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Bootstrap estimation of the variance of the error term in monotonic regression models
Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Statistics.
Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Statistics.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .ORCID iD: 0000-0003-1836-4200
2013 (English)In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 83, no 4, 625-638 p.Article in journal (Refereed) Published
Abstract [en]

The variance of the error term in ordinary regression models and linear smoothers is usually estimated by adjusting the average squared residual for the trace of the smoothing matrix (the degrees of freedom of the predicted response). However, other types of variance estimators are needed when using monotonic regression (MR) models, which are particularly suitable for estimating response functions with pronounced thresholds. Here, we propose a simple bootstrap estimator to compensate for the over-fitting that occurs when MR models are estimated from empirical data. Furthermore, we show that, in the case of one or two predictors, the performance of this estimator can be enhanced by introducing adjustment factors that take into account the slope of the response function and characteristics of the distribution of the explanatory variables. Extensive simulations show that our estimators perform satisfactorily for a great variety of monotonic functions and error distributions.

Place, publisher, year, edition, pages
Taylor & Francis Group, 2013. Vol. 83, no 4, 625-638 p.
Keyword [en]
uncertainty estimation; bootstrap; monotonic regression; pool-adjacent-violators algorithm
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-78858DOI: 10.1080/00949655.2011.631138ISI: 000317276900003OAI: oai:DiVA.org:liu-78858DiVA: diva2:536280
Available from: 2012-06-21 Created: 2012-06-21 Last updated: 2017-12-07
In thesis
1. Monotonic regression for large multivariate datasets
Open this publication in new window or tab >>Monotonic regression for large multivariate datasets
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Monoton regression för stora multivariata datamateriaI
Abstract [en]

Monotonic regression is a non-parametric statistical method that is designed especially for applications in which the expected value of a response variable increases or decreases in one or more explanatory variables. Such applications can be found in business, physics, biology, medicine, signal processing, and other areas. Inasmuch as many of the collected datasets can contain a very large number of multivariate observations, there is a strong need for efficient numerical algorithms. Here, we present new methods that make it feasible to fit monotonic functions to more than one hundred thousand data points. By simulation, we show that our algorithms have high accuracy and represent  considerable improvements with respect to computational time and memory requirements. In particular , we demonstrate how segmentation of a large-scale problem can greatly improve the performance of existing algorithms. Moreover, we show how the uncertainty of a monotonic regression model can be estimated. One of the procedures we developed can be employed to estimate the variance of the random error present in the observed response. Other procedures are based on resampling  techniques and can provide confidence intervals for the expected response at given levels of a set of predictors.

Abstract [sv]

Monoton regression är en icke-parametrisk statistisk metod som är utvecklad speciellt för tillämpningar i vilka det förväntade värdet aven responsvariabel ökar eller minskar med en eller flera förklaringsvariabler. Sådana tillämpningar finns inom företagsekonomi, fysik, biologi, medicin, signalbehandling och andra områden. Eftersom många insamlade datamaterial kan innehålla ett mycket stort antal multivariata observationer finns ett starkt behov av effektiva numeriska algoritmer. Här presenterar vi nya metoder som gör det möjligt att anpassa monotona funktioner till mer än 100000 datapunkter. Genom simulering visar vi. att våra algoritmer har hög noggrannhet och innebär betydande förbättringar med avseende på beräkningstid och krav på minnesutrymme. Speciellt visar vi hur segmentering av ett storskaligt problem starkt kan förbättra existerande algoritmer. Dessutom visar vi hur osäkerheten aven monoton regressions modell kan uppskattas. En av de metoder vi utvecklat kan användas för att uppskatta variansen för de slumpkomponenter som kan finnas i den observerade responsvariabeln. Andra metoder, baserade på s.k. återsampling, kan ge konfidensintervall för den förväntade responsen för givna värden på ett antal prediktorer.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2010. 75 p.
Series
Linköping Studies in Statistics, ISSN 1651-1700 ; 11Linköping Studies in Arts and Science, ISSN 0282-9800 ; 514
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-65349 (URN)978-91-7393-412-1 (ISBN)
Public defence
2010-04-16, Glashuset, Building B, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Available from: 2011-02-04 Created: 2011-02-04 Last updated: 2012-11-08Bibliographically approved

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Sysoev, OlegGrimvall, AndersBurdakov, Oleg

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