Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data

Duembgen, Lutz; Huesler, Andre; Rufibach, Kaspar

doi:10.48550/arxiv.0707.4643

Cited by 12 publications

(18 citation statements)

References 7 publications

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“…Best and Chakravarti [3] or Dümbgen et al [8], sometimes combined with the ordinary PAVA in a particular fashion. It will be shown that all these algorithms find the exact solution in finitely many steps, at least when Q(•) is an arbitrary quadratic and strictly convex function.…”

Section: Introductionmentioning

confidence: 99%

Least squares and shrinkage estimation under bimonotonicity constraints

Beran

Dümbgen

2009

Stat Comput

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In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r × s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i, j) ∈ {1, . . . , r} × {1, . . . , s}. Various numerical examples illustrate our methods.

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Section: Introductionmentioning

confidence: 99%

Least squares and shrinkage estimation under bimonotonicity constraints

Beran

Dümbgen

2009

Stat Comput

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“…The first part can be transformed into a convex optimization problem, where the unique optimum φ ∈ Φ can be found very quickly by an active set algorithm implemented in the R package logcondens . More details on its implementation can be found in Dümbgen, Hüsler and Rufibach (2011).…”

Section: Computational Issues and Numerical Properties 41 Computation...mentioning

confidence: 99%

“…The nonparametric log-concave maximum likelihood density estimator was studied in the i.i.d. setting by Walther (2002), Pal, Woodroofe and Meyer (2007), Dümbgen and Rufibach (2009), Balabdaoui, Rufibach and Wellner (2009), Cule, Samworth and Stewart (2010), , Schuhmacher, Hüsler and Dümbgen (2011) and Dümbgen, Hüsler and Rufibach (2011). These references contain characterizations of the estimator, asymptotics and algorithms for its computation.…”

Section: Introductionmentioning

confidence: 99%

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

Chen

2014

Scandinavian J Statistics

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This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its ConsistencyYining Chen AbstractWe study semiparametric time series models with innovations following a log-concave distribution. We propose a general maximum likelihood framework which allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well-known models, including ARMA, GARCH and ARMA-GARCH. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA-GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log-return of FTSE 100 index and the rabbit population.

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“…More details on recent research on log-concave density estimation is provided in the appendix. Rate of uniform convergence for the univariate log-concave maximum likelihood estimate was derived in Dümbgen and Rufibach (2009) and its computation is described in Rufibach (2007), Dümbgen et al (2010), and. The corresponding software is available from CRAN as the R package logcondens (Rufibach and Dümbgen, 2011).…”

Section: Log-concave Density Estimationmentioning

confidence: 99%

A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates

Rufibach¹

2012

The International Journal of Biostatistics

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We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the logconcavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou and Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios.

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Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data

Abstract: We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indicate an EM algorithm to treat arbitrarily censored or binned data.

Cited by 12 publications

References 7 publications

Least squares and shrinkage estimation under bimonotonicity constraints

Least squares and shrinkage estimation under bimonotonicity constraints

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates

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