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

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