Point and Interval Estimation of Powers of Scale Parameters for Two Normal Populations with a Common Mean

The problems of point and interval estimation of powers of scale parameter $$(\sigma _{1}^{2c})$$ ( σ 1 2 c ) have been considered when samples are available from two normal populations with a common mean. Maximum likelihood estimators (MLEs) and plug-in estimators using some of the popular estimators of the common mean have been proposed. A sufficient condition for improving affine equivariant estimators using the quadratic loss function is derived. Moreover, we propose several interval estimators, such as the asymptotic confidence interval, bootstrap confidence intervals, HPD credible interval, and intervals based on generalized pivot variables. Interestingly, some of the well-known estimators for the common mean have been used in constructing the generalized confidence intervals. A numerical comparison among all the proposed estimators has been made in terms of risk (in the case of point estimation) values using the quadratic loss function and coverage probabilities, and average lengths (in the case of interval estimation). Based on our simulation results, some recommendations are given for the use of the estimators. A real-life example has been considered to demonstrate the estimation methods.