In various applied areas such as reliability engineering, molecular biology, finance, etc., the measure of uncertainty of a probability distribution plays an important role. In the present work, we consider the estimation of a function of the scale parameter, namely entropy of many exponential distributions having unknown and unequal location parameters with a common scale parameter. For this estimation problem, we have considered bowl-shaped location invariant loss functions. The inadmissibility of the minimum risk invariant estimator (MRIE) is proved by proposing a non-smooth improved estimator. Also, we have obtained a smooth estimator which improves upon the MRIE. As an application, we have obtained explicit expressions of improved estimators for two well-known loss functions namely squared error loss and linex loss. Further, we have shown that these estimators can be derived for other important censored sampling schemes. At first, we obtained the results for the complete and i.i.d. sample. We have seen that the results can be applied for (i) record values, (ii) type-II censoring, and (iii) progressive Type-II censoring. Finally, a simulation study has been carried out to compare the risk performance of the proposed improved estimators.

翻译：在可靠性工程、分子生物学、金融等多个应用领域中，概率分布的不确定性度量起着重要作用。本文考虑尺度参数函数的估计问题，即具有未知且不相等的位置参数、共用一个尺度参数的多个指数分布的熵。针对该估计问题，我们采用碗形位置不变损失函数。通过提出一个非光滑改进估计量，证明了最小风险不变估计量（MRIE）的不可容许性。此外，我们获得了一个优于MRIE的光滑估计量。作为应用，我们给出了两种常用损失函数——平方误差损失和Linex损失——下改进估计量的显式表达式。进一步表明，这些估计量可推广到其他重要的截尾抽样方案。我们首先针对完整独立同分布样本获得结果，并发现这些结果可应用于：(i) 记录值数据、(ii) 第II类截尾数据，以及 (iii) 渐进第II类截尾数据。最后，通过模拟研究比较了所提出改进估计量的风险表现。