Research Article
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Year 2020, , 815 - 829, 01.09.2020

Bulent Altunkaynak , Hamza Gamgam , Merve Bağçacı

https://doi.org/10.35378/gujs.566936

Abstract

References

  • Terpstra, J.T., "The asymptotic normality and consistency of kendall's test against trend, when ties are present in one ranking", Indagationes Mathematicae, 14(3): 327 - 333, (1952).
  • Jonckheere, A.R., "A Distribution-Free k-Sample Test Against Ordered Alternatives", Biometrika, 41(1/2): 133-145, (1954).
  • Chacko, V.J., "Testing Homogeneity Against Ordered Alternatives", Ann. Math. Statist., 34(3): 945-956, (1963).
  • Puri, M.L., "Some distribution-free K-sample rank tests of homogeneity against ordered alternatives", Communications on Pure and Applied Mathematics, 18(1‐2): 51-63, (1965).
  • May, R.B. and Konkin, P.R., "A Nonparametric Test of an Ordered Hypothesis for k Independent Samples", 30(2): 251-257, (1970).
  • Odeh, R.E., "On Jonckheere's k-Sample Test Against Ordered Alternatives", Technometrics, 13(4): 912-918, (1971).
  • Cuzick, J., "A wilcoxon-type test for trend", Stat Med., 4(1): 87-90, (1985).
  • Hettmansperger, T.P. and Norton, R.M., "Tests for Patterned Alternatives in k-Sample Problems", Journal of the American Statistical Association, 82(397): 292-299, (1987).
  • Beier, F. and Buning, H., "An adaptive test against ordered alternatives", Computational Statistics & Data Analysis, 25(4): 441-452, (1997).
  • Neuhäuser, M., Liu, P.-Y., and Hothorn, L.A., "Nonparametric Tests for Trend: Jonckheere's Test, a Modification and a Maximum Test", Biometrical Journal, 40(8): 899-909, (1998).
  • Buning, H. and Kossler, W., "Adaptive tests for umbrella alternatives", Biometrical Journal, 40(5): 573-587, (1998).
  • Terpstra, J.T. and Magel, R.C., "A new nonparametric test for the ordered alternative problem", Journal of Nonparametric Statistics, 15(3): 289-301, (2003).
  • Chang, C.H. and Yen, C.H., "A Nonparametric Test for the Ordered Alternative Based on Fast Discrete Fourier Transform Coefficient", Journal of Testing and Evaluation, 39(6): 1131-1143, (2011).
  • Terpstra, J.T., Chang, C.H., and Magel, R.C., "On the use of Spearman's correlation coefficient for testing ordered alternatives", Journal of Statistical Computation and Simulation, 81(11): 1381-1392, (2011).
  • Chang, C.H., Chin, C.C., Yu, W.W., and Huang, Y.Y., "Using the Bernoulli trial approaches for detecting ordered alternatives", Bmc Medical Research Methodology, 13((2013).
  • Shan, G.G., Young, D., and Kang, L., "A New Powerful Nonparametric Rank Test for Ordered Alternative Problem", Plos One, 9(11): (2014).
  • Gaur, A., "A class of k-sample distribution-free tests for location against ordered alternatives", Communications in Statistics-Theory and Methods, 46(5): 2343-2353, (2017).
  • Mack, G.A. and Wolfe, D.A., "K-Sample Rank-Tests for Umbrella Alternatives", Journal of the American Statistical Association, 76(373): 175-181, (1981).
  • Kossler, W., "Some c-sample rank tests of homogeneity against umbrella alternatives with unknown peak", Journal of Statistical Computation and Simulation, 76(1): 57-74, (2006).
  • Gaur, A., Mahajan, K.K., and Arora, S., "New nonparametric tests for testing homogeneity of scale parameters against umbrella alternative", Statistics & Probability Letters, 82(9): 1681-1689, (2012).
  • Shi, N.Z., "Rank Test Statistics for Umbrella Alternatives", Communications in Statistics-Theory and Methods, 17(6): 2059-2073, (1988).
  • Cohen, A. and Sackrowitz, H.B., "Tests for the umbrella alternative under normality", Communications in Statistics-Theory and Methods, 25(11): 2807-2817, (1996).
  • Pan, G.H., "Distribution-free tests for umbrella alternatives", Communications in Statistics-Theory and Methods, 25(12): 3185-3194, (1996).
  • Hartlaub, B.A. and Wolfe, D.A., "Distribution-free ranked-set sample procedures for umbrella alternatives in the m-sample setting", Environmental and Ecological Statistics, 6(2): 105-118, (1999).
  • Hayter, A.J. and Liu, W., "A new test against an umbrella alternative and the associated simultaneous confidence intervals", Computational Statistics & Data Analysis, 30(4): 393-401, (1999).
  • Lee, W.C. and Chen, Y.I., "Weighted Kaplan-Meier tests for umbrella alternatives", Annals of the Institute of Statistical Mathematics, 53(4): 835-852, (2001).
  • Magel, R.C. and Qin, L., "A non-parametric test for umbrella alternatives based on ranked-set sampling", Journal of Applied Statistics, 30(8): 925-937, (2003).
  • Singh, P. and Liu, W., "A test against an umbrella ordered alternative", Computational Statistics & Data Analysis, 51(3): 1957-1964, (2006).
  • Alvo, M., "Nonparametric tests of hypotheses for umbrella alternatives", Canadian Journal of Statistics-Revue Canadienne De Statistique, 36(1): 143-156, (2008).
  • Bhat, S.V. and Patil, A.B., "A class of k-sample distribution-free tests for the umbrella alternatives", International Journal of Agricultural and Statistical Sciences, 4(1): 53-61, (2008).
  • Basso, D. and Salmaso, L., "A permutation test for umbrella alternatives", Statistics and Computing, 21(1): 45-54, (2011).
  • Altunkaynak, B. and Gamgam, H., "Comparing the performance of nonparametric tests for equality of location against ordered alternatives", Communications in Statistics - Simulation and Computation, 1-22, (2019).
  • Bradley, J.V., "Robustness?", British Journal of Mathematical and Statistical Psychology, 31(2): 144-152, (1978).

A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives

Year 2020, , 815 - 829, 01.09.2020

Bulent Altunkaynak , Hamza Gamgam , Merve Bağçacı

https://doi.org/10.35378/gujs.566936

Abstract

In this study, a distribution free new statistic is introduced to test the equality of locations against the umbrella alternative hypotheses. The Shan test known for the ordered alternatives hypotheses is arranged for the umbrella alternative hypotheses. This statistic can be considered as an extension of the sign and Mann-Whitney statistics. Using a comprehensive simulation design, the proposed test was compared with the Hettmansperger and Norton and, Mack-Wolfe tests according to the criteria of the power and type I error rate of the test. In the simulation outcomes, it was seen that the robustness condition for Bradley's type I error rate were ensured for all tests. The power comparison outcomes also showed that the proposed test is more powerful than the other tests.

Keywords

Nonparametric statistics Umbrella alternative Monte Carlo Simulation

References

  • Terpstra, J.T., "The asymptotic normality and consistency of kendall's test against trend, when ties are present in one ranking", Indagationes Mathematicae, 14(3): 327 - 333, (1952).
  • Jonckheere, A.R., "A Distribution-Free k-Sample Test Against Ordered Alternatives", Biometrika, 41(1/2): 133-145, (1954).
  • Chacko, V.J., "Testing Homogeneity Against Ordered Alternatives", Ann. Math. Statist., 34(3): 945-956, (1963).
  • Puri, M.L., "Some distribution-free K-sample rank tests of homogeneity against ordered alternatives", Communications on Pure and Applied Mathematics, 18(1‐2): 51-63, (1965).
  • May, R.B. and Konkin, P.R., "A Nonparametric Test of an Ordered Hypothesis for k Independent Samples", 30(2): 251-257, (1970).
  • Odeh, R.E., "On Jonckheere's k-Sample Test Against Ordered Alternatives", Technometrics, 13(4): 912-918, (1971).
  • Cuzick, J., "A wilcoxon-type test for trend", Stat Med., 4(1): 87-90, (1985).
  • Hettmansperger, T.P. and Norton, R.M., "Tests for Patterned Alternatives in k-Sample Problems", Journal of the American Statistical Association, 82(397): 292-299, (1987).
  • Beier, F. and Buning, H., "An adaptive test against ordered alternatives", Computational Statistics & Data Analysis, 25(4): 441-452, (1997).
  • Neuhäuser, M., Liu, P.-Y., and Hothorn, L.A., "Nonparametric Tests for Trend: Jonckheere's Test, a Modification and a Maximum Test", Biometrical Journal, 40(8): 899-909, (1998).
  • Buning, H. and Kossler, W., "Adaptive tests for umbrella alternatives", Biometrical Journal, 40(5): 573-587, (1998).
  • Terpstra, J.T. and Magel, R.C., "A new nonparametric test for the ordered alternative problem", Journal of Nonparametric Statistics, 15(3): 289-301, (2003).
  • Chang, C.H. and Yen, C.H., "A Nonparametric Test for the Ordered Alternative Based on Fast Discrete Fourier Transform Coefficient", Journal of Testing and Evaluation, 39(6): 1131-1143, (2011).
  • Terpstra, J.T., Chang, C.H., and Magel, R.C., "On the use of Spearman's correlation coefficient for testing ordered alternatives", Journal of Statistical Computation and Simulation, 81(11): 1381-1392, (2011).
  • Chang, C.H., Chin, C.C., Yu, W.W., and Huang, Y.Y., "Using the Bernoulli trial approaches for detecting ordered alternatives", Bmc Medical Research Methodology, 13((2013).
  • Shan, G.G., Young, D., and Kang, L., "A New Powerful Nonparametric Rank Test for Ordered Alternative Problem", Plos One, 9(11): (2014).
  • Gaur, A., "A class of k-sample distribution-free tests for location against ordered alternatives", Communications in Statistics-Theory and Methods, 46(5): 2343-2353, (2017).
  • Mack, G.A. and Wolfe, D.A., "K-Sample Rank-Tests for Umbrella Alternatives", Journal of the American Statistical Association, 76(373): 175-181, (1981).
  • Kossler, W., "Some c-sample rank tests of homogeneity against umbrella alternatives with unknown peak", Journal of Statistical Computation and Simulation, 76(1): 57-74, (2006).
  • Gaur, A., Mahajan, K.K., and Arora, S., "New nonparametric tests for testing homogeneity of scale parameters against umbrella alternative", Statistics & Probability Letters, 82(9): 1681-1689, (2012).
  • Shi, N.Z., "Rank Test Statistics for Umbrella Alternatives", Communications in Statistics-Theory and Methods, 17(6): 2059-2073, (1988).
  • Cohen, A. and Sackrowitz, H.B., "Tests for the umbrella alternative under normality", Communications in Statistics-Theory and Methods, 25(11): 2807-2817, (1996).
  • Pan, G.H., "Distribution-free tests for umbrella alternatives", Communications in Statistics-Theory and Methods, 25(12): 3185-3194, (1996).
  • Hartlaub, B.A. and Wolfe, D.A., "Distribution-free ranked-set sample procedures for umbrella alternatives in the m-sample setting", Environmental and Ecological Statistics, 6(2): 105-118, (1999).
  • Hayter, A.J. and Liu, W., "A new test against an umbrella alternative and the associated simultaneous confidence intervals", Computational Statistics & Data Analysis, 30(4): 393-401, (1999).
  • Lee, W.C. and Chen, Y.I., "Weighted Kaplan-Meier tests for umbrella alternatives", Annals of the Institute of Statistical Mathematics, 53(4): 835-852, (2001).
  • Magel, R.C. and Qin, L., "A non-parametric test for umbrella alternatives based on ranked-set sampling", Journal of Applied Statistics, 30(8): 925-937, (2003).
  • Singh, P. and Liu, W., "A test against an umbrella ordered alternative", Computational Statistics & Data Analysis, 51(3): 1957-1964, (2006).
  • Alvo, M., "Nonparametric tests of hypotheses for umbrella alternatives", Canadian Journal of Statistics-Revue Canadienne De Statistique, 36(1): 143-156, (2008).
  • Bhat, S.V. and Patil, A.B., "A class of k-sample distribution-free tests for the umbrella alternatives", International Journal of Agricultural and Statistical Sciences, 4(1): 53-61, (2008).
  • Basso, D. and Salmaso, L., "A permutation test for umbrella alternatives", Statistics and Computing, 21(1): 45-54, (2011).
  • Altunkaynak, B. and Gamgam, H., "Comparing the performance of nonparametric tests for equality of location against ordered alternatives", Communications in Statistics - Simulation and Computation, 1-22, (2019).
  • Bradley, J.V., "Robustness?", British Journal of Mathematical and Statistical Psychology, 31(2): 144-152, (1978).
There are 33 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Statistics
Authors

Bulent Altunkaynak 0000-0002-7571-2155

Hamza Gamgam 0000-0002-9595-9315

Merve Bağçacı This is me 0000-0003-2367-1889

Publication Date September 1, 2020
Published in Issue Year 2020

Cite

APA Altunkaynak, B., Gamgam, H., & Bağçacı, M. (2020). A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives. Gazi University Journal of Science, 33(3), 815-829. https://doi.org/10.35378/gujs.566936
AMA Altunkaynak B, Gamgam H, Bağçacı M. A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives. Gazi University Journal of Science. September 2020;33(3):815-829. doi:10.35378/gujs.566936
Chicago Altunkaynak, Bulent, Hamza Gamgam, and Merve Bağçacı. “A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives”. Gazi University Journal of Science 33, no. 3 (September 2020): 815-29. https://doi.org/10.35378/gujs.566936.
EndNote Altunkaynak B, Gamgam H, Bağçacı M (September 1, 2020) A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives. Gazi University Journal of Science 33 3 815–829.
IEEE B. Altunkaynak, H. Gamgam, and M. Bağçacı, “A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives”, Gazi University Journal of Science, vol. 33, no. 3, pp. 815–829, 2020, doi: 10.35378/gujs.566936.
ISNAD Altunkaynak, Bulent et al. “A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives”. Gazi University Journal of Science 33/3 (September 2020), 815-829. https://doi.org/10.35378/gujs.566936.
JAMA Altunkaynak B, Gamgam H, Bağçacı M. A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives. Gazi University Journal of Science. 2020;33:815–829.
MLA Altunkaynak, Bulent et al. “A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives”. Gazi University Journal of Science, vol. 33, no. 3, 2020, pp. 815-29, doi:10.35378/gujs.566936.
Vancouver Altunkaynak B, Gamgam H, Bağçacı M. A New Nonparametric Test for Testing Equality of Locations Against Umbrella Alternatives. Gazi University Journal of Science. 2020;33(3):815-29.