Consider two normal populations that share a common mean, denoted as \(\mu ,\) but have different variances, \(\sigma _{x}^{2}\) and \(\sigma _{y}^{2}.\) It is known that the variances follow a specific ordering, such that \(\sigma _{x}^{2} \le \sigma _{y}^{2}.\) Our goal is to estimate the quantile vector \(\mathop {\theta }\limits _{\sim }= (\mu + \eta \sigma _x, \mu + \eta \sigma _y)\) using a decision-theoretic approach. Given the condition \(\sigma _{x}^{2} \le \sigma _{y}^{2},\) new estimators for the quantile vector have been proposed, which are shown to outperform their older counterparts–those estimators that do not account for the ordering of the variances–in terms of risk using the affine invariant loss function. An inadmissibility condition for estimators invariant under the affine class has been derived, leading to the proposal of improved estimators. The percentage improvements in risk for these new estimators, compared to their older counterparts, have been computed numerically and found to be significant. A numerical comparison among the improved estimators has also been conducted in terms of their risk values, along with recommendations for their application. Finally, some real-life data analyses have been performed to demonstrate the potential application of the proposed model.
