A chi-squabe statistic for goodies-of-fit tests within the exponential family

When the class boundaries used in constructing a chi-square goodness-of-fit statistic are predetermined and the unknown parameters are estimated by maximum likelihood from the ungrouped data, the resulting statistic does not have a limiting X2-distribution but instead is asymptotically distributed as a linear function of chi-square variables. The same result applies in the more realistic and useful case where only the number of classes and their probability content are predetermined. It is shown here that in both of the above cases, in the case of exponential family, the quadratic form of the asymptotic multinomial conditional distribution of the class frequencies given the parameter estimates can be used to test the goodness-of-fit, The simulated distribution of the statistic agrees with the X2-distribution with degrees of freedom one less than the number of classes after grouping, regardless of the number of parameters estimated.