# A Poisson Parable: Bias in Linear Least Squares Estimation

• Wm. A. Wheaton
Conference paper

## Abstract

A standard problem in high-energy astronomy data analysis is the decomposition of a set of I observed counts, ni, described by Poisson statistics, for i = 1,…, I, according to some known J-component linear model,
$$\overline {{n_i}} = E\left[ {{n_i}} \right] = \sum\limits_{j = 1}^J {{A_{ij}}{r_j}}$$
(41.1)

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 ni, 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$$\hat r = {N \mathord{\left/{\vphantom {N T}} \right.\kern-\nulldelimiterspace} T} = {{\sum {{n_i}} } \mathord{\left/ {\vphantom {{\sum {{n_i}} } {\sum {{t_i}} }}} \right.\kern-\nulldelimiterspace} {\sum {{t_i}} }}$$

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### References

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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. The Astrophysical Journal, 438(322). jan 1995.