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- Belyaev, Yuri K, et al.
(author)
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Approaching in distribution with applications to resampling of stochastic processes
- 2000
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In: Scandinavian Journal of Statistics. - : Wiley-VCH Verlagsgesellschaft. - 0303-6898 .- 1467-9469. ; 27:2, s. 371-384
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Journal article (peer-reviewed)abstract
- We introduce the notion of weak approaching and conditionally weak approaching sequences of random processes. This notion generalizes the conventional weak convergence, and has been proposed for real valued random variables in Belyaev (1995). Some of the standard tools for an investigation of the behaviour of weak approaching sequences of random elements in metric spaces are developed. The spaces of smoothed and right-continuous functions with left-hand limits are considered. This technique allows us to use the resampling approach for an evaluation of distributions of continuous functionals on realizations of sum of an increasing number of independent random processes. Two numerical examples are presented for such functionals as supremum and number of level crossings.
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- Belyaev, Yuri K
(author)
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Assessing accuracy of statistical inferences by resamplings
- 2010
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In: Mathematical and Statistical Models and Methods in Reliability. - New York : Birkhäuser Verlag. - 9780817649708 - 9780817649715 ; , s. 193-206
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Conference paper (other academic/artistic)abstract
- 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.
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- Belyaev, Yuri K
(author)
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Asymptotical properties of nonparametric point estimators based on complexly structured reliability data with right-censoring
- 1991
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In: Statistics (Berlin). - : Taylor & Francis. - 0233-1888 .- 1029-4910. ; 22:4, s. 589-612
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Journal article (peer-reviewed)abstract
- This paper presents a general approach to nonparametric estimation of unknown distribution functions and related characteristics such as cumulative hazard functions. It is based on the notion of portions of statistical data and on the property of discertely separated distributions of statistical data General assumptions are given under which the corresponding generalized maximum likelihood estimators are consistent and their deviations have asymptotically normal distributions, if the number of portions increases to indinity.
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