Estimating Cumulative Distribution Function Using Gamma Kernel

In this article, we propose the gamma kernel estimator for the cumulative distribution functions with nonnegative support. We derive the asymptotic bias and variance of the proposed estimator in both boundary and interior regions and show that it is free of boundary bias. We also obtain the optimal smoothing parameter which minimizes the mean integrated square error (MISE). In addition to consistency, we prove the almost sure convergence of the proposed estimator and show that it follows the same approximate normal distribution as empirical distribution. We presented a simulation study to compare the performance of the proposed estimator with other estimators. We use the proposed estimator to estimate the cumulative probability distribution function of the food expenses for urban households in Iran.