Least squares and shrinkage estimation under bimonotonicity constraints

Beran, Rudolf; Dümbgen, Lutz

doi:10.1007/s11222-009-9124-0

Cited by 10 publications

(17 citation statements)

References 7 publications

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“…isotonic with respect to bimonotone order relation on a set X , if whenever x 1 x 2 one has g(x 1 ) ≤ g(x 2 ), cf. [7]. A real valued function h(x) is called a bimonotone decreasing function, if whenever x 1…”

Section: Application To Bimonotone Probability Mass Function and Regrmentioning

confidence: 99%

“…[16]. In this setting we would also like to mention [7], that studied algorithms resulting from the minimisation of a smooth criterion function under bimonotonicity constraints. In the regression setting we are able to derive the limit distributions for the isotonic regression of functions that are monotone with respect to the matrix pre-order on Z d + , for arbitrary d > 1, cf.…”

Section: Introductionmentioning

confidence: 99%

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The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

Anevski

Pastukhov

2018

Electron. J. Statist.

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We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.MSC 2010 subject classifications: 62F30, 62G20, 62G08.

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Section: Application To Bimonotone Probability Mass Function and Regrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

Anevski

Pastukhov

2018

Electron. J. Statist.

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“…Algorithms for isotonic regressions on multidimensional orderings, without imposing restrictions such as additive models, have concentrated on 2 dimensions [5,12,14,25,30,31,36]. For the L 2 metric and 2-dimensional points, the fastest known algorithms take Θ(n 2 ) time if the points are in a grid [36] and Θ(n 2 log n) time for points in general position [39], while for L 1 the times are Θ(n log n) and Θ(n log 2 n), respectively [39].…”

Section: Multidimensional Orderingmentioning

confidence: 99%

Isotonic Regression for Multiple Independent Variables

Stout

2013

Algorithmica

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This paper gives algorithms for determining isotonic regressions for weighted data at a set of points P in multidimensional space with the standard componentwise ordering. The approach is based on an orderpreserving embedding of P into a slightly larger directed acyclic graph (dag) G, where the transitive closure of the ordering on P is represented by paths of length 2 in G. Algorithms are given which, compared to previous results, improve the time by a factor ofΘ(|P |) for the L 1 , L 2 , and L ∞ metrics. A yet faster algorithm is given for L 1 isotonic regression with unweighted data. L ∞ isotonic regression is not unique, and algorithms are given for finding L ∞ regressions with desirable properties such as minimizing the number of large regression errors.

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“…Recent papers considering additional information issues are, for example, Beran and Dümbgen (2010) and Oh and Shin (2011). It is frequent that this information tells us that the observations from one of the populations, for example Π 1 , take higher (or lower) values than those coming from the other, i.e.…”

Section: Introductionmentioning

confidence: 99%

Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial

Conde

Salvador

Rueda

et al. 2013

Statistical Applications in Genetics and Molecular Biology

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Classification rules that incorporate additional information usually present in discrimination problems are receiving certain attention during the last years as they perform better than the usual rules in poor discrimination problems. Fernández et al (2006) proved that these rules have a lower unconditional misclassification probability than the usual Fisher's rule but they did not consider the estimation of the conditional error probability when a training sample is given (the so-called true error rate) which is a very interesting parameter in practice.In this paper we consider the problem of estimating the true error rate of these classification rules in the classical topic of discrimination among two normal populations. We prove theoretical results on the apparent error rate of the rules that expose the need of new estimators of their true error rate. Our proposal is to also consider the additional information in the definition of the true error rate estimators. We propose four such new estimators. Two of them are defined incorporating the additional information into the leave-one-out-bootstrap. The other two are the corresponding cross-validation after bootstrap versions. We compare these new estimators with the usual ones in a simulation study and in a cancer trial application, showing the very good behavior, in terms of mean square error, of the leave-one-out bootstrap estimators that incorporate the available additional information.

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Least squares and shrinkage estimation under bimonotonicity constraints

Cited by 10 publications

References 7 publications

The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

Isotonic Regression for Multiple Independent Variables

Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial

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