Nonparametric Estimation Method for the Distribution Function Using Various Types of Ranked Set Sampling
Abstract
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
References
- Abdallah, M. S., & Al-Omari, A. I. (2022). ON THE NONPARAMETRIC ESTIMATION OF THE ODDS AND DISTRIBUTION FUNCTION USING MOVING EXTREME RANKED SET SAMPLING. Investigacin Operacional, 43(1), 90-102.
- AL_Rahman, R., & Mohammad, S. (2022). Generalized ratio-cum-product type exponential estimation of the population mean in median ranked set sampling. Iraqi Journal of Statistical Sciences, 19(1), 54-66.
- Al-Saleh, M. F., & Ahmad, D. M. R. (2019). Estimation of the distribution function using moving extreme ranked set sampling (MERSS). In Ranked Set Sampling (pp. 43-58). Academic Press.
- Fan, J., Gijbels, I., Hu, T. C., & Huang, L. S. (1996). A study of variable bandwidth selection for local polynomial regression. Statistica Sinica, 113-127.
- Frey, J. (2012). Constrained nonparametric estimation of the mean and the CDF using ranked-set sampling with a covariate. Annals of the Institute of Statistical Mathematics, 64(2), 439-456.
- Gulati, S. (2004). Smooth nonparametric estimation of the distribution function from balanced ranked set samples. Environmetrics, 15(5), 529-539.
- HA, M. (1997). Median ranked set sampling. J Appl Stat Sci, 6, 245-255.
- Halls, L. K., & Dell, T. R. (1966). Trial of ranked-set sampling for forage yields. Forest Science, 12(1), 22-26.
- McIntyre, G. A. (1952). A method for unbiased selective sampling, using ranked sets. Australian journal of agricultural research, 3(4), 385-390.
- Stokes, S. L., & Sager, T. W. (1988). Characterization of a ranked-set sample with application to estimating distribution functions. Journal of the American Statistical Association, 83(402), 374-381.
- Takahasi, K., & Wakimoto, K. (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the institute of statistical mathematics, 20(1), 1-31.
- Zamanzade, E., Mahdizadeh, M., & Samawi, H. M. (2020). Efficient estimation of cumulative distribution function using moving extreme ranked set sampling with application to reliability. AstA Advances in Statistical Analysis, 104(3), 485-502.
