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A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters

Year 2018, Volume: 31 Issue: 2, 628 - 641, 01.06.2018

Abstract

Analysis of Reciprocals F-test
developed by Miura [1] is used to test the equality of Inverse Gaussian (IG)
means based on the assumption of homogeneity of scale parameters. However this
method is not valid when this assumption is not satisfied. There are some
method developed for comparing the equality of the IG means under heteroscedasticity
of scale parameters. In this study, we compare the performance of the five commonly
used tests in the literature via Monte Carlo simulation study. The tests
considered are analysis of reciprocals (ANORE) F-test, Parametric Bootstrap Approach
(PBA), Generalized p-Value Approach proposed by Tian (GPT), Generalized p-Value
Approach proposed by Shi and Lv (GPS) and Computational Approach Test (CAT), The
goal of this study is to compare these methods under different combinations of
parameters and various sample sizes.

References

  • [1] Miura, C.K., “Tests for the mean of the inverse Gaussian distribution”, Scandinavian Journal of Statistics, 5(4): 200-204, (1978).
  • [2] Welch, B.L., “On the comparison of several mean values: an alternative approach”, Biometrika, 38(3/4): 330-336, (1951).
  • [3] Şenoğlu, B. and Tiku, M.L., “Analysis of variance in experimental design with nonnormal error distributions”, Communications in Statistics-Theory and Methods, 30(7): 1335-1352, (2001).
  • [4] Yigit, E. and Gokpinar, F., “A simulation study on tests for one-way ANOVA under the unequal variance assumption”, Commun Fac Sci Univ Ankara, Ser A, 1: 15-34, (2010).
  • [5] Gokpinar, E.Y. and Gokpinar, F., “A test based on the computational approach for equality of means under the unequal variance assumption”, Hacettepe Jour. of Math. and Static, 41(4): 605-613, (2012).
  • [6] Mutlu, H.T., Gökpinar, F., Gökpinar, E., Gül, H.H. and Güven, G., “A New Computational Approach Test for One-Way ANOVA under Heteroscedasticity”, Communications in Statistics-Theory and Methods, (just-accepted), (2016). Doi: 10.1080/03610926.2016.1177082.
  • [7] Gokpinar, E. and Gokpinar, F., “Testing equality of variances for several normal populations”, Communications in Statistics-Simulation and Computation, 46(1): 38-52, (2017).
  • [8] Gokpinar, F. and Gokpinar, E., “Testing the equality of several log-normal means based on a computational approach”, Communications in Statistics-Simulation and Computation, 46(3): 1998-2010, (2017).
  • [9] Schrödinger, E., “Zur theorie der fall-und steigversuche an teilchen mit brownscher bewegung”, Physikalische Zeitschrift, 16(1915): 289-295, (1915).
  • [10] von Smoluchowski, M., “Notiz uiber die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikanschen Versuchsanordning”, Phys. Z, 16, 318-321. (1915).
  • [11] Basak, P. and Balakrishnan, N., “Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring”, Journal of Statistical Computation and Simulation, 82(7): 1055-1072, (2012).
  • [12] Takagi, K., Kumagai, S. and Kusaka, Y., “Application of inverse Gaussian distribution to occupational exposure data”, The Annals of Occupational Hygiene, 41(5): 505-514, (1997).
  • [13] Chang, M., You, X. and Wen, M., “Testing the homogeneity of inverse Gaussian scale-like parameters”, Statistics & Probability Letters, 82(10): 1755-1760, (2012).
  • [14] Fahidy, T.Z., “Applying the inverse Gaussian distribution to the assessment of chemical reactor performance”, International Journal of Chemistry, 4(2): 26, (2012).
  • [15] Chhikara, R.S. and Folks, J.L., The Inverse Gaussian distribution., Marcel Decker. Inc., New York, (1989).
  • [16] Seshadri, V., The inverse Gaussian distribution: a case study in exponential families, Oxford University Press, (1993).
  • [17] Seshadri, V., The inverse Gaussian distribution: statistical theory and applications. Springer, New York, (1999).
  • [18] Tweedie, M.C., “Statistical Properties of Inverse Gaussian Distributions. I”, The Annals of Mathematical Statistics, 28(2): 362-377, (1957).
  • [19] Folks, J.L. and Chhikara, R.S., “The inverse Gaussian distribution and its statistical application--a review”, Journal of the Royal Statistical Society. Series B (Methodological), 40(3): 263-289, (1978).
  • [20] Wasan, M.T., “Monograph on Inverse Gaussian Distribution”, Department of Mathematics, (1966).
  • [21] Mudholkar, G.S. and Natarajan, R., “The inverse Gaussian models: analogues of symmetry, skewness and kurtosis”, Annals of the Institute of statistical Mathematics, 54(1): 138-154, (2002).
  • [22] Bardsley, W.E., “Note on the use of the inverse Gaussian distribution for wind energy applications”, Journal of Applied Meteorology, 19(9): 1126-1130, (1980).
  • [23] Balakrishnan, N. and Rahul, T., “Inverse Gaussian distribution for modeling conditional durations in finance”, Communications in Statistics-Simulation and Computation, 43(3): 476-486, (2014).
  • [24] Chhikara, R.S. “Optimum Tests for Comparison of Two Inverse Gaussian Distribution Means1”, Australian and New Zealand Journal of Statistics, 17(2): 77-83, (1975).
  • [25] Davis, A.S. “Use of the likelihood ratio test on the inverse Gaussian distribution”, The American Statistician, 34(2): 108-110, (1980).
  • [26] Samanta, M., “On tests of equality of two inverse Gaussian distributions”, South African Statistical Journal, 19(2): 83-95, (1985).
  • [27] Tian, L., “Testing equality of inverse Gaussian means under heterogeneity, based on generalized test variable”, Computational statistics & data analysis, 51(2): 1156-1162. (2006).
  • [28] Ma, C.X. and Tian, L., “A parametric bootstrap approach for testing equality of inverse Gaussian means under heterogeneity”, Communications in Statistics-Simulation and Computation, 38(6): 1153-1160, (2009).
  • [29] Ye, R.D., Ma, T.F. and Wang, S.G., “Inferences on the common mean of several inverse Gaussian populations”, Computational Statistics & Data Analysis, 54(4): 906-915, (2010).
  • [30] Lin, S.H. and Wu, I.M., “On the common mean of several inverse Gaussian distributions based on a higher order likelihood method”. Applied Mathematics and Computation, 217(12): 5480-5490, (2011).
  • [31] Shi, J.H. and Lv, J.L., “A new generalized p-value for testing equality of inverse Gaussian means under heterogeneity”, Statistics & Probability Letters, 82(1): 96-102. (2012).
  • [32] Gökpınar, E.Y., Polat, E., Gokpınar, F. and Günay, S., “A new computational approach for testing equality of inverse Gaussian means under heterogeneity”, Hacettepe journal of Mathematics and Statistics, 42(5): 581-590, (2013).
  • [33] Pal, N., Lim, W.K. and Ling, C.H., “A computational approach to statistical inferences”, Journal of Applied Probability & Statistics, 2(1): 13-35, (2007).
  • [34] Chang, C.H. and Pal, N., “A revisit to the Behrens–Fisher problem: comparison of five test methods”, Communications in Statistics—Simulation and Computation, 37(6): 1064-1085, (2008).
  • [35] Chang, C.H., Pal, N., Lim, W.K. and Lin, J.J., “Comparing several population means: a parametric bootstrap method, and its comparison with usual ANOVA F test as well as ANOM”, Computational Statistics, 25(1): 71-95, (2010).
  • [36] Gokpinar, F. and Gokpinar, E., “A computational approach for testing equality of coefficients of variation in k normal populations”, Hacettepe Journal of Mathematics and Statistics, 44(5): 1197-1213, (2015).
  • [37] Jafari, A.A. and Abdollahnezhad, K., “Inferences on the Means of Two Log-Normal Distributions: A Computational Approach Test”, Communications in Statistics-Simulation and Computation, 44(7): 1659-1672, (2015).
  • [38] Jafari, A.A. and Kazemi, M.R., “Computational approach test for inference about several correlation coefficients: Equality and common”, Communications in Statistics-Simulation and Computation, 46(3): 2043-2056, (2017).
Year 2018, Volume: 31 Issue: 2, 628 - 641, 01.06.2018

Abstract

References

  • [1] Miura, C.K., “Tests for the mean of the inverse Gaussian distribution”, Scandinavian Journal of Statistics, 5(4): 200-204, (1978).
  • [2] Welch, B.L., “On the comparison of several mean values: an alternative approach”, Biometrika, 38(3/4): 330-336, (1951).
  • [3] Şenoğlu, B. and Tiku, M.L., “Analysis of variance in experimental design with nonnormal error distributions”, Communications in Statistics-Theory and Methods, 30(7): 1335-1352, (2001).
  • [4] Yigit, E. and Gokpinar, F., “A simulation study on tests for one-way ANOVA under the unequal variance assumption”, Commun Fac Sci Univ Ankara, Ser A, 1: 15-34, (2010).
  • [5] Gokpinar, E.Y. and Gokpinar, F., “A test based on the computational approach for equality of means under the unequal variance assumption”, Hacettepe Jour. of Math. and Static, 41(4): 605-613, (2012).
  • [6] Mutlu, H.T., Gökpinar, F., Gökpinar, E., Gül, H.H. and Güven, G., “A New Computational Approach Test for One-Way ANOVA under Heteroscedasticity”, Communications in Statistics-Theory and Methods, (just-accepted), (2016). Doi: 10.1080/03610926.2016.1177082.
  • [7] Gokpinar, E. and Gokpinar, F., “Testing equality of variances for several normal populations”, Communications in Statistics-Simulation and Computation, 46(1): 38-52, (2017).
  • [8] Gokpinar, F. and Gokpinar, E., “Testing the equality of several log-normal means based on a computational approach”, Communications in Statistics-Simulation and Computation, 46(3): 1998-2010, (2017).
  • [9] Schrödinger, E., “Zur theorie der fall-und steigversuche an teilchen mit brownscher bewegung”, Physikalische Zeitschrift, 16(1915): 289-295, (1915).
  • [10] von Smoluchowski, M., “Notiz uiber die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikanschen Versuchsanordning”, Phys. Z, 16, 318-321. (1915).
  • [11] Basak, P. and Balakrishnan, N., “Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring”, Journal of Statistical Computation and Simulation, 82(7): 1055-1072, (2012).
  • [12] Takagi, K., Kumagai, S. and Kusaka, Y., “Application of inverse Gaussian distribution to occupational exposure data”, The Annals of Occupational Hygiene, 41(5): 505-514, (1997).
  • [13] Chang, M., You, X. and Wen, M., “Testing the homogeneity of inverse Gaussian scale-like parameters”, Statistics & Probability Letters, 82(10): 1755-1760, (2012).
  • [14] Fahidy, T.Z., “Applying the inverse Gaussian distribution to the assessment of chemical reactor performance”, International Journal of Chemistry, 4(2): 26, (2012).
  • [15] Chhikara, R.S. and Folks, J.L., The Inverse Gaussian distribution., Marcel Decker. Inc., New York, (1989).
  • [16] Seshadri, V., The inverse Gaussian distribution: a case study in exponential families, Oxford University Press, (1993).
  • [17] Seshadri, V., The inverse Gaussian distribution: statistical theory and applications. Springer, New York, (1999).
  • [18] Tweedie, M.C., “Statistical Properties of Inverse Gaussian Distributions. I”, The Annals of Mathematical Statistics, 28(2): 362-377, (1957).
  • [19] Folks, J.L. and Chhikara, R.S., “The inverse Gaussian distribution and its statistical application--a review”, Journal of the Royal Statistical Society. Series B (Methodological), 40(3): 263-289, (1978).
  • [20] Wasan, M.T., “Monograph on Inverse Gaussian Distribution”, Department of Mathematics, (1966).
  • [21] Mudholkar, G.S. and Natarajan, R., “The inverse Gaussian models: analogues of symmetry, skewness and kurtosis”, Annals of the Institute of statistical Mathematics, 54(1): 138-154, (2002).
  • [22] Bardsley, W.E., “Note on the use of the inverse Gaussian distribution for wind energy applications”, Journal of Applied Meteorology, 19(9): 1126-1130, (1980).
  • [23] Balakrishnan, N. and Rahul, T., “Inverse Gaussian distribution for modeling conditional durations in finance”, Communications in Statistics-Simulation and Computation, 43(3): 476-486, (2014).
  • [24] Chhikara, R.S. “Optimum Tests for Comparison of Two Inverse Gaussian Distribution Means1”, Australian and New Zealand Journal of Statistics, 17(2): 77-83, (1975).
  • [25] Davis, A.S. “Use of the likelihood ratio test on the inverse Gaussian distribution”, The American Statistician, 34(2): 108-110, (1980).
  • [26] Samanta, M., “On tests of equality of two inverse Gaussian distributions”, South African Statistical Journal, 19(2): 83-95, (1985).
  • [27] Tian, L., “Testing equality of inverse Gaussian means under heterogeneity, based on generalized test variable”, Computational statistics & data analysis, 51(2): 1156-1162. (2006).
  • [28] Ma, C.X. and Tian, L., “A parametric bootstrap approach for testing equality of inverse Gaussian means under heterogeneity”, Communications in Statistics-Simulation and Computation, 38(6): 1153-1160, (2009).
  • [29] Ye, R.D., Ma, T.F. and Wang, S.G., “Inferences on the common mean of several inverse Gaussian populations”, Computational Statistics & Data Analysis, 54(4): 906-915, (2010).
  • [30] Lin, S.H. and Wu, I.M., “On the common mean of several inverse Gaussian distributions based on a higher order likelihood method”. Applied Mathematics and Computation, 217(12): 5480-5490, (2011).
  • [31] Shi, J.H. and Lv, J.L., “A new generalized p-value for testing equality of inverse Gaussian means under heterogeneity”, Statistics & Probability Letters, 82(1): 96-102. (2012).
  • [32] Gökpınar, E.Y., Polat, E., Gokpınar, F. and Günay, S., “A new computational approach for testing equality of inverse Gaussian means under heterogeneity”, Hacettepe journal of Mathematics and Statistics, 42(5): 581-590, (2013).
  • [33] Pal, N., Lim, W.K. and Ling, C.H., “A computational approach to statistical inferences”, Journal of Applied Probability & Statistics, 2(1): 13-35, (2007).
  • [34] Chang, C.H. and Pal, N., “A revisit to the Behrens–Fisher problem: comparison of five test methods”, Communications in Statistics—Simulation and Computation, 37(6): 1064-1085, (2008).
  • [35] Chang, C.H., Pal, N., Lim, W.K. and Lin, J.J., “Comparing several population means: a parametric bootstrap method, and its comparison with usual ANOVA F test as well as ANOM”, Computational Statistics, 25(1): 71-95, (2010).
  • [36] Gokpinar, F. and Gokpinar, E., “A computational approach for testing equality of coefficients of variation in k normal populations”, Hacettepe Journal of Mathematics and Statistics, 44(5): 1197-1213, (2015).
  • [37] Jafari, A.A. and Abdollahnezhad, K., “Inferences on the Means of Two Log-Normal Distributions: A Computational Approach Test”, Communications in Statistics-Simulation and Computation, 44(7): 1659-1672, (2015).
  • [38] Jafari, A.A. and Kazemi, M.R., “Computational approach test for inference about several correlation coefficients: Equality and common”, Communications in Statistics-Simulation and Computation, 46(3): 2043-2056, (2017).
There are 38 citations in total.

Details

Journal Section Statistics
Authors

Gamze Guven

Publication Date June 1, 2018
Published in Issue Year 2018 Volume: 31 Issue: 2

Cite

APA Guven, G. (2018). A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters. Gazi University Journal of Science, 31(2), 628-641.
AMA Guven G. A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters. Gazi University Journal of Science. June 2018;31(2):628-641.
Chicago Guven, Gamze. “A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale Parameters”. Gazi University Journal of Science 31, no. 2 (June 2018): 628-41.
EndNote Guven G (June 1, 2018) A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters. Gazi University Journal of Science 31 2 628–641.
IEEE G. Guven, “A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters”, Gazi University Journal of Science, vol. 31, no. 2, pp. 628–641, 2018.
ISNAD Guven, Gamze. “A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale Parameters”. Gazi University Journal of Science 31/2 (June 2018), 628-641.
JAMA Guven G. A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters. Gazi University Journal of Science. 2018;31:628–641.
MLA Guven, Gamze. “A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale Parameters”. Gazi University Journal of Science, vol. 31, no. 2, 2018, pp. 628-41.
Vancouver Guven G. A Comparison of Five Tests for the Equality of Inverse Gaussian Means under Heteroscedasticity of Scale parameters. Gazi University Journal of Science. 2018;31(2):628-41.