Approximate inference for the factor loading of a simple factor analysis model (Q2739871)

The observations are i.i.d. bivariate normal vectors with unknown mean and the covariance matrix NEWLINE\[NEWLINE \left( \begin{matrix} \sigma_t^2+\sigma^2 & \beta\sigma_f^2\\ \beta\sigma^2_f & \beta\sigma^2_f+\sigma^2 \end{matrix} \right), NEWLINE\]NEWLINE where \(\beta\) is the factor loading, \(\sigma^2_f\) is the factor variance and \(\sigma^2\) is the variance of the random error in the simple factor analysis model. These parameters are also unknown. The authors derive a modified signed log likelihood ratio statistic for the inference on \(\beta\) which is asymptotically \(N(0,1)\) distributed with third order accuracy. It is used to construct a confidence interval for \(\beta\). Performance of this method is investigated via a simulation study.