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SUMMARY:Some optimization-theoretic issues in analysis of interval-valued
data
DTSTART:20220627T093000Z
DTEND:20220627T100000Z
DTSTAMP:20240226T122000Z
UID:indico-contribution-334@conferences.enbis.org
DESCRIPTION:Speakers: Michal Černý (Prague University of Economics & Bus
iness)\, Sokol Ondřej (Prague University of Economics & Business)\n\nInte
rval-valued data are often encountered in practice\, namely when only uppe
r and lower bounds on observations are available. As a simple example\, co
nsider a random sample $x_1\, \\dots\, x_n$ from a distribution $\\Phi$\;
the task is to estimate some of the characteristics of $\\Phi$\, such as m
oments or quantiles. Assume that the data $x_1\, \\dots\, x_n$ are not obs
ervable\; we have only bounds $\\underline{x}_i \\leq x_i \\leq \\overline
{x}_i$ a.s. and all estimators and statistics are allowed to be functions
of $\\underline{x}_i\, \\overline{x}_i$ only\, but not $x_i$. The analysis
very much depends on whether we are able to make additional assumptions a
bout the joint distribution of $(\\underline{x}_i\, x_i\, \\overline{x}_i)
$ (for example\, a strong distributional assumption could have the form\n$
\\mathsf{E}[x_i|\\underline{x}_i\,\\overline{x}_i]=\\frac{1}{2}(\\underlin
e{x}_i+\\overline{x}_i)$). Without such assumptions\, a statistic $S(x_1\,
\\dots\, x_n)$ can only be replaced by the pair of tight bounds $\\overli
ne{S} = \\sup\\{S(\\xi_1\,\\dots\,\\xi_n)|\\underline{x}_i\\leq\\xi_i\\leq
\\overline{x}_i\\ \\forall i\\}$ and\n$\\underline{S} = \\inf\\{S(\\xi_1\,
\\dots\,\\xi_n)|\\underline{x}_i\\leq\\xi_i\\leq\\overline{x}_i\\ \\forall
i\\}$. We report some of our recent results on the algorithms for the com
putation of $\\underline{S}\, \\overline{S}$. In particular\, when $S$ is
the sample variance\, it can be shown that the computation of $\\overline{
S}$ is an NP-hard problem. We study a method based on Ferson et al.\, whic
h works in exponential time in the worst case\, while it is almost linear
on average (under certain regularity assumptions)\, showing that the NP-ha
rdness result need not be too restrictive for practical data analysis.\n\n
https://conferences.enbis.org/event/18/contributions/334/
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RELATED-TO:indico-event-18@conferences.enbis.org
URL:https://conferences.enbis.org/event/18/contributions/334/
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