Nonparametric Estimation Method for the Distribution Function Using Various Types of Ranked Set Sampling

The purpose of this research is to estimate the cumulative distribution function  using the local polynomial regression  and compare it to parameter estimation using the method of moments and the maximum likelihood method to calculate both the mean square error and the bias using the ranked sets sample  and the median ranked sets sample . As well as  frequently produces more exact estimates than simple random sampling  for the same sample size. By ranking samples based on some easily measurable characteristic, the variability within each set is decreased, resulting in more accurate estimations. We investigated three different degrees of local polynomial regression: the first, second, and third. The simulation analysis demonstrated that the second degree outperforms the other degrees. Also, when  is used to analyze  data, it takes advantage of the reduced variability within each ranked set, resulting in more precise and reliable regression function estimates. Following that, we investigated several degrees of bandwidth (0.1, 0.2, … and 0.9) and discovered that the bandwidth of degree 0.8 is superior to the other degrees based on a simulation study. Finally, we analyzed the relative efficiency of each of the three approaches: , , and , and we discovered that  is more efficient than the other methods for estimating the  in different kernels (normal (gaussian), epanechinkov). The numerical results provide that the suggested estimator  based on  is more efficient than other methods, as predicted by the simulation analysis