Research Article
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Year 2023, Volume: 52 Issue: 1, 209 - 228, 15.02.2023
https://doi.org/10.15672/hujms.1031959

Abstract

References

  • [1] M.V. Alba-Fernández, A. Batsidis, M.D. Jiménez-Gamero and P. Jodrá, A class of tests for the two-sample problem for count data, J. Comput. Appl. Math. 318, 220- 229, 2017.
  • [2] I.G. Bairamov and Y.I. Petunin, Statistical tests based on training samples, Cybernet. Systems Anal. 27 (3), 408-413, 1991.
  • [3] H.M. Barakat, On moments of bivariate order statistics, Ann. Inst. Statist. Math. 51 (2), 351-358, 1999.
  • [4] I. Bayramoglu and S. Eryilmaz, Order statistics of dependent sequences consisting of two different sets of exchangeable variables, J. Comput. Appl. Math. 286, 1-6, 2015.
  • [5] I. Bayramoglu and G. Kemalbay, Some novel discrete distributions under fourfold sampling schemes and conditional bivariate order statistics, J. Comput. Appl. Math. 248, 1-14, 2013.
  • [6] A. Bücher, H. Dette and S. Volgushev, A test for archimedeanity in bivariate copula models, J. Multivariate Anal. 110, 121–132, 2012.
  • [7] A.R. da Rocha Neto, R. Sousa, G.D.A. Barreto and J.S. Cardoso, Diagnostic of pathology on the vertebral column with embedded reject option, in: Iberian Conference on Pattern Recognition and Image Analysis, 588-595, Springer, Berlin, Heidelberg, 2011.
  • [8] A. Erem, Bivariate two sample test based on exceedance statistics, Comm. Statist. Simulation Comput. 49 (9), 2020.
  • [9] A. Erem and I. Bayramoglu, Exact and asymptotic distributions of exceedance statistics for bivariate random sequences, Statist. Probab. Lett. 125, 181-188, 2017.
  • [10] S. Eryılmaz and I.G. Bairamov, On a new sample rank of an order statistics and its concomitant, Statist. Probab. Lett. 63 (2), 123-131, 2003.
  • [11] V.A. Fernández, M.J. Gamero, and J.M. Garcia, A test for the two-sample problem based on empirical characteristic functions, Comput. Statist. Data Anal. 52 (7), 3730- 3748, 2008.
  • [12] G. Fumera, F. Roli and G. Giacinto, Reject option with multiple thresholds, Pattern Recognit. 33 (12), 2099-2101, 2000.
  • [13] R. Herbei and M.H. Wegkamp, Classification with reject option, Canad. J. Statist. 34 (4), 709–721, 2006.
  • [14] W. Homenda, A. Jastrzebska and W. Pedrycz, Unsupervised mode of rejection of foreign patterns, Appl. Soft Comput. 57, 615-626, 2017.
  • [15] M.D. Jiménez-Gamero, M.V. Alba-Fernández, P. Jodrá and I. Barranco-Chamorro, Fast tests for the two-sample problem based on the empirical characteristic function, Math. Comput. Simulation 137, 390-410, 2017.
  • [16] G. Kemalbay and I. Bayramoglu, Joint distribution of new sample rank of bivariate order statistics, J. Appl. Stat. 42 (10), 2280-2289, 2015.
  • [17] D. Lin, L. Sun, K.A. Toh, J.B. Zhang and Z. Lin, Biomedical image classification based on a cascade of an SVM with a reject option and subspace analysis, Comput. Biol. Med. 96, 128-140, 2018.
  • [18] D.P. Mesquita, L.S. Rocha, J.P.P. Gomes and A.R.R. Neto, Classification with reject option for software defect prediction, Appl. Soft Comput. 49, 1085-1093, 2016.
  • [19] R.A. Mohammadpour, S.M. Abedi, S. Bagheri and A. Ghaemian, Fuzzy rule-based classification system for assessing coronary artery disease, Comput. Math. Methods Med., Doi: 10.1155/2015/564867, 2015.
  • [20] T. Nishino and H. Murakami, The generalized Cucconi test statistic for the two-sample problem, J. Korean Statist. Soc. 48 (4), 593-612, 2019.
  • [21] I. Pillai, G. Fumera and F. Roli, Multi-label classification with a reject option, Pattern Recognit. 46 (8), 2256-2266, 2013.
  • [22] B. Rémillard and O. Scaillet, Testing for equality between two copulas, J. Multivariate Anal. 100 (3), 377-386, 2009.
  • [23] E. Stoimenova, The power of exceedance-type tests under lehmann alternatives, Comm. Statist. Theory Methods 40 (4), 731-744, 2011.
  • [24] E. Stoimenova and N. Balakrishnan, A class of exceedance-type statistics for the twosample problem, J. Statist. Plann. Inference 141 (9), 3244-3255, 2011.
  • [25] E. Stoimenova and N. Balakrishnan, Sidak-type tests for the two-sample problem based on precedence and exceedance statistics, Statistics 51 (2), 247-264, 2017.
  • [26] S.O. Susam and B. Hudaverdi Ucer, Testing independence for Archimedean copula based on Bernstein estimate of Kendall distribution function, J. Stat. Comput. Simul. 88 (13), 2589-2599, 2018.
  • [27] D.M. Tax and R.P. Duin, Growing a multi-class classifier with a reject option, Pattern Recognit. Lett. 29 (10), 1565-1570, 2008.

A consistent statistical test based on bivariate random samples

Year 2023, Volume: 52 Issue: 1, 209 - 228, 15.02.2023
https://doi.org/10.15672/hujms.1031959

Abstract

We propose a consistent test for testing the distribution of bivariate random samples. The probability of type I, type II errors and probability of making no decisions under null and alternative hypotheses are obtained based on copula functions. The consistency of the proposed test is discussed under some null and alternative hypotheses. An unbiased, consistent estimator is proposed for probability of making no decision. Moreover, a simulation study is performed for showing the consistency of the proposed test for some well-known copulas such as independent, Clayton, Gumbel, Frank and Farlie-Gumbel-Morgenstern.

References

  • [1] M.V. Alba-Fernández, A. Batsidis, M.D. Jiménez-Gamero and P. Jodrá, A class of tests for the two-sample problem for count data, J. Comput. Appl. Math. 318, 220- 229, 2017.
  • [2] I.G. Bairamov and Y.I. Petunin, Statistical tests based on training samples, Cybernet. Systems Anal. 27 (3), 408-413, 1991.
  • [3] H.M. Barakat, On moments of bivariate order statistics, Ann. Inst. Statist. Math. 51 (2), 351-358, 1999.
  • [4] I. Bayramoglu and S. Eryilmaz, Order statistics of dependent sequences consisting of two different sets of exchangeable variables, J. Comput. Appl. Math. 286, 1-6, 2015.
  • [5] I. Bayramoglu and G. Kemalbay, Some novel discrete distributions under fourfold sampling schemes and conditional bivariate order statistics, J. Comput. Appl. Math. 248, 1-14, 2013.
  • [6] A. Bücher, H. Dette and S. Volgushev, A test for archimedeanity in bivariate copula models, J. Multivariate Anal. 110, 121–132, 2012.
  • [7] A.R. da Rocha Neto, R. Sousa, G.D.A. Barreto and J.S. Cardoso, Diagnostic of pathology on the vertebral column with embedded reject option, in: Iberian Conference on Pattern Recognition and Image Analysis, 588-595, Springer, Berlin, Heidelberg, 2011.
  • [8] A. Erem, Bivariate two sample test based on exceedance statistics, Comm. Statist. Simulation Comput. 49 (9), 2020.
  • [9] A. Erem and I. Bayramoglu, Exact and asymptotic distributions of exceedance statistics for bivariate random sequences, Statist. Probab. Lett. 125, 181-188, 2017.
  • [10] S. Eryılmaz and I.G. Bairamov, On a new sample rank of an order statistics and its concomitant, Statist. Probab. Lett. 63 (2), 123-131, 2003.
  • [11] V.A. Fernández, M.J. Gamero, and J.M. Garcia, A test for the two-sample problem based on empirical characteristic functions, Comput. Statist. Data Anal. 52 (7), 3730- 3748, 2008.
  • [12] G. Fumera, F. Roli and G. Giacinto, Reject option with multiple thresholds, Pattern Recognit. 33 (12), 2099-2101, 2000.
  • [13] R. Herbei and M.H. Wegkamp, Classification with reject option, Canad. J. Statist. 34 (4), 709–721, 2006.
  • [14] W. Homenda, A. Jastrzebska and W. Pedrycz, Unsupervised mode of rejection of foreign patterns, Appl. Soft Comput. 57, 615-626, 2017.
  • [15] M.D. Jiménez-Gamero, M.V. Alba-Fernández, P. Jodrá and I. Barranco-Chamorro, Fast tests for the two-sample problem based on the empirical characteristic function, Math. Comput. Simulation 137, 390-410, 2017.
  • [16] G. Kemalbay and I. Bayramoglu, Joint distribution of new sample rank of bivariate order statistics, J. Appl. Stat. 42 (10), 2280-2289, 2015.
  • [17] D. Lin, L. Sun, K.A. Toh, J.B. Zhang and Z. Lin, Biomedical image classification based on a cascade of an SVM with a reject option and subspace analysis, Comput. Biol. Med. 96, 128-140, 2018.
  • [18] D.P. Mesquita, L.S. Rocha, J.P.P. Gomes and A.R.R. Neto, Classification with reject option for software defect prediction, Appl. Soft Comput. 49, 1085-1093, 2016.
  • [19] R.A. Mohammadpour, S.M. Abedi, S. Bagheri and A. Ghaemian, Fuzzy rule-based classification system for assessing coronary artery disease, Comput. Math. Methods Med., Doi: 10.1155/2015/564867, 2015.
  • [20] T. Nishino and H. Murakami, The generalized Cucconi test statistic for the two-sample problem, J. Korean Statist. Soc. 48 (4), 593-612, 2019.
  • [21] I. Pillai, G. Fumera and F. Roli, Multi-label classification with a reject option, Pattern Recognit. 46 (8), 2256-2266, 2013.
  • [22] B. Rémillard and O. Scaillet, Testing for equality between two copulas, J. Multivariate Anal. 100 (3), 377-386, 2009.
  • [23] E. Stoimenova, The power of exceedance-type tests under lehmann alternatives, Comm. Statist. Theory Methods 40 (4), 731-744, 2011.
  • [24] E. Stoimenova and N. Balakrishnan, A class of exceedance-type statistics for the twosample problem, J. Statist. Plann. Inference 141 (9), 3244-3255, 2011.
  • [25] E. Stoimenova and N. Balakrishnan, Sidak-type tests for the two-sample problem based on precedence and exceedance statistics, Statistics 51 (2), 247-264, 2017.
  • [26] S.O. Susam and B. Hudaverdi Ucer, Testing independence for Archimedean copula based on Bernstein estimate of Kendall distribution function, J. Stat. Comput. Simul. 88 (13), 2589-2599, 2018.
  • [27] D.M. Tax and R.P. Duin, Growing a multi-class classifier with a reject option, Pattern Recognit. Lett. 29 (10), 1565-1570, 2008.
There are 27 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Ayşegül Erem 0000-0002-7713-5005

Ismihan Bayramoglu This is me 0000-0002-8575-8405

Publication Date February 15, 2023
Published in Issue Year 2023 Volume: 52 Issue: 1

Cite

APA Erem, A., & Bayramoglu, I. (2023). A consistent statistical test based on bivariate random samples. Hacettepe Journal of Mathematics and Statistics, 52(1), 209-228. https://doi.org/10.15672/hujms.1031959
AMA Erem A, Bayramoglu I. A consistent statistical test based on bivariate random samples. Hacettepe Journal of Mathematics and Statistics. February 2023;52(1):209-228. doi:10.15672/hujms.1031959
Chicago Erem, Ayşegül, and Ismihan Bayramoglu. “A Consistent Statistical Test Based on Bivariate Random Samples”. Hacettepe Journal of Mathematics and Statistics 52, no. 1 (February 2023): 209-28. https://doi.org/10.15672/hujms.1031959.
EndNote Erem A, Bayramoglu I (February 1, 2023) A consistent statistical test based on bivariate random samples. Hacettepe Journal of Mathematics and Statistics 52 1 209–228.
IEEE A. Erem and I. Bayramoglu, “A consistent statistical test based on bivariate random samples”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 1, pp. 209–228, 2023, doi: 10.15672/hujms.1031959.
ISNAD Erem, Ayşegül - Bayramoglu, Ismihan. “A Consistent Statistical Test Based on Bivariate Random Samples”. Hacettepe Journal of Mathematics and Statistics 52/1 (February 2023), 209-228. https://doi.org/10.15672/hujms.1031959.
JAMA Erem A, Bayramoglu I. A consistent statistical test based on bivariate random samples. Hacettepe Journal of Mathematics and Statistics. 2023;52:209–228.
MLA Erem, Ayşegül and Ismihan Bayramoglu. “A Consistent Statistical Test Based on Bivariate Random Samples”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 1, 2023, pp. 209-28, doi:10.15672/hujms.1031959.
Vancouver Erem A, Bayramoglu I. A consistent statistical test based on bivariate random samples. Hacettepe Journal of Mathematics and Statistics. 2023;52(1):209-28.