An X-ray background is measured in 8 spectral bins

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

An X-ray background is measured in 8 spectral bins, expected to be a flat (constant) level, with recorded counts nin_i: 2,4,3,5,1,3,4,22, 4, 3, 5, 1, 3, 4, 2. ((a)) Compute Neyman's chi-squared estimate of the level, the harmonic mean μ^N=M/∑i(1/ni)\hat\mu_{\rm N} = M/\sum_i(1/n_i) with M=8M=8 bins. ((b)) Compute Pearson's chi-squared estimate, μ^P=n2‾\hat\mu_{\rm P} = \sqrt{\overline{n^2}} where n2‾=1M∑ini2\overline{n^2} = \frac1M\sum_i n_i^2. ((c)) Compute the Poisson-likelihood estimate, the arithmetic mean μ^L=nˉ\hat\mu_{\rm L} = \bar n. ((d)) State which estimate is preferred for counts this small, and why.

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