Abstract
In this study, we propose simultaneous tests for the mean and variance under the normality assumption. With formulating the null hypothesis and its alternative, we derive the joint test statistics using the likelihood ratio (LR) principle and combine the individual partial tests into a single one with combining functions including the union-intersection (UI) approach. Then we compare our proposed tests with other one by obtaining empirical powers through a simulation study. We also apply these simultaneous tests to the statistical process control (SPC) with discussion for running the combined charts with an industrial example. We discuss some interesting features as concluding remarks. Finally, in the appendix, we briefly derive the correlation coefficient between two individual LR statistics.
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Alfieri, R., A. Bonnini, C. Brombin, C. Castro, and L. Salmaso. 2012. Iterated combination-based paired permutation tests to determine shape effects of chemotherapy in patients with esophageal cancer. Statistical Methods in Medical Research. doi:10.1177/0962280212461981
Alloway, J. A., and M. Raghavachari. 1991. Control chart based on the Hodges–Lehmann estimator. Journal of Quality Technology 23: 336–47.
Arnold, B. C., and R. M. Shavelle. 1998. Joint confidence sets for the mean and variance of a normal distribution. American Statistician 52: 133–40.
Berger, R. L. 1996. Likelihood ratio tests and intersection–union tests. Raleigh, NC: Institute of Statistics Mimeo Series Number 2288, Department of Statistics, North Carolina State University.
Bickel, P. J., and K. A. Doksum. 1977. Mathematical statisics. San Francisco, CA: Basic Ideas and Selected Topics, Holden-Day.
Chao, M. T., and S. W. Cheng. 1996. Semicircle control chart for variables data. Quality Engineering 8: 441–46. doi:10.1080/08982119608904646
Chen, G., and S. W. Cheng. 1998. Max chart: Combining X-bar chart and S chart. Statistica Sinica 8: 263–71.
Choudhari, P., D. Kundu, and N. Misra. 2001. Likelihood ratio test for simultaneous testing of the mean and the variance of a normal distribution. Journal of Statistical Computation and Simulation 71: 313–33. doi:10.1080/00949650108812151
Corain, L., and L. Salmaso. 2013. Nonparametric permutation and combination-based multivariate control charts with applications in microelectronics. Applied Stochastic Models in Business and Industry 29: 334–49. doi:10.1002/asmb.1976
Duran, B. S., W. S. Tsai, and T. O. Lewis. 1976. A class of location-scale tests. Biometrika 63: 173–76.
Fisher, R. A. 1932. Statistical methods for research workers, 4th ed. Edinburgh, UK: Oliver and Boyd.
Frey, J., O. Marrero, and D. Norton. 2009. Minimum-area confidence sets for a normal distribution. Journal of Statistical Planning and Inference 139: 1023–32. doi:10.1016/j.jspi.2008.06.005
Hawkins, D. M., and Q. Deng. 2009. Combined charts for mean and variance information. Journal of Quality Technology 41: 415–25.
Hayter, A. 2014. Identifying common normal distributions. Test 23: 135–52. doi:10.1007/s11749-013-0345-3
Kendall, M. G., and A. Stuart. 1961. The advanced theory of statistics, vol. 2. London, UK: Charles Griffin.
Lepage, Y. 1971. A combination of Wilcoxon’s and Ansari-Bradley’s statistics. Biometrika 58: 213–17. doi:10.1093/biomet/58.1.213
Lepage, Y. 1973. A table for a combined Wilcoxon Ansari-Bradley statistic. Biometrika 60: 113–16. doi:10.1093/biomet/60.1.113
Mood, A. M., F. A. Graybill, and D. C. Boes. 1974. Introduction to the theory of statistics, 3rd. ed. (international student edition). Tokyo, Japan: McGraw-Hill.
Neuhäuser, M., A.-K. Leuchs, and D. Ball. 2011. A new location-scale test based on a combination of the ideas of Levene and Lepage. Biometrical Journal 53: 525–34. doi:10.1002/bimj.201000162
Owen, A. B. 2009. Karl Pearson’s meta-analysis revisited. Annals of Statistics 37: 3867–92. doi:10.1214/09-AOS697
Park, H.-I. 2013. On the study for the simultaneous test. Communications for Statistical Applications and Methods 20: 241–46. doi:10.5351/CSAM.2013.20.4.241
Pesarin, F., and L. Salmaso. 2010. Permutation tests for complex data. New York, NY: Wiley.
Randles, R. H., and D. A. Wolfe. 1979. Introduction to the theory of nonparametric statistics. New York, NY: Wiley.
Roy, S. N. 1957. Some aspects of multivariate analysis. New York, NY: Wiley.
Rublik, F. 2009. Critical values for testing location-scale hypothesis. Em Measurement Science Review 9: 9–15.
Won, S., N. Morris, Q. Lu, and R. C. Elston. 2009. Choosing an optimal method to combine p-values. Statistics in Medicine 28: 1537–53. doi:10.1002/sim.v28:11
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Park, HI. Simultaneous test for the mean and variance with an application to the statistical process control. J Stat Theory Pract 9, 868–881 (2015). https://doi.org/10.1080/15598608.2015.1034387
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DOI: https://doi.org/10.1080/15598608.2015.1034387