On the Comparison of Several Goodness of Fit tests under Simple Random Sampling and Ranked Set Sampling

Authors: F. Azna A. Shahabuddin, Kamarulzaman Ibrahim, Abdul Aziz Jemain

Abstract:

Many works have been carried out to compare the efficiency of several goodness of fit procedures for identifying whether or not a particular distribution could adequately explain a data set. In this paper a study is conducted to investigate the power of several goodness of fit tests such as Kolmogorov Smirnov (KS), Anderson-Darling(AD), Cramer- von- Mises (CV) and a proposed modification of Kolmogorov-Smirnov goodness of fit test which incorporates a variance stabilizing transformation (FKS). The performances of these selected tests are studied under simple random sampling (SRS) and Ranked Set Sampling (RSS). This study shows that, in general, the Anderson-Darling (AD) test performs better than other GOF tests. However, there are some cases where the proposed test can perform as equally good as the AD test.

Keywords: Empirical distribution function, goodness-of-fit, order statistics, ranked set sampling

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1072455

References:

[1] G.A. McIntyre, "A method for unbiased selective sampling using ranked sets", Australian Journal of Agricultural Research, No.3, 1952, pp.385-390.
[2] Z. Chen, Z. Bai & B.K. Sinha. Ranked Set Sampling: Theory and Applications. Springer. 2004.
[3] L.S.Hall and T.R.Dell, "Trial of ranked set sampling for forage yields". Forest Science, No. 22, 1966, pp. 22-26.
[4] K. Takahashi and K. Wakimoto, "On unbiased estimates of the population mean based on the sample stratified by means of ordering", Annals of the Institute of Statistical Mathematics, No. 20, 1968, pp.1-31.
[5] T.R. Dale and J.L. Clutter, "Ranked set sampling theory with order statistics background", Biometrika, No.28, 1972, pp.545-555.
[6] S.H. Chen, "Ranked set sampling theory with selective probability vector", Journal of Statistical Planning and Inference, No. 8, 1983, pp. 161-174.
[7] S.L. Stokes and T.W. Sager, "Characterization of a ranked set sample with application to estimating distribution functions", Journal of the American Statistical Association, No. 83, 1988, pp. 374-381.
[8] G.P. Patil. Editorial: "Ranked set sampling", Environmental and Ecological Statistics, No.2, 1995, pp.271-285.
[9] G.P. Patil, A. K. Sinha, and C. Tallie, "Ranked set sampling: a bibliography", Environmental and Ecological Statistics, No.6, 1999, pp.91-98.
[10] Z.D. Bai and Z. Chen, "On the theory of ranked set sampling and its ramifications". Journal of Statistical Planning and Inference. No. 109, 2003, pp. 81-99.
[11] A.A. Jemain, A. Al-Omari, and K. Ibrahim, "Modified ratio estimator for the population mean using double median ranked set sampling", Pakistan Journal of Science, No. 24(3), 2008, pp. 217-226.
[12] X12. R.B. D-Agostino and M.A. Stephens, Goodness of fit techniques, Marcel Dekker, New York, 1986.
[13] J. Zhang, "Powerful goodness of fit tests based on the likelihood ratio". Journal of Royal Statist. Soc. B, No. 64, Part 2, 2002, pp. 281- 294.
[14] J.R. Green and Y.A.S. Hegazy, "Powerful Modified Goodness of fit tests", Journal of the American Statistical Association, Vol. 71, No.353, 1976, pp. 204-209.