$χ$^2 and Poissonian Data: Biases Even in the High-Count Regime and How to Avoid Them TeX Export

@article{citeulike:3576166, abstract = {We demonstrate that two approximations to the chi<SUP>2</SUP> statistic as popularly employed by observational astronomers for fitting Poisson-distributed data can give rise to intrinsically biased model parameter estimates, even in the high-count regime, unless care is taken over the parameterization of the problem. For a small number of problems, previous studies have shown that the fractional bias introduced by these approximations is often small when the counts are high. However, we show that for a broad class of problem, unless the number of data bins is far smaller than \sqrt{N\_c}, where N<SUB>c</SUB> is the total number of counts in the data set, the bias will still likely be comparable to, or even exceed, the statistical error. Conversely, we find that fits using Cash's C-statistic give comparatively unbiased parameter estimates when the counts are high. Taking into account their well-known problems in the low-count regime, we conclude that these approximate chi<SUP>2</SUP> methods should not routinely be used for fitting an arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of counts per bin, and instead the C-statistic should be adopted. We discuss several practical aspects of using the C-statistic in modeling real data. We illustrate the bias for two specific problems---measuring the count rate from a light curve and obtaining the temperature of a thermal plasma from its X-ray spectrum measured with the Chandra X-ray observatory. In the context of X-ray astronomy, we argue the bias could give rise to systematically miscalibrated satellites and a \~{}5-10\% shift in galaxy cluster scaling relations.}, archivePrefix = {arXiv}, author = {{Humphrey}, P. J. and {Liu}, W. and {Buote}, D. A.}, citeulike-article-id = {3576166}, citeulike-linkout-0 = {http://arxiv.org/abs/0811.2796}, citeulike-linkout-1 = {http://arxiv.org/pdf/0811.2796}, citeulike-linkout-2 = {http://dx.doi.org/10.1088/0004-637X/693/1/822}, citeulike-linkout-3 = {http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2009ApJ...693..822H}, day = {17}, doi = {10.1088/0004-637X/693/1/822}, eprint = {0811.2796}, journal = {Astrophys. J.}, keywords = {chisq, fitting, statistics}, month = {March}, pages = {822--829}, posted-at = {2008-11-19 02:51:25}, priority = {2}, title = {{\$\chi\$}^{2} and Poissonian Data: Biases Even in the High-Count Regime and How to Avoid Them}, url = {http://dx.doi.org/10.1088/0004-637X/693/1/822}, volume = {693}, year = {2009} }

TY - JOUR ID - citeulike:3576166 L3 - citeulike-article-id:3576166 N2 - We demonstrate that two approximations to the chi<SUP>2</SUP> statistic as popularly employed by observational astronomers for fitting Poisson-distributed data can give rise to intrinsically biased model parameter estimates, even in the high-count regime, unless care is taken over the parameterization of the problem. For a small number of problems, previous studies have shown that the fractional bias introduced by these approximations is often small when the counts are high. However, we show that for a broad class of problem, unless the number of data bins is far smaller than \sqrt{N_c}, where N<SUB>c</SUB> is the total number of counts in the data set, the bias will still likely be comparable to, or even exceed, the statistical error. Conversely, we find that fits using Cash's C-statistic give comparatively unbiased parameter estimates when the counts are high. Taking into account their well-known problems in the low-count regime, we conclude that these approximate chi<SUP>2</SUP> methods should not routinely be used for fitting an arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of counts per bin, and instead the C-statistic should be adopted. We discuss several practical aspects of using the C-statistic in modeling real data. We illustrate the bias for two specific problems---measuring the count rate from a light curve and obtaining the temperature of a thermal plasma from its X-ray spectrum measured with the Chandra X-ray observatory. In the context of X-ray astronomy, we argue the bias could give rise to systematically miscalibrated satellites and a ~5-10% shift in galaxy cluster scaling relations. JF - Astrophys. J. EP - 829 TI - {$\chi$}^{2} and Poissonian Data: Biases Even in the High-Count Regime and How to Avoid Them VL - 693 SP - 822 KW - chisq KW - fitting KW - statistics AU - {Humphrey}, PJ AU - {Liu}, W AU - {Buote}, DA PY - 2009/March/17/ UR - http://dx.doi.org/10.1088/0004-637X/693/1/822 ER -