Weakly approaching sequences of random distributions

Belyaev, Yuri K.; Luna, Sara Sjöstedt de

doi:10.1239/jap/1014842838

Cited by 28 publications

(33 citation statements)

References 13 publications

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“…Translating our results into those alternative languages might facilitate various generalizations. -Another generalization of the traditional notion of weak convergence is Belyaev's notion of weakly approaching sequences of random distributions (Belyaev and Sjöstedt-de Luna 2000). When comparing the LSPM with Oracle III, we limited ourselves to stating the absence of weak convergence and calculating the asymptotics of 1-dimensional distributions; Belyaev's definition is likely to lead to more precise results.…”

Section: Resultsmentioning

confidence: 99%

Nonparametric predictive distributions based on conformal prediction

et al. 2018

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This paper applies conformal prediction to derive predictive distributions that are valid under a nonparametric assumption. Namely, we introduce and explore predictive distribution functions that always satisfy a natural property of validity in terms of guaranteed coverage for IID observations. The focus is on a prediction algorithm that we call the Least Squares Prediction Machine (LSPM). The LSPM generalizes the classical Dempster-Hill predictive distributions to nonparametric regression problems. If the standard parametric assumptions for Least Squares linear regression hold, the LSPM is as efficient as the Dempster-Hill procedure, in a natural sense. And if those parametric assumptions fail, the LSPM is still valid, provided the observations are IID.

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Section: Resultsmentioning

confidence: 99%

Nonparametric predictive distributions based on conformal prediction

et al. 2018

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“…Aiming our statistical applications, we mainly consider weak and conditionally weak approaching in probability although conditional a.s.-approaching can be introduced and studied in a similar way (cf. Belyaev, 1996). It follows directly by the de®nitions and the Lebesgue dominated convergence theorem, that (X n j Z n ) 23 ad(P) Y n implies X n 6 ad Y n .…”

Section: Introductionmentioning

confidence: 88%

“…Assume that fY n g n>1 is uniformly tight and fG n ( X )g n>1 is uniformly continuous in n and x P R 1 . It is shown in Belyaev (1996) …”

Section: Propositionmentioning

confidence: 99%

“…Spectral resampling for stationary random processes is considered in Nordgaard (1992) (see also Hjorth, 1994). For realvalued and vector-valued random elements, approaching and conditional approaching were introduced and investigated in Belyaev (1995Belyaev ( , 1996Belyaev ( , 1997, and Belyaev & Sjo Èstedt (1996, respectively. The theory of weak convergence and the weak approaching of sequences have a lot of useful applications in probability theory and statistics.…”

Section: Introductionmentioning

confidence: 99%

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Approaching in Distribution with Applications to Resampling of Stochastic Processes

Belyaev

Seleznjev

2000

Scandinavian J Statistics

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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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“…As will be argued below, an appropriate concept to study conditional confidence intervals is merging, a concept that generalizes weak convergence. To the best of our knowledge, except for Belyaev and Sjöstedt-De Luna (2000), the present work is the only one to study merging in the context of conditional distributions. Moreover, we seem to be the first to employ merging of conditional distributions in time series.…”

Section: Introductionmentioning

confidence: 99%

Bootstrap inference for conditional risk measures

Heinemann¹

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Weakly approaching sequences of random distributions

Cited by 28 publications

References 13 publications

Nonparametric predictive distributions based on conformal prediction

Nonparametric predictive distributions based on conformal prediction

Approaching in Distribution with Applications to Resampling of Stochastic Processes

Bootstrap inference for conditional risk measures

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