Tests of normality based on EDF statistics using partially rank ordered set sampling designs

Abstract

In this study, we considered five goodness-of-fit (GOF) tests based on empirical distribution function (EDF) which are Kolmogorov-Smirnov (D), Kuiper (V ), Cram´er-von Mises (W2), Watson (U2), Anderson-Darling (A2) tests to assess normality. Thus, we first suggested the EDFs based partially rank ordered set (PROS) sampling designs which are known as PROS Level-0, Level-1 and Level-2 sampling designs. Then, we discussed the relative efficiencies of the suggested EDFs w.r.t their counterparts of simple random sampling (SRS) and ranked set sampling (RSS). The main idea of this study is to compare the performances of the five different GOF tests based on PROS sampling designs with the GOF tests based on SRS and RSS. For this purpose, we investigated the power of the suggested GOF tests based on PROS sampling designs by performing simulations. In addition to the simulations, a real data set is considered to illustrate the GOF tests based on PROS sampling designs. According to the results, it can be seen that the EDFs based on PROS sampling designs are more efficient than the EDFs based on SRS and RSS. Also, it is clearly appeared that the GOF tests, Kolmogorov-Smirnov (D), Cram´er-von Mises (W2) and Anderson- Darling (A2), based on PROS sampling designs has the best power performance.

Keywords

• Ranked set sampling
• Partially rank ordered set
• Sampling designs
• Empirical distribution function
• Type I error;
• Goodness-of-fit tests
• Power of test

References

1. McIntyre, G.A. (1952). A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research, 3(4), 385-390.
2. Takahasi, K. and 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.
3. Dell, T. and Clutter, J. (1972). Ranked set sampling theory with order statistics background. Biometrics, 545-555.
4. Kaur, A., Patil, G., Sinha, A. and Taillie, C. (1995). Ranked set sampling: an annotated bibliography. Environmental and Ecological Statistics, 2(1), 25-54.
5. Chen, Z., Bai, Z. and Sinha, B. (2003). Ranked set sampling: theory and applications. Springer Science & Business Media.
6. Al-Omari, A. I. and Bouza, C. N. (2014). Review of ranked set sampling: modifications and applications. Revista Investigaci´on Operacional, 3, 215-240.
7. Deshpande, J. V. and Frey, J. and Ozturk, O. (2006). Nonparametric ranked-set sampling confidence intervals for quantiles of a finite population. Environmental and Ecological Statistics, 13(1), 25-40.
8. Al-Saleh, M. F. and Samawi, H. M. (2007). A note on inclusion probability in ranked set sampling and some of its variations. Test, 16(1), 198-209.

9. Ozdemir, Y. A. and Gokpinar, F. (2007). A generalized formula for inclusion probabilities in ranked set sampling. Hacettepe Journal of Mathematics and Statistics, 36(1), 89-99.
10. Ozdemir, Y. A. and Gokpinar, F. (2008). A new formula for inclusion probabilities in median ranked set sampling. Communication in Statistics- Theory and Methods, 37(13), 2022-2033.
11. Gokpinar, F, Ozdemir, Y. A. (2010). Generalization of inclusion probabilities in ranked set sampling. Hacettepe Journal of Mathematics and Statistics, 39(1), 89-95.
12. Frey, J. (2011). Recursive computation of inclusion probabilities in ranked set sampling. Journal of Statistical Planning and Inference, 141(11), 3632-3639.
13. Jafari Jozani, M., Johnson, B. C. (2011). Design based estimation for ranked set sampling in finite population. Environmental and Ecological Statistics, 18(4), 663-685.
14. Jafari Jozani, M., Johnson, B. C. (2011). Randomized nomination sampling in finite populations. Journal of Statistical Planning and inference, 142(7), 2103-2115.
15. Stokes, S L. and 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.
16. Frey, J. andWang, L. (2014). EDF-based goodness-of-fit tests for ranked-set sampling. Canadian Journal of Statistics, 42(3), 451-469.
17. Nazari, S., Jafari Jozani, M. and Kharrati-Kopaei, M. (2014). Nonparametric density estimation using partially rank-ordered set samples with application in estimating the distribution of wheat yield. Electronic Journal of Statistics, 8(1), 738-761.
18. Sevil, Y. C. and Yildiz, T. O. (2017). Power comparison of the Kolmogorov–Smirnov test under ranked set sampling and simple random sampling. Journal of Statistical Computation and Simulation, 87(11), 2175-2185.
19. Yildiz, T. O., and Sevil, Y. C. (2018). Performances of some goodness-of-fit tests for sampling designs in ranked set sampling. Journal of Statistical Computation and Simulation, 88(9), 1702-1716.
20. Yildiz, T. O., and Sevil, Y. C. (2019). Empirical distribution function estimators based on sampling designs in a finite population using single auxiliary variable. Journal of Applied Statistics, 46(16), 2962- 2974.
21. Sevil, Y. C. and Yildiz, T. O. (2020). Performances of the distribution function estimators based on ranked set sampling using body fat data. Turkiye Klinikleri Journal of Biostatistics, 12(2), 218-228.
22. Ozturk, O. (2011). Sampling from partially rank-ordered sets. Environmental and Ecological statistics, 18(4), 757-779.
23. Ozturk, O. (2012a). Combining ranking information in judgment post stratified and ranked set sampling designs. Environmental and Ecological Statistics, 19(1), 73-93.
24. Ozturk, O. (2012b). Quantile inference based on partially rank-ordered set samples. Journal of Statistical Planning and Inference, 142(7), 2116-2127.
25. Ozturk, O. (2014). Estimation of population mean and total in a finite population setting using multiple auxiliary variables. Journal of Agricultural, Biological, and Environmental Statistics, 19(2), 161-184.
26. Ozturk, O. and Jafari Jozani, M. (2014). Inclusion probabilities in partially rank ordered set sampling. Computational Statistics & Data Analysis, 69, 122-132.
27. Hatefi, A., Jafari Jozani, M. and Ozturk, O. (2015). Mixture model analysis of partially rank-ordered set samples: age groups of fish from length-frequency data. Scandinavian Journal of Statistics, 42(3), 848-871.
28. Wang, X., Wang, K., and Lim, J. (2012). Isotonized CDF estimation from judgment poststratification data with empty strata. Biometrics, 68(1), 194-202.
29. D’Agostino, R. B. (1986). Goodness-of-fit techniques. CRC press.
30. Penrose, K. W., Nelson, A. G., and Fisher, A. G. (1985). Generalized body composition prediction equation for men using simple measurement techniques. Medicine & Science in Sports & Exercise, 17(2), 189.