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A Kiefer--Wolfowitz theorem for convex densities
Statistics Theory · Submitted 5 Jan 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Kiefer and Wolfowitz [Z. Wahrsch. Verw. | background 1.00 | 0.02 | 0.01 | unclear 0.98 |
| s1 | 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. | background 0.99 | 0.03 | 0.09 | asserted 1.00 |
| s2 | We review their result and give an updated version of their proof. | contribution 0.91 | 0.86 | 0.03 | asserted 0.95 |
| s3 | 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. | result 0.81 | 0.95 | 0.85 | asserted 1.00 |
| s4 | 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. | method 0.96 | 0.13 | 0.02 | asserted 1.00 |
| s5 | Mech. 13 (1964) 827--835]. | background 0.92 | 0.03 | 0.01 | unclear 0.76 |
| s6 | 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]. | background 0.94 | 0.05 | 0.02 | asserted 1.00 |
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Original abstract
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].
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