Exact and approximate distributions of the maximum likelihood estimate in a simple factor analysis model

We study a factor analysis model with two normally distributed observations and one factor, In the case when the errors have equal variance, the maximum likelihood estimate of the factor loading is given in closed form, Exact and approximate distributions of the maximum likelihood estimate are considered. The exact distribution function is given in a complex form that involves the incomplete Beta function, Approximations to the distribution function are given for the cases of large sample sizes and small error variances, The accuracy of the approximations is discussed.