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 several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped location invariant loss functions. The inadmissibility of the minimum risk invariant estimator (MRIE) is proved by deriving 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 special loss functions: squared error loss and linex loss. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. Finally, a simulation study was carried out to compare the risk performance of the proposed estimators.

翻译：在可靠性工程、分子生物学、金融等众多应用领域中，概率分布的不确定性度量具有重要作用。本文考虑在一般碗状位置不变损失函数类下，对具有未知且不等位置参数、公共尺度参数的多个指数分布的熵（即尺度参数函数）进行估计。通过构造非光滑改进估计量，证明了最小风险不变估计量(MRIE)的不可容许性。此外，我们获得了优于MRIE的光滑估计量。作为应用，针对平方误差损失和Linex损失两种特殊损失函数，给出了改进估计量的显式表达式。进一步表明这些估计量可适用于四种重要抽样方案：(i)完全独立同分布样本，(ii)记录值，(iii)II型删失，以及(iv)渐进II型删失。最后通过模拟研究比较了所提估计量的风险表现。