Abstract
Kernel density estimation is a technique for estimating the probability density function, when data are obtained from unknown data generating processes. Because the kernel estimator is a good alternative to the histogram utilized as a relative estimator for the probability density function, it can supply us with the probability of an event of interest. In this note, we contribute to this subject through an extensive study of the hyperbolic secant kernel density estimator. We derived some properties of the obtained estimator, such as bias, variance, optimal bandwidth, and mean squared error. Finally, its performance is investigated using three practical data sets, two of them have both negative and positive values. In addition, a significant smooth bandwidth was proposed during the discussion.