Document Type : Original Paper


Shahid Chamran University of Ahvaz, Ahvaz, Iran


In this article, we propose the gamma kernel estimator for the cumulative distribution function 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, the almost sure convergence of the proposed estimator is proven, and it is shown 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.