Statistical Challenges in Modern Astronomy II pp 441-444 | Cite as

# A Poisson Parable: Bias in Linear Least Squares Estimation

## Abstract

*I*observed counts,

*ni*, described by Poisson statistics, for i = 1,…,

*I*, according to some known

*J*-component linear model,

with underlying physical count rates *r* _{ j }, or fluxes which are to be estimated from the data, the *A* _{ ij }, being known experiment constants. This problem is often solved by Linear Least Squares (LLSQ), but limited to situations where the number of counts per bin i is not too small.

For the simplest possible case,*J =* 1, which is just a counting experiment with no background, it is interesting to attempt a direct application of the weighted average formula using\(\sqrt {{n_i}} \approx {\sigma _i} \cdot \)However, the resulting formula is completely wrong! Using, instead of the observed n_{i}, the *expected count*,\(E\left[ {{n_i}} \right] = \sigma _i^2 = r{t_i}\)in the weighting, where *t* _{ i }
is the observing time in bin i, it turns out that the unknown rate r cancels from the weighted average sums, and we recover the obviously correct estimate\(\hat r = {N \mathord{\left/{\vphantom {N T}} \right.\kern-\nulldelimiterspace} T} = {{\sum {{n_i}} } \mathord{\left/
{\vphantom {{\sum {{n_i}} } {\sum {{t_i}} }}} \right.\kern-\nulldelimiterspace} {\sum {{t_i}} }}\)

## Keywords

Count Rate Poisson Estimation Counting Experiment Weighted Average Method True Count## Preview

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## References

- [EDJ+71]W. T. Eadie, D. Drijard. F. E. James, M. Roos. and B. Sadoulet.
*Statistical Methods in Experimental Physics*. North-Holland, Amsterdam, 1971. See especially chapters 7 and 8.MATHGoogle Scholar - [Leh59]E. L. Lehmann.
*Testing Statistical Hypotheses*. John Wiley and Sons, New York, 1959.MATHGoogle Scholar - [WDJ+95]Wm. A. Wheaton, Alfred D. Dunklee, Allen S. Jacobson, James C. Ling, William A. Mahoney. and Robert G. Radocinski. Multiparameter linear least squares fitting to poisson data one count at a time.
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