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|a (DE-627)JST055255124
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|a (JST)20461482
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|a DE-627
|b ger
|c DE-627
|e rakwb
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|a eng
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|a Balabdaoui, Fadoua
|e verfasserin
|4 aut
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|a A Kiefer-Wolfowitz Theorem for Convex Densities
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|c 2007
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|a Text
|b txt
|2 rdacontent
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|a Computermedien
|b c
|2 rdamedia
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|a Online-Ressource
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|a 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 $\widehat{F}_{n}$ , which is, in fact, the least concave majorant of the empirical distribution function ${\Bbb F}_{n}$ , differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1}\text{log}n)^{2/3}$ 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 $\widehat{F}_{n}$ of F replaced by the least squares estimator $\tilde{F}_{n}$ : if X₁,..., $X_{n}$ are sampled from a distribution function F with strictly convex density f, then the least squares estimator $\tilde{F}_{n}$ of F and the empirical distribution function ${\Bbb F}_{n}$ differ in the uniform norm by no more than a constant times $(n^{-1}\text{log}n)^{3/5}$ 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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|a Copyright 2007 Institute of Mathematical Statistics
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| 650 |
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|a Primary 62G10
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|a Primary 62G20
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| 650 |
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|a secondary 62G30
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|a Brownian bridge
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| 650 |
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|a convex density
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|a distance
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|a empirical distribution
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|a invelope process
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|a monotone density
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|a optimality theory
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|a shape constraints
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| 650 |
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|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Descriptive statistics
|x Statistical distributions
|x Distribution functions
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| 650 |
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4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Inferential statistics
|x Statistical estimation
|x Estimation methods
|x Estimators
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| 650 |
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4 |
|a Mathematics
|x Mathematical analysis
|x Numerical analysis
|x Interpolation
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| 650 |
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4 |
|a Mathematics
|x Applied mathematics
|x Analytics
|x Analytical estimating
|x Maximum likelihood estimation
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| 650 |
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4 |
|a Mathematics
|x Mathematical expressions
|x Mathematical theorems
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| 650 |
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4 |
|a Information science
|x Information analysis
|x Data analysis
|x Regression analysis
|x Linear regression
|x Least squares
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| 650 |
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4 |
|a Physical sciences
|x Physics
|x Mechanics
|x Density
|x Density measurement
|x Density estimation
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| 650 |
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4 |
|a Mathematics
|x Applied mathematics
|x Game theory
|x Minimax
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Statistical theories
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| 650 |
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4 |
|a Mathematics
|x Mathematical procedures
|x Approximation
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Descriptive statistics
|x Statistical distributions
|x Distribution functions
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Inferential statistics
|x Statistical estimation
|x Estimation methods
|x Estimators
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| 650 |
|
4 |
|a Mathematics
|x Mathematical analysis
|x Numerical analysis
|x Interpolation
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Analytics
|x Analytical estimating
|x Maximum likelihood estimation
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| 650 |
|
4 |
|a Mathematics
|x Mathematical expressions
|x Mathematical theorems
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| 650 |
|
4 |
|a Information science
|x Information analysis
|x Data analysis
|x Regression analysis
|x Linear regression
|x Least squares
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| 650 |
|
4 |
|a Physical sciences
|x Physics
|x Mechanics
|x Density
|x Density measurement
|x Density estimation
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Game theory
|x Minimax
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Statistical theories
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| 650 |
|
4 |
|a Mathematics
|x Mathematical procedures
|x Approximation
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| 655 |
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4 |
|a research-article
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| 700 |
1 |
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|a Wellner, Jon A.
|e verfasserin
|4 aut
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| 773 |
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|i Enthalten in
|t Lecture Notes-Monograph Series
|d Institute of Mathematical Statistics, 1982
|g 55(2007) vom: Jan., Seite 1-31
|w (DE-627)583817815
|w (DE-600)2460925-0
|x 07492170
|7 nnns
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| 773 |
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|g volume:55
|g year:2007
|g month:01
|g pages:1-31
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|u https://www.jstor.org/stable/20461482
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|d 55
|j 2007
|c 01
|h 1-31
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