10.1214/074921707000000256 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0701179 | |
| dc.identifier | http://arxiv.org/abs/math/0701179 | |
| dc.identifier | IMS Lecture Notes Monograph Series 2007, Vol. 55, 1-31 | |
| dc.identifier | doi:10.1214/074921707000000256 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/139678 | |
| dc.subject | Statistics Theory | |
| dc.subject | 62G10, 62G20 (Primary); 62G30 (Secondary) | |
| dc.title | A Kiefer--Wolfowitz theorem for convex densities | |
| dc.type | text |