on bias reduction for probability density function estimation based on a kernel estimator

The probability density function is a fundamental concept in statistics‎. ‎this study focuses on estimating the probability density function using nonparametric kernel methods. ‎initially‎, ‎the usual kernel method is introduced‎. ‎subsequently‎, ‎we present the two new estimates of the probability density function‎, ‎termed the biased reduced kernel estimate‎, ‎the repeat of the biased reduced kernel estimate, and the proposed biased reduced kernel estimate obtained by subtracting the bias value from the kernel estimator‎. ‎the paper explores theoretical properties, including the selection of the smoothness parameter, bias, variance, and mean squared error of the proposed estimator. ‎the accuracy of the biased reduced kernel estimate is scrutinized through monte carlo simulations‎. ‎moreover‎, ‎the mentioned methods were employed using the five real datasets‎. ‎the findings reveal that the proposed biased reduced kernel method exhibits a further reduction in bias compared to the usual kernel‎, ‎biased reduced kernel, and repeated biased reduced kernel methods.