Obtaining moment estimators using the empirical distribution function

The empirical distribution function is used as an estimate of the cumulative probability distribution function of a random variable. The empirical distribution function has a fundamental role in many statistical inferences, which are little known in some cases. In this article, the empirical probability function is introduced as a derivative of the empirical distribution function, and it is shown that moment estimators such as sample mean, sample median, sample variance, and sample correlation coefficient result from replacing the random variable density function with the empirical probability function in the theoretical definitions. In addition, the kernel probability density function estimator is used to estimate the population parameters and a new method for bandwidth estimation in the kernel density estimation is introduced. Keywords: Empirical distribution function, moment estimate, kernel estimator, bandwidth.