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International Encyclopedia of Statistical Science pp 246–250Cite as

Chi-Square Goodness-of-Fit Tests: Drawbacks and Improvements

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The famous chi-squared goodness-of-fit test was discovered by Karl Pearson in 1900. If the partition of a sample space is such that observations are grouped over r disjoined intervals Δ i , and denoting ν i observed frequencies and np i (θ) expected that correspond to a multinomial scheme, the Pearson’s sum is written

$${\chi }^{2} = {X}_{ n}^{2}(\theta ) ={ \sum }_{i=1}^{r}\frac{{({\nu }_{i} - n{p}_{i}(\theta ))}^{2}} {n{p}_{i}(\theta )} ={ \bf V}^{T}(\theta ){\bf V}(\theta ),$$
(1)

where V(θ) is a vector with components v i (θ) = (ν i − np i (θ))(np i (θ)) − 1 ∕ 2, i = 1, …, r. If the number of observations n → ∞, the statistic (1) for a simple null hypothesis, specifying the true value of θ, will follow chi-squared probability distribution with r − 1 degrees of freedom.

Until 1934, Pearson believed that the limit distribution of his chi-squared statistic would be the same if unknown parameters of the null hypothesis were replaced by estimates based on a sample (Stigler (2008), p....

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Author information

Authors and Affiliations

  1. Rosstat’s StatisticalResearch Institute, Moscow, Russia

    Vassiliy Voinov

  2. Université Victor Segalen, Bordeaux, France

    Mikhail Nikulin

Authors
  1. Vassiliy Voinov
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  2. Mikhail Nikulin
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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Voinov, V., Nikulin, M. (2011). Chi-Square Goodness-of-Fit Tests: Drawbacks and Improvements. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_172

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