The X2 Goodness-of-Fit Test for Normal Distributions

Chernoff & Lehmann (1954) have shown that if, in making a X2 goodness-of-fit test of a continuous distribution, the parameters are estimated efficiently from the sample and not from the cell frequencies, the statistic X2* is asymptotically distributed as X2b−a−1+λ1y12+λaya2 where 0≤λ1, …, λ2≤1, y1…, y2 being independent standard normal variables. They used k fixed class intervals, defined without reference to the sample. In the present paper, the particular case of the normal distribution is examined when the class intervals are chosen to contain, with reference to the sample mean and variance, constant probabilities and so vary with the sampling. The X2 statistic is again found to be distributed in the Chernoff & Lehmann form (s = 2). Explicit formulae are given for λ1 and λ2, which tend to zero rapidly as k increases. It is suggested that at least ten class intervals be used in practice so that the tabular points of X2 may be used with an error of less than 1 %.