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it is well-known that the Chi-square test between an observed distributionOand an expected distributionEcan be interpreted as a test based on (twice) the second order Taylor approximation of the Kullback-Leibler divergence, i.e.: [tex]2\,\mathcal{D}_{KL}(O \| E) \approx \sum_i \frac{(O_i-E_i)^2}{E_i} = \chi^2[/tex]

whereiis the bin of the histogram (or contigency table). A proof is given here (page 5).

The question is: how do we know that each of the error terms [itex]\frac{(O_i-E_i)^2}{E_i}[/itex] on the right side of the above equation follows a normal distributionN(0,1)? There is probably some some assumption to be made...?

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# Chi-square test: why does it follow a Chi-square distribution

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