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smallNumberStatistics.bib
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0_6JP86KR3 0_PRQDQ4Q9 0_PGQD2IT2 0_8NI8IVIF 0_GATRIE8E 0_ECURHW5R 0_MEGAFUTG 0_2G9WE82W 0_3WUF3ENV 0_QS7KKW5G
@article{ebeling2003,
title = {Improved approximations of Poissonian errors for high confidence levels},
volume = {340},
url = {http://adsabs.harvard.edu/abs/2003MNRAS.340.1269E},
abstract = {We present improved numerical approximations to the exact Poissonian confidence limits for small numbers n of observed events following the approach of Gehrels. Analytic descriptions of all parameters used in the approximations are provided to allow their straightforward inclusion in computer algorithms for processing of large data sets. Our estimates of the upper (lower) Poisson confidence limits are accurate to better than 1 per cent for n{\textless}= 100 and values of S, the derived significance in units of Gaussian standard deviations, of up to 7 (5). In view of the slow convergence of the commonly used Gaussian approximations towards the correct Poissonian values, in particular for higher values of S, we argue that, for n{\textless}= 40, Poissonian statistics should be used in most applications, unless errors of the order of, or exceeding, 10 per cent are acceptable.},
journal = {Monthly Notices of the Royal Astronomical Society},
author = {Ebeling, Harald},
month = apr,
year = {2003},
keywords = {{METHODS:} {NUMERICAL}, methods: statistical},
pages = {1269--1278},
file = {Improved approximations of Poissonian errors for high confidence levels:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/G94TEIGR/2003MNRAS.340.html:text/html}
}
@article{kraft1991,
title = {Determination of confidence limits for experiments with low numbers of counts},
volume = {374},
url = {http://adsabs.harvard.edu/abs/1991ApJ...374..344K},
doi = {DOI: 10.1086/170124},
abstract = {Two different methods, classical and Bayesian, for determining
confidence intervals involving Poisson-distributed data are compared.
Particular consideration is given to cases where the number of counts
observed is small and is comparable to the mean number of background
counts. Reasons for preferring the Bayesian over the classical method
are given. Tables of confidence limits calculated by the Bayesian method
are provided for quick reference.},
journal = {The Astrophysical Journal},
author = {Kraft, Ralph P. and Burrows, David N. and Nousek, John A.},
month = jun,
year = {1991},
keywords = {{BAYES} {THEOREM}, {CHARGE} {COUPLED} {DEVICES}, {CONFIDENCE} {LIMITS}, {ERROR} {ANALYSIS}, {GAMMA} {RAY} {ASTRONOMY}, {PHOTON} {DENSITY}, {POISSON} {DENSITY} {FUNCTIONS}, {PROBABILITY} {DISTRIBUTION} {FUNCTIONS}, {SUPERNOVA} {1987A}},
pages = {344--355},
file = {NASA/ADS Full Text PDF:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/2JI3EB8E/Kraft et al. - 1991 - Determination of confidence limits for experiments.pdf:application/pdf}
}
@article{mighell1999,
title = {Parameter Estimation in Astronomy with Poisson-distributed Data. {I.The} chi{\textasciicircum}2\_gamma Statistic},
volume = {518},
url = {http://adsabs.harvard.edu/abs/1999ApJ...518..380M},
doi = {DOI: 10.1086/307253; eprintid: arXiv:astro-ph/9903093},
abstract = {Applying the standard weighted mean formula, {[SUM\_i} n\_isigma{\textasciicircum}-2\_i] /
{[SUM\_i} sigma{\textasciicircum}-2\_i], to determine the weighted mean of data, n\_i, drawn
from a Poisson distribution, will, on average, underestimate the true
mean by {\textasciitilde}1 for all true mean values larger than {\textasciitilde}3 when the common
assumption is made that the error of the ith observation is
sigma\_i=max(sqrt(n\_i), 1). This small, but statistically significant
offset, explains the long-known observation that chi-square minimization
techniques which use the modified Neyman's chi{\textasciicircum}2 statistic, {chi{\textasciicircum}2\_N=}
{SUM\_i} (n\_i-y\_i){\textasciicircum}2/max(n\_i,1), to compare Poisson-distributed data with
model values, y\_i, will typically predict a total number of counts that
underestimates the true total by about 1 count per bin. Based on my
finding that the weighted mean of data drawn from a Poisson distribution
can be determined using the formula {[SUM\_i} [n\_i+min(n\_i, 1)](n\_i +
1){\textasciicircum}-1] / {[SUM\_i} (n\_i + 1){\textasciicircum}-1], I propose that a new chi{\textasciicircum}2 statistic,
chi{\textasciicircum}2\_gamma= {SUM\_i} [n\_i+min(n\_i, 1) - y\_i]{\textasciicircum}2 / [n\_i + 1], should always
be used to analyze Poisson-distributed data in preference to the
modified Neyman's chi{\textasciicircum}2 statistic. I demonstrate the power and
usefulness of chi{\textasciicircum}2\_gamma minimization by using two statistical fitting
techniques and five chi{\textasciicircum}2 statistics to analyze simulated X-ray
power-law 15 channel spectra with large and small counts per bin. I show
that chi{\textasciicircum}2\_gamma minimization with the Levenberg-Marquardt or Powell's
method can produce excellent results (mean slope errors {\textless}{\textasciitilde}3\%) with
spectra having as few as 25 total counts.},
journal = {The Astrophysical Journal},
author = {Mighell, Kenneth J.},
month = jun,
year = {1999},
keywords = {{METHODS:} {NUMERICAL}, methods: statistical, X-{RAYS:} {GENERAL}},
pages = {380--393},
file = {NASA/ADS Full Text PDF:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/NZD7BTSI/Mighell - 1999 - Parameter Estimation in Astronomy with Poisson-dis.pdf:application/pdf}
}
@misc{cousins2007,
title = {Evaluation of three methods for calculating statistical significance when incorporating a systematic uncertainty into a test of the background-only hypothesis for a Poisson process},
url = {http://adsabs.harvard.edu/abs/2007physics...2156C},
abstract = {Hypothesis tests for the presence of new sources of Poisson counts
amidst background processes are frequently performed in high energy
physics {(HEP)}, gamma ray astronomy {(GRA)}, and other branches of science.
While there are conceptual issues already when the mean rate of
background is precisely known, the issues are even more difficult when
the mean background rate has non-negligible uncertainty. After
describing a variety of methods to be found in the {HEP} and {GRA}
literature, we consider in detail three classes of algorithms and
evaluate them over a wide range of parameter space, by the criterion of
how close the ensemble-average Type I error rate (rejection of the
background-only hypothesis when it is true) compares with the nominal
significance level given by the algorithm. We recommend wider use of an
algorithm firmly grounded in frequentist tests of the ratio of Poisson
means, although for very low counts the over-coverage can be severe due
to the effect of discreteness. We extend the studies of Cranmer, who
found that a popular Bayesian-frequentist hybrid can undercover severely
when taken to high Z values. We also examine the profile likelihood
method, which has long been used in {GRA} and {HEP;} it provides an
excellent approximation in much of the parameter space, as previously
studied by Rolke and collaborators.},
author = {Cousins, Robert D. and Linnemann, James T. and Tucker, Jordan},
month = feb,
year = {2007},
keywords = {Physics - Data Analysis, Statistics and Probability}
}
@article{gehrels1986,
title = {Confidence limits for small numbers of events in astrophysical data},
volume = {303},
url = {http://adsabs.harvard.edu/abs/1986ApJ...303..336G},
abstract = {The calculation of limits for small numbers of astronomical counts is
based on standard equations derived from Poisson and binomial
statistics; although the equations are straightforward, their direct use
is cumbersome and involves both table-interpolations and several
mathematical operations. Convenient tables and approximate formulae are
here presented for confidence limits which are based on such Poisson and
binomial statistics. The limits in the tables are given for all
confidence levels commonly used in astrophysics.},
journal = {The Astrophysical Journal},
author = {Gehrels, N.},
month = apr,
year = {1986},
keywords = {{APPROXIMATION}, {ASTROPHYSICS}, {BINOMIAL} {THEOREM}, {CONFIDENCE} {LIMITS}, {DATA} {PROCESSING}, {POISSON} {DENSITY} {FUNCTIONS}, {TABLES} {(DATA)}},
pages = {336--346},
file = {Confidence limits for small numbers of events in astrophysical data:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/R3UJSPVF/1986ApJ...303..html:text/html;Confidence limits for small numbers of events in astrophysical data:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/VK56GUM3/1986ApJ...303..html:text/html}
}
@article{cameron2011,
title = {On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach},
volume = {28},
shorttitle = {On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy},
url = {http://adsabs.harvard.edu/abs/2011PASA...28..128C},
doi = {10.1071/AS10046;},
abstract = {I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small to intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, active galactic nucleus fraction, supermassive black hole fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals - the `normal approximation' and the Clopper \& Pearson approach - are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (i) to reveal the ease with which {(Bayesian)} binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. r, matlab, mathematica, idl, python); and (ii) to demonstrate convincingly the major flaws of both the `normal approximation' and the Clopper \& Pearson approach for error estimation.},
urldate = {2012-11-21},
journal = {Publications of the Astronomical Society of Australia},
author = {Cameron, Ewan},
month = jun,
year = {2011},
keywords = {methods: data analysis, methods: statistical},
pages = {128--139},
file = {NASA/ADS Full Text PDF:/home/tim/.mozilla/firefox/ds0skv6s.default/zotero/storage/PZ94FFZB/Cameron - 2011 - On the Estimation of Confidence Intervals for Bino.pdf:application/pdf}
}
@article{clopper1934,
title = {{THE} {USE} {OF} {CONFIDENCE} {OR} {FIDUCIAL} {LIMITS} {ILLUSTRATED} {IN} {THE} {CASE} {OF} {THE} {BINOMIAL}},
volume = {26},
doi = {10.1093/biomet/26.4.404},
journal = {Biometrika},
author = {Clopper, C and Pearson, E},
year = {1934},
pages = {404--413}
}