Confidence intervals on the ratio of expected mean squares \((\theta _ 1-d\theta _ 2)/\theta _ 3\) (Q1116583)

Let \(n_ 1S^ 2_ 1/\theta_ 1\), \(n_ 2S^ 2_ 2/\theta_ 2\), and \(n_ 3S^ 2_ 3/\theta_ 3\) represent jointly independent chi-squared random variables. Three methods for constructing confidence intervals on the parameter \(\theta =(\theta_ 1-d\theta_ 2)/\theta_ 3\) are proposed and compared using numerical integration. Cases are considered where \(\theta\geq 0\) and where the sign of \(\theta\) is unknown. Recommendations are provided for selecting an appropriate method.