Goodness of Fit Test Based on Modified Log-Likelihood Ratio for Normal Distribution
Abstract
This research presents a goodness of fit test based on modified log-likelihood ratio for normal distribution and compares the efficiency of the proposed test with modified Anderson-Darling test and modified Cramer-von-Mises test which were developed by Zhang (2002), and the original Anderson-Darling tests. The critical values of the tests are obtained through simulation and the efficiency of test is considered as probability of type I error and power of the tests. There are four types of distribution; near normal distribution, symmetric distribution, asymmetric distribution with more than three of kurtosis, and asymmetric distribution with less than three of kurtosis. The results show that the four tests can control type I error probability and the proposed test is the most powerful test for near normal and symmetric distributions with all sizes of sample. The modified Anderson-Darling test is the most powerful test for asymmetric distribution both more than three and less than three of kurtosis with all sizes of sample. Keywords : goodness of fit test statistic, modified likelihood ratio , power of the testReferences
Anderson, T. W., & Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association,
49(268), 765-769.
Cochran, W. G. (1952). The X2 test of goodness of fit. Annals of Mathematical Statistics, 23, 315-345.
Cressie, N., & Read, R.C.T. (1984). Multinomial goodness-of-fit tests. The Journal of the Royal Statistical
Society. Series B (Methodological), 46(3), 440-464.
Pinthongpan, A. & Thongteeraparp, A. (2015). Comparison of the Goodness of Fit Tests for Normality.
Academic Journal Pranakorn Rajabhat University, 6(1), 240-249. (in Thai)
Potas, N., Sarısoy, E.E. & Kara, M. (2014). A modified goodness-of-fit test based on likelihood ratiofor the skew-
normal distribution. Applied Mathematical Sciences, 8, 3869 – 3887.
Razali, N. M., & Wah, Y. B. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and
Anderson–Darling tests. Journal of Statistical Modeling and Analytics, 2(1), 21-33.
Rinnakorn, C., & Kamon, B. (2007). A Power Comparison of Goodness-of-fit Tests for Normality Based on the
Likelihood Ratio and the Non-likelihood Ratio. Thailand Statistician, 5, 57-68.
Saethow, P. & Neamvonk, J. (2014). Modified Anderson-Darling test based on the likelihood ratio for skewed
distributions. In Proceedings of the International Conference on Applied Statistics 2014, (pp. 27-34),
KhonKaen, Thailand.
Wasinra, S. (2006). Comparison of the test for Normality using goodness of fit test based on the likelihood ratio.
Master thesis in Statistics, Kasetsart University. (in Thai)
Wichitchan, S. (2010). Effects of statistical analysis under normal distribution assumption on data from logistic
distribution. Master thesis in Statistics, Chulalongkorn University. (in Thai)
Yap, B. W. & Sim, C. H. (2011). Comparisons of various types of normality tests. Journal of Statistical
Computation and Simulation, 81(12), 2141–2155.
Zhang, J. (2002). Powerful goodness-of-fit tests based on the likelihood ratio. Journal of the Royal Statistical
Society. Series B (Methodological), 64(2), 281-294.
49(268), 765-769.
Cochran, W. G. (1952). The X2 test of goodness of fit. Annals of Mathematical Statistics, 23, 315-345.
Cressie, N., & Read, R.C.T. (1984). Multinomial goodness-of-fit tests. The Journal of the Royal Statistical
Society. Series B (Methodological), 46(3), 440-464.
Pinthongpan, A. & Thongteeraparp, A. (2015). Comparison of the Goodness of Fit Tests for Normality.
Academic Journal Pranakorn Rajabhat University, 6(1), 240-249. (in Thai)
Potas, N., Sarısoy, E.E. & Kara, M. (2014). A modified goodness-of-fit test based on likelihood ratiofor the skew-
normal distribution. Applied Mathematical Sciences, 8, 3869 – 3887.
Razali, N. M., & Wah, Y. B. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and
Anderson–Darling tests. Journal of Statistical Modeling and Analytics, 2(1), 21-33.
Rinnakorn, C., & Kamon, B. (2007). A Power Comparison of Goodness-of-fit Tests for Normality Based on the
Likelihood Ratio and the Non-likelihood Ratio. Thailand Statistician, 5, 57-68.
Saethow, P. & Neamvonk, J. (2014). Modified Anderson-Darling test based on the likelihood ratio for skewed
distributions. In Proceedings of the International Conference on Applied Statistics 2014, (pp. 27-34),
KhonKaen, Thailand.
Wasinra, S. (2006). Comparison of the test for Normality using goodness of fit test based on the likelihood ratio.
Master thesis in Statistics, Kasetsart University. (in Thai)
Wichitchan, S. (2010). Effects of statistical analysis under normal distribution assumption on data from logistic
distribution. Master thesis in Statistics, Chulalongkorn University. (in Thai)
Yap, B. W. & Sim, C. H. (2011). Comparisons of various types of normality tests. Journal of Statistical
Computation and Simulation, 81(12), 2141–2155.
Zhang, J. (2002). Powerful goodness-of-fit tests based on the likelihood ratio. Journal of the Royal Statistical
Society. Series B (Methodological), 64(2), 281-294.
Downloads
Published
2018-02-16
Issue
Section
Research Article
