Components of the Pearson-Fisher chi-squared statistic

The Pearson-Fisher chi-squared test can be used to evaluate the goodnessof-fit of categorized continuous data with known bin endpoints compared to a continuous distribution, in the presence of unknown (nuisance) distribution parameters. Rayner and McAlevey [11] and Rayner and Best [9],[10] demonstrate that in this case, component tests of the Pearson-Fisher chi-squared test statistic can be obtained by equating it to the Neyman smooth score test for a categorized composite null hypothesis under certain restrictions. However, only Rayner and McAlevey [11] provide even brief details as to how these restrictions can be used to obtain any kind of decomposition. More importantly, the relationship between the range of possible decompositions and the interpretation of the corresponding test statistic components has not previously been investigated. This paper provides the necessary details, as well as an overview of the decomposition options available, and revisits two published examples.