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Assessing accuracy of statistical inferences by resamplings
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)In: Mathematical and statistical models and methods in reliability: applications to medicine, finance, and quality control / [ed] V. V. Rykov, N. Balakrishnan, M. S. Nikulin, New York: Birkhäuser Verlag, 2010, 193-206 p.Chapter in book (Other academic)
Abstract [en]

Suppose that a list of explanatory variables and corresponding random responses was obtained during a series of regression experiments. The characteristic of interest is the mean value of responses considered as a regression function of corresponding values of explanatory variables. For example, if responses are failure times of tested elements, then the conditional mean value of life time given the value of explanatory variable is one of the important reliability characteristics of the tested elements. The analysis of this type of data can be realized in the framework of linear heteroscedastic regression models. Here, one of the central problems is a consistent estimation of the unknown regression function when the size of data grows unboundedly. The problems related to analysis of regression data attracted many researches, see Wu [Ann. Statist. 14, 1261–1350 (1986)]. We give an approach to consistent solution of the problems under the assumption that values of explanatory variables are real numbers and the regression function is a polynomial with unknown degree and coefficients. The selection of regression function is based on resamplings from terms in the sum of the residuals estimated by the ordinary least squares method with various values of polynomial degree. In a similar way, resamplings from the weighted estimated residuals are used for consistent estimation of the deviations distributions of estimated coefficients from their true unknown values. The consistency of applied resamplings methods holds under certain assumptions, e.g. it is assumed that the residuals distributions have uniformly integrable second moments (assumption AW 2). Given in Appendix a variant of the Central Limit Resampling Theorem is used in the proofs of Theorems 1 and 2.

Place, publisher, year, edition, pages
New York: Birkhäuser Verlag, 2010. 193-206 p.
, Statistics for Industry and Technology
Keyword [en]
Asymptotic normality, Distributions of deviations, Least squares estimators, Linear heteroscedastic regression, Overparametrisation, Resampled sums of weighted estimated residuals, Selection of regression function
National Category
Probability Theory and Statistics
URN: urn:nbn:se:umu:diva-87900DOI: 10.1007/978-0-8176-4971-5_14ISI: 000292954700014ISBN: 978-0-8176-4970-8ISBN: 978-0-8176-4971-5OAI: diva2:712156
Available from: 2014-04-14 Created: 2014-04-14 Last updated: 2015-09-25Bibliographically approved

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Belyaev, Yuri K
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