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A more complicated, but more widely used form of Péarson's chi-square test arises in the case where the null hypothesis of interest includes unknown paraméters. For instance we may wish to test whether some data follows a normal distribution but without specifying a mean or variance. In this situation the unknown paraméters need to be estimated by the data, typically by maximum likelihood estimation, and these estimates are then used to calculate the expected values in the Péarson statistic. It is commonly stated that the degrees of freedom for the chi-square distribution of the statistic are then k − 1 − r, where r is the number of unknown paraméters. This result is valid when the original data was multinomial and hence the estimated paraméters are efficient for minimizing the chi-square statistic. More generally however, when maximum likelihood estimation does not coincide with minimum chi-square estimation, the distribution will lie somewhere between a chi-square distribution with k − 1 − r and k − 1 degrees of freedom (See for instance Chernoff and Lehmann 1954).