kernel estimation of the cumulative distribution function (CDF), when the support of the data is bounded, suffers from bias at the boundaries. To solve this problem, we introduce a new estimator for the CDF with support (0,1) based on the beta kernel function. By studying the asymptotic properties of the proposed estimator, we show that it is consistent and free from boundary bias. We conducted an extensive simulation to illustrate the performance of the proposed estimator. The results demonstrate the superiority of the proposed estimator over other commonly used estimators. As an application, we use the estimated CDF for nonparametric simulation. Using a numerical study, we show that the performance of the kernel probability density function (PDF) estimation in which a large sample simulated from the estimated CDF is employed can be noticeably improved. We also use the proposed estimator to estimate the CDF of the household health cost in Iran in 2019.

In this paper, a new method called adaptive bandwidth in the kernel function has been used for two-dimensional upscaling of reservoir data. Bandwidth in the kernel can be considered as a variability parameter in porous media. Given that the variability of the reservoir characteristics depends on the complexity of the system, either in terms of geological structure or the specific feature distribution, variations must be considered differently for upscaling from a fine model to a coarse one. The upscaling algorithm, introduced in this paper, is based on the kernel function bandwidth, written in combination with the A* search algorithm and the first-depth search algorithm. In this algorithm, each cell in its x and y neighborhoods as well as the optimal bandwidth, obtained in two directions will be able to be merged with its adjacent cells. The upscaling process is performed on artificial data with 30×30 grid dimensions and SPE-10 model as real data. Four modes are used to start the point of upscaling and the process is performed according to the desired pattern, and in each case, the upscaling error and the number of final upscaled blocks are obtained. Based on the number of coarsen cells as well as the upscaling error, the first pattern is selected as the optimal pattern for synthetic data and the second pattern is selected as the optimal simulator model for real data. In this model, the number of cells was 236 and 3600, and the upscaling errors for synthetic and real data were 0. 4183 and 12. 2, respectively. The results of the upscaling in the real data were compared with the normalization method and showed that the upscaling error of the normalization method was 15 times the upscaling error of the kernel bandwidth algorithm.