The Gumbel kernel for estimating the probability density function with application to hydrology data

The in-depth study of data, such as their precise characteristics or detection of anomalies, necessitates the estimation of the underlying probability density. The kernel estimation method, which consists of developing estimators based on special functions called kernels, is one of the most popular methods for reaching this aim. Despite the existence of numerous kernel-type estimators, there is still a high demand for new efficient kernel estimation strategies that can reveal hidden features in the data of important phenomena. In this regard, the development of modern distribution and their manageable implementation are beneficial. The Gumbel type 2 distribution is one of these modern distributions that has the feature of realizing a distributional compromise between the famous Fréchet and Weibull distributions. Among others, it presents distributional functions with high levels of flexibility. With this in mind, we define the Gumbel type 2 kernel function, which uses some analytical properties of the Gumbel type 2 distribution to achieve the aim of efficiency. This claim is proved through the theoretical analysis of the corresponding bias, variance, and rate of convergence under the mean square error. The question of how to select the estimator’s underlying bandwidth in practice is addressed. Finally, the performance of the proposed kernel estimator is evaluated by using two real-world hydrology data sets of bimodal nature. For these data sets, it performs better visually than the Weibull, gamma 1, inverse Gaussian, beta prime, and Erlang kernel estimators.