A. V. Prokhorov, N. G. Ushakov, “On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum”, Teor. Veroyatnost. i Primenen., 46:3 (2001), 449–462; Theory Probab. Appl., 46:3 (2002), 420

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Teoriya Veroyatnostei i ee Primeneniya, 2001, Volume 46, Issue 3, Pages 449–462
DOI: https://doi.org/10.4213/tvp3895 (Mi tvp3895)

This article is cited in 9 scientific papers (total in 9 papers)

On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum

A. V. Prokhorov^a, N. G. Ushakov^b

^a M. V. Lomonosov Moscow State University
^b Institute of Microelectronics Technology and High Purity Materials, Russian Academy of Sciences

Full-text PDF (1410 kB) Citations (9)

DOI: https://doi.org/10.4213/tvp3895

Abstract: The uniqueness and stability conditions of reconstructing a distribution of independent identically distributed random variables

X_{1}, \dots, X_{m}

$X_1,\dots,X_m$ by a distribution of the sum

S = X_{1} + \dots + X_{m}

$S=X_1+\dots+X_m$ for fixed

m

$m$ are given. This paper considers two generalizations of the problem of reconstructing the random variables

X_{j}

$X_j$ : by the distribution

S = γ_{1} X_{1} + \dots + γ_{m} X_{m}

$S=\gamma_1X_1+\dots+\gamma_mX_m$ , where the random variables

γ_{j}

$\gamma_j$ take values 0 and 1 with some fixed probabilities, and bythe distribution of the sum

S_{N} = X_{1} + \dots + X_{N}

$S_N=X_1+\dots+X_N$ of the random number

N

$N$ of summands

X_{j}

$X_j$ . In these problems there are given not only sufficient stability conditions of reconstructing but quantitative stability estimators.

Keywords: summands distribution, stability, sum of a random number of summands, linear combinations, characteristic function, Poisson distribution, geometric distribution.

Received: 02.09.1999

English version:
Theory of Probability and its Applications, 2002, Volume 46, Issue 3, Pages 420–430
DOI: https://doi.org/10.1137/S0040585X97979202

Bibliographic databases:

Language: Russian

Citation: A. V. Prokhorov, N. G. Ushakov, “On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum”, Teor. Veroyatnost. i Primenen., 46:3 (2001), 449–462; Theory Probab. Appl., 46:3 (2002), 420–430

Citation in format AMSBIB

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\issue 3

\pages 449--462

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\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1978662}

\zmath{https://zbmath.org/?q=an:1032.60010}

\transl

\jour Theory Probab. Appl.

\yr 2002

\vol 46

\issue 3

\pages 420--430

\crossref{https://doi.org/10.1137/S0040585X97979202}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000179228700003}

Linking options:

https://www.mathnet.ru/eng/tvp3895

https://doi.org/10.4213/tvp3895

https://www.mathnet.ru/eng/tvp/v46/i3/p449

This publication is cited in the following 9 articles:

Vexler A., “Univariate Likelihood Projections and Characterizations of the Multivariate Normal Distribution”, J. Multivar. Anal., 179 (2020), 104643
D. V. Belomestny, A. V. Prokhorov, “Stability of characterization the independence of random variables by the independence of the linear statistics”, Theory Probab. Appl., 59:4 (2015), 672–677
Vexler A., Liu A., Schisterman E., “Nonparametric deconvolution of density estimation based on observed sums”, Journal of Nonparametric Statistics, 22:1 (2010), 23–39
Vexler A., Schisterman E.F., Liu A., “Estimation of ROC curves based on stably distributed biomarkers subject to measurement error and pooling mixtures”, Statistics in Medicine, 27:2 (2008), 280–296
Bondell H.D., Liu A., Schisterman E.F., “Statistical inference based on pooled data: A moment–based estimating equation approach”, Journal of Applied Statistics, 34:2 (2007), 129–140
Gordienko E., “Comparing the distributions of sums of independent random vectors”, Kybernetika, 41:4 (2005), 519–529
D. V. Belomestny, “Reconstruction of the general distribution by the distribution of some statistics”, Theory Probab. Appl., 49:1 (2005), 1–15
Denis Belomestny, “Constraints on distributions imposed by properties of linear forms”, ESAIM: PS, 7 (2003), 313
D. V. Belomestny, “On the Problem of Reconstructing a Summands Distribution by Their Sum”, Theory Probab. Appl., 46:2 (2002), 336–341

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Theory of Probability and its Applications

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