This paper introduces a novel kernel density estimator (KDE) based on the generalised exponential (GE) distribution, designed specifically for positive continuous data. The proposed GE KDE offers a mathematically tractable form that avoids the use of special functions, for instance, distinguishing it from the widely used gamma KDE, which relies on the gamma function. Despite its simpler form, the GE KDE maintains similar flexibility and shape characteristics, aligning with distributions such as the gamma, which are known for their effectiveness in modelling positive data. We derive the asymptotic bias and variance of the proposed kernel density estimator, and formally demonstrate the order of magnitude of the remaining terms in these expressions. We also propose a second GE KDE, for which we are able to show that it achieves the optimal mean integrated squared error, something that is difficult to establish for the former. Through numerical experiments involving simulated and real data sets, we show that GE KDEs can be an important alternative and competitive to existing KDEs.

翻译：本文提出了一种基于广义指数分布的新型核密度估计器，专门针对正连续数据设计。所提出的广义指数核密度估计器具有数学上易于处理的形式，避免了特殊函数的使用，例如与广泛使用的伽马核密度估计器（依赖于伽马函数）形成鲜明对比。尽管形式更为简洁，广义指数核密度估计器保持了相似的灵活性和形状特征，与伽马等分布保持一致，这些分布以在正数据建模中的有效性而闻名。我们推导了所提出核密度估计器的渐近偏差和方差，并正式证明了这些表达式中剩余项的阶数。我们还提出了第二种广义指数核密度估计器，能够证明其达到了最优均方积分误差，而这一点对于前者难以确立。通过涉及模拟和真实数据集的数值实验，我们表明广义指数核密度估计器可以成为现有核密度估计器的重要替代方案，并具有竞争力。