Estimation of a k-monotone density: limit distribution theory and the Spline connection
By Fadoua Balabdaoui, Jon A. Wellner. 2005
102 citations recorded by the source corpus.
We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a kmonotone density g0 at a fixed point x0 when k> 2. We find that the jth derivative of the estima- tors at x0 converges at the rate n −(k−j)/(2k+1) for j =0 ,...,k � 1. The limiting distribution depends on an almost surely uniquely de- fined stochastic process Hk that stays above (below) the k-fold in- tegral of Brownian motion plus a deterministic drift, when k is even (odd). Both the MLE and LSE are known to be splines of degree k � 1 with simple knots. Establishing the order of the random gap τ + n � τ − n where τ ± n denote two successive knots, is a key ingredient of the proof of the main results. We show that this "gap problem" can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.
Subfields: Applied Mathematics, Finance, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/3d80960817f85e1aececaa2b164b7de80108fea0
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In Applied Mathematics
- Yining Chen
- Ryan Martin
- Moritz Hardt
- Ilias Diakonikolas
- Madeleine Cule
- Richard J. Samworth
- Pierre C. Bellec
In Finance
In Statistics and Probability
- Richard J. Samworth
- Yining Chen
- Ilias Diakonikolas
- Ludwig Schmidt
- Ryan Martin
- Moritz Hardt
- Madeleine Cule
- Pierre C. Bellec
Citations in the corpus
- Smoothed log-concave maximum likelihood estimation with applications (2011, methodology): “Other methods for enforcing various shape constraints have been studied in Braun and Hall (2001), Groeneboom, Jongbloed and Wellner (2001), Balabdaoui and Wellner (2007), Balabdaoui and Wellner (2010), Pavlides and Wellner (2012) and Carroll, Delaigle and Hall (2011).”
- An imprecise-probabilistic characterization of frequentist statistical inference (2021, methodology): “Estimation of a monotone density is a classical problem, and the nonparametric maximum likelihood estimator was developed by Grenander (1956) and extensively studied by Prakasa Rao (1969), Groeneboom (1985), and Balabdaoui and Wellner (2007), among others. In the context of regression, testing for m”
- Differentially Private Learning of Structured Discrete Distributions (2015, background): “There is a long history of research in statistics on estimating structured families of distributions going back to the 1950’s [10, 11, 12, 13], and it is still a very active research area [14, 15, 16].”
- Sharp oracle inequalities for Least Squares estimators in shape restricted regression (2015, background): “For density estimation, m-monotone functions have been studied in [3, 4].”
- Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density (2009, methodology): “The properties of the Grenander estimator are now quite well understood, thanks to the works of Marshall and Proschan (1965), Prakasa Rao (1969), Devroye (1987), Birgé (1989), van de Geer (1993) and Balabdaoui et al.”