Consider the limit distribution of the Chi-squared goodness-offit statistic for testing normality if using the maximum likelihood estimators to estimate the unknown parameters. Specifically, suppose X1,...,Xn are i.i.d. and the problem is to test whether the underlying distribution is N(θ, 1) for some

θ. Group the observations into just 2 groups: positive observations and negative observations. Derive the limit distribution of the Chi-squared statistic using the sample mean to estimate θ and show it is not Chi-squared.

Section 14.4