Modified generalized p-value and confidence interval by Fisher's fiducial approach

Year 2017, Volume: 46 Issue: 2, 339 - 360, 01.04.2017

Evren Ozkip Berna Yazici , Ahmet Sezer

Abstract

In this study, we develop two simple generalized confidence intervals for the difference between means of two normal populations with heteroscedastic variances which is usually referred to as the Behrens-Fisher problem. The developed confidence intervals are compared with the generalized confidence interval in the literature. We also propose modified fiducial based approach using Fisher's fiducial inference for comparing the mean of two lognormal distributions and compare them with the other tests in the literature. A Monte Carlo simulation study is
conducted to evaluate performances of the proposed methods under different scenarios. The simulation results indicate that the developed
confidences intervals for the Behrens-Fisher problem have shorter interval lengths and they give better coverage accuracy in some cases. The
modified ducial based approach is the best to provide satisfactory results in respect to its type error and power in all sample sizes. The modified test is applicable to small samples and is easy to compute and implement. The methods are also applied to two real-life examples.

Keywords

Modified fiducial based test, Generalized approach, Parametric bootstrap approach, Lognormal distribution, The Behrens-Fisher problem

References

• Fisher, R.A. The fiducial argument in statistical inference, Annals of Eugenics 6, 391-398, 1935.
• Fisher, R.A. The asymptotic approach to Behrens' integral with further tables for the d test of significance, Annals of Eugenics 11, 141-172, 1941.
• Welch, B.L. The signicance of the difference between two means when the population vari- ances are unequal, Biometrika 29, 350-362, 1938.
• Welch, B.L. The generalization of 'Student's' problem when several different population variances are involved, Biometrika 34, 28-35, 1947.
• Aspin, A.A. An examination and further development of a formula arising in the problem of comparing two mean values, Biometrika 35, 88-96, 1948.
• Cochran, W.G., Cox, G.M. Experiment Designs, John Wiley and Sons, New York, 1950.
• Jing, B.Y. Two-sample empirical likelihood method, Statistics and Probability Letters 24, 315-319, 1995.
• Kim, S.H., Cohen, A.S. On the Behrens-Fisher problem: A review, Journal of Educational and Behavioral Statistics 23, 356-377, 1998.
• Singh, P., Saxena, K.K., Srivastava, O.P. Power comparisons of solutions to the Behrens- Fisher problem, American Journal of Mathematical and Management Sciences 22, 233-250, 2002.
• Dong, L.B. The Behrens-Fisher problem: an empirical likelihood approach, Econometrics Working Paper EWP0404, University of Victoria, ISSN 1485-6441, 2004.
• Tsui, K. and Weerahandi, S. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association 84, 602-607, 1989.
• Weerahandi, S. Generalized confidence intervals, Journal of the American Statistical Asso- ciation 88, 899-905, 1993.
• Weerahandi, S. Exact Statistical Methods for Data Analysis, Springer, 1995.
• Krishnamoorthy, K., Mathew, T. Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals, Journal of Statistical Planning and Inference 115, 103-121, 2003.
• Krishnamoorthy, K., Lu, Y. Inferences on the common mean of several normal populations based on the generalized variable method, Biometrics 59, 237-247, 2003.
• Krishnamoorthy, K., Lu, F., Mathew, T. A parametric bootstrap approach for ANOVA with unequal variances: fixed and random models, Computational Statistics and Data Analysis 51, 5731-5742, 2007.
• Lee, J.C, Lin, S.H. Generalized confidence interval for the ratio of means of two normal populations, Journal of Statistical Planning and Inference 123, 49-60, 2004.
• Chang, C.H., Pal, N. A revisit to the Behrens-Fisher problem: comparison of ve test methods, Communications in Statistics-Simulation and Computation 37, 1064-1085, 2008.
• Zheng, S., Shi, N.Z., Ma, W. Statistical inference on difference or ratio of means from heteroscedastic normal populations, Journal of Statistical Planning and Inference 140, 1236- 1242, 2010.
• Ozkip, E., Yazici, B., Sezer, A. A simulation study on tests for the Behrens-Fisher problem, Turkiye Klinikleri: Journal of Biostatistics 6, 59-66, 2014.
• Sezer, A., Ozkip, E., Yazici, B. Comparison of confidence intervals for the Behrens-Fisher problem, Communications in Statistics-Simulation and Computation, accepted manuscript, 2015.
• Ye, R.D., Ma, T.F., Luo, K. Inferences on the reliability in balanced and unbalanced one-way random models, Journal of Statistical Computation and Simulation 84, 1136-1153, 2014.
• Gunasekera, S., Ananda, M. Generalized variable method inference for the location pa- rameter of the general half-normal distribution, Journal of Statistical Computation and Simulation 85, 2115-2132, 2015.
• Zhao, G., Xu, X. The p-values for one-sided hypothesis testing in univariate linear calibra- tions, Communications in Statistics - Simulation and Computation, accepted manuscript, 2016.
• Krishnamoorthy, K., Mathew, T., Ramachandran, G. Generalized p-values and confidence intervals: A novel approach for analyzing lognormally distributed exposure data, , Journal of Occupational and Environmental Hygiene 3, 642-650, 2006.
• Shen, H., Brown, L., Hui, Z. Efficient estimation of log-normal means with application to pharmacokinetics data, Statistics in Medicine 25, 3023-3038, 2006.
• Land, C.E. Confidence intervals for linear functions of the lognormal mean and variance, Annals of Mathematical Statistics 42, 1187-1205, 1971.
• Land, C.E. An evaluation of approximate confidence interval estimation methods for log- normal means, Technometrics 14, 145-158, 1972.
• Land, C.E. Standard confidence limits for linear functions of the normal mean and variance, Journal of the American Statistical Association 68, 960-963, 1973.
• Angus ,J.E. Inferences on the lognormal mean for complete samples, Comm. Statist. Sim- ulation Comput. 17, 1307-1331, 1988.
• Angus ,J.E. Bootstrap one-sided confidence intervals for the lognormal mean, Statistician 43, 395-401, 1994.
• Zhou, X.H., Gao, S., Hui, S.L. Methods for comparing the means of two independent log- normal samples, Biometrics 53, 1129-1135, 1997.
• Wu, J., Jiang, G., Wong, A.C.M., Sun, X. Likelihood analysis for the ratio of means of two independent log-normal distributions, Biometrics 58, 463-469, 2002.
• Krishnamoorthy, K., Mathew, T. Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals, Journal of Statistical Planning and Inference 115, 130-121, 2003.