July  2022, 9(3): 283-297. doi: 10.3934/jdg.2022014

Stress-strength reliability with dependent variables based on copula function

Department of Statistics, Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Sabzevar, Iran

*Corresponding author: Mohammad Bolbolian Ghalibaf

Received  June 2021 Revised  April 2022 Published  July 2022 Early access  June 2022

The stress-strength model is a basic model in the field of reliability, but it still has some obvious limitations in many applications. Therefore, considering different aspects and using different methods to investigate the stress-strength model has been one of the main research directions of reliability. In this paper, we suppose the problem of evaluating reliability considering the stress and strength as dependent variables when the association is modeled by the copula function. By using Monte Carlo simulation, we estimate the reliability measure $ R $ for dependent margins by choosing various copulas and known marginal distributions to belong to a specific class of parametric models, here the Dagum family. Finally, the application of the copula-based approach in reliability modelling is illustrated using two medical data sets. The results of these two data sets show the effectiveness of this method in reliability modelling.

Citation: Mohammad Bolbolian Ghalibaf. Stress-strength reliability with dependent variables based on copula function. Journal of Dynamics and Games, 2022, 9 (3) : 283-297. doi: 10.3934/jdg.2022014
References:
[1]

B. Abdous, C. Genest and B. Rémillard, Dependence properties of meta-elliptical distributions, In Statistical Modeling and Analysis for Complex Data Problems, (2005), 1–15. doi: 10.1007/0-387-24555-3_1.

[2] N. Balakrishnan and C. D. Lai, Continuous Bivariate Distributions, 2$^{nd}$ edition, Springer, Dordrecht, 2009. 
[3]

T. BurzykowskiG. Molenberghs and M. Buyse, The validation of surrogate endpoints by using data from randomized clinical trials: A case study in advanced colorectal cancer, J. Roy. Statist. Soc. Ser. A, 167 (2004), 103-124.  doi: 10.1111/j.1467-985X.2004.00293.x.

[4]

U. Cherubini, E. Luciano and W. Vecchiato, Copula Methods in Finance, Wiley Finance Series. John Wiley & Sons, Ltd., Chichester, 2004. doi: 10.1002/9781118673331.

[5]

D. G. Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika, 65 (1978), 141-151.  doi: 10.1093/biomet/65.1.141.

[6]

D. A. Conway, Plackett family of distributions, In Encyclopedia of Statistical Sciences, Kotz, S. and Johnson, N. L. (eds), 7 (1986), 1–5.

[7]

R. D. Cook and M. E. Johnson, A family of distributions for modeling non-elliptically symmetric multivariate data, J. Roy. Statist. Soc. Ser. B, 43 (1981), 210-218. 

[8]

C. Dagum, A model of income distribution and the conditions of existence of moments of finite order, Bulletin of the International Statistical Institute, 46 (1975), 199-205. 

[9]

C. Dagum, A new model of personal income distribution: Specification and estimation, Economie Appliquée, 30 (1977), 413-437. 

[10]

J. Dobrić and F. Schmid, Testing goodness of fit for parametric families of copulas: Application to financial data, Comm. Statist. Simulation Comput., 34 (2005), 1053-1068.  doi: 10.1080/03610910500308685.

[11]

F. Domma and S. Giordano, A stress-strength model with dependent variables to measure household financial fragility, Stat Methods Appl., 21 (2012), 375-389.  doi: 10.1007/s10260-012-0192-5.

[12]

F. Domma and S. Giordano, A copula-based approach to account for dependence in stress-strength models, Stat. Papers, 54 (2013), 807-826.  doi: 10.1007/s00362-012-0463-0.

[13]

F. DommaS. Giordano and M. Zenga, Maximum likelihood estimation in Dagum distribution with censored samples, J. Appl. Stat., 38 (2011), 2971-2985.  doi: 10.1080/02664763.2011.578613.

[14]

F. Durante and C. Sempi, Copula theory: An introduction, Copula Theory and Its Applications. Lecture Notes in Statistics-Proceedings, 198 (2010), 3-31.  doi: 10.1007/978-3-642-12465-5_1.

[15]

H. B. FangK. T. Fang and S. Kotz, The meta-elliptical distributions with given marginals, J. Multivariate Anal., 82 (2002), 1-16.  doi: 10.1006/jmva.2001.2017.

[16]

H. B. FangK. T. Fang and S. Kotz, Corrigendum to: "The meta-elliptical distributions with given marginals" [J. Multivariate Anal. 82: 1-16 (2002)], J. Multivariate Anal., 94 (2005), 222-223.  doi: 10.1016/j.jmva.2004.10.001.

[17]

D. Faraggi and E. L. Korn, Competing risks with frailty models when treatment affects only one failure type, Biometrika, 83 (1996), 467-471.  doi: 10.1093/biomet/83.2.467.

[18]

G. FrahmM. Junker and A. Szimayer, Elliptical copulas: Applicability and limitations, Statist. Probab. Lett., 63 (2003), 275-286.  doi: 10.1016/S0167-7152(03)00092-0.

[19]

M. J. Frank, On the simultaneous associativity of $F(x, y)$ and $x+y-F(x, y)$, Aequationes Math., 19 (1979), 194-226.  doi: 10.1007/BF02189866.

[20]

E. W. Frees and E. A. Valdez, Understanding relationships using copulas, N. Am. Actuar. J., 2 (1998), 1-25.  doi: 10.1080/10920277.1998.10595667.

[21]

C. Genest, Frank's family of bivariate distributions, Biometrika, 74 (1987), 549-555.  doi: 10.1093/biomet/74.3.549.

[22]

C. GenestB. Rémillard and D. Beaudoin, Goodness-of-fit tests for copulas: A review and a power study, Insurance Math. Econom., 44 (2009), 199-213.  doi: 10.1016/j.insmatheco.2007.10.005.

[23]

É. J. Gumbel, Distributions des valeurs extrêmes en plusieurs dimensions, Publ. Inst. Statist. Univ. Paris, 9 (1960), 171-173. 

[24]

R. C. GuptaM. E. Ghitany and D. K. Al-Mutairi, Estimation of reliability from a bivariate log-normal data, J. Stat. Comput. Simul., 83 (2013), 1068-1081.  doi: 10.1080/00949655.2011.649284.

[25]

R. C. Gupta and S. Subramanian, Estimation of reliability in a bivariate normal distribution with equal coefficients of variation, Commun. Stat. Simul. Comput., 27 (1998), 675-698. 

[26]

D. D. Hanagal, Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers, 38 (1997), 453-459.  doi: 10.1007/BF02926000.

[27]

P. Hougaard, A class of multivanate failure time distributions, Biometrika, 73 (1986), 671-678.  doi: 10.2307/2336531.

[28]

T. P. Hutchinson and C. D. Lai, The Engineering Statistician's Guide to Continuous Bivariate Distributions, Rumsby Scientific Publishing, Adelaide, 1991.

[29]

E. S. Jeevanand, Bayes estimation of $P(X_1 < X_2)$ for a bivariate Pareto distribution, Statistician, 46 (1997), 93-99. 

[30]

H. Joe, Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability, 73. Chapman & Hall, London, 1997. doi: 10.1201/b13150.

[31]

R. E. Keith and E. Merrill, The effects of vitamin C on maximum grip strength and muscular endurance, J. Sports Medicine and Physical Fitness, 23 (1983), 253-256. 

[32]

C. Kleiber, A guide to the Dagum distributions, In Modeling Income Distributions and Lorenz Curves, Springer, New York, (2008), 97–117.

[33]

C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2003. doi: 10.1002/0471457175.

[34]

S. Klugman and R. Parsa, Fitting bivariate loss distributions with copulas, Insurance Math. Econom., 24 (1999), 139-148.  doi: 10.1016/S0167-6687(98)00039-0.

[35]

S. Kotz, Y. Lumelskii and M. Pensky, The Stress-Strength Model and its Generalizations, Theory and applications. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564511.

[36]

E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153.  doi: 10.1214/aoms/1177699260.

[37]

Y. Malevergne and D. Sornette, Testing the Gaussian copula hypothesis for financial assets dependences, Quant. Finance, 3 (2003), 231-250.  doi: 10.1088/1469-7688/3/4/301.

[38] A. J. McNeilR. Frey and P. Embrechts, Quantitative Risk Management, Princeton University Press, Princeton, NJ, 2005. 
[39]

S. Nadarajah, Reliability for some bivariate beta distributions, Math. Problems Eng., 2005 (2005), 101-111.  doi: 10.1155/MPE.2005.101.

[40]

S. Nadarajah, Reliability for some bivariate gamma distributions, Math. Probl. Eng., 2005 (2005), 151-163.  doi: 10.1155/MPE.2005.151.

[41]

S. Nadarajah and S. Kotz, Reliability for some bivariate exponential distributions, Math. Probl. Eng., 2006 (2006), Art. ID 41652, 14 pp. doi: 10.1155/MPE/2006/41652.

[42]

R. B. Nelsen, Properties of a one-parameter family of bivariate distributions with specified marginals, Comm. Statist. A—Theory Methods, 15 (1986), 3277-3285.  doi: 10.1080/03610928608829309.

[43] R. B. Nelsen, An Introduction to Copulas, 2$^{nd}$ edition, Springer, New York, 2006. 
[44]

D. Oakes, Bivariate survival models induced by frailties, J. Amer. Statist. Assoc., 84 (1989), 487-493. 

[45]

D. Oakes, A model for association in bivariate survival data, J. Roy. Statist. Soc. Ser. B, 44 (1982), 414-422. 

[46]

R. L. Plackett, A class of bivariate distributions, J. Amer. Statist. Assoc., 60 (1965), 516-522. 

[47]

R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2021. Available from: https://www.R-project.org/.

[48] B. F. RyanB. L. Joiner and T. A. Ryan Jr, Minitab Handbook, 2$^{nd}$ edition, Duxbury Press, Boston, 1985. 
[49]

A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8 (1959), 229-231. 

[50]

W. Wang and M. T. Wells, Model selection and semiparametric inference for bivariate failure-time data, J. Amer. Statist. Assoc., 95 (2000), 62-72.  doi: 10.2307/2669523.

show all references

References:
[1]

B. Abdous, C. Genest and B. Rémillard, Dependence properties of meta-elliptical distributions, In Statistical Modeling and Analysis for Complex Data Problems, (2005), 1–15. doi: 10.1007/0-387-24555-3_1.

[2] N. Balakrishnan and C. D. Lai, Continuous Bivariate Distributions, 2$^{nd}$ edition, Springer, Dordrecht, 2009. 
[3]

T. BurzykowskiG. Molenberghs and M. Buyse, The validation of surrogate endpoints by using data from randomized clinical trials: A case study in advanced colorectal cancer, J. Roy. Statist. Soc. Ser. A, 167 (2004), 103-124.  doi: 10.1111/j.1467-985X.2004.00293.x.

[4]

U. Cherubini, E. Luciano and W. Vecchiato, Copula Methods in Finance, Wiley Finance Series. John Wiley & Sons, Ltd., Chichester, 2004. doi: 10.1002/9781118673331.

[5]

D. G. Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika, 65 (1978), 141-151.  doi: 10.1093/biomet/65.1.141.

[6]

D. A. Conway, Plackett family of distributions, In Encyclopedia of Statistical Sciences, Kotz, S. and Johnson, N. L. (eds), 7 (1986), 1–5.

[7]

R. D. Cook and M. E. Johnson, A family of distributions for modeling non-elliptically symmetric multivariate data, J. Roy. Statist. Soc. Ser. B, 43 (1981), 210-218. 

[8]

C. Dagum, A model of income distribution and the conditions of existence of moments of finite order, Bulletin of the International Statistical Institute, 46 (1975), 199-205. 

[9]

C. Dagum, A new model of personal income distribution: Specification and estimation, Economie Appliquée, 30 (1977), 413-437. 

[10]

J. Dobrić and F. Schmid, Testing goodness of fit for parametric families of copulas: Application to financial data, Comm. Statist. Simulation Comput., 34 (2005), 1053-1068.  doi: 10.1080/03610910500308685.

[11]

F. Domma and S. Giordano, A stress-strength model with dependent variables to measure household financial fragility, Stat Methods Appl., 21 (2012), 375-389.  doi: 10.1007/s10260-012-0192-5.

[12]

F. Domma and S. Giordano, A copula-based approach to account for dependence in stress-strength models, Stat. Papers, 54 (2013), 807-826.  doi: 10.1007/s00362-012-0463-0.

[13]

F. DommaS. Giordano and M. Zenga, Maximum likelihood estimation in Dagum distribution with censored samples, J. Appl. Stat., 38 (2011), 2971-2985.  doi: 10.1080/02664763.2011.578613.

[14]

F. Durante and C. Sempi, Copula theory: An introduction, Copula Theory and Its Applications. Lecture Notes in Statistics-Proceedings, 198 (2010), 3-31.  doi: 10.1007/978-3-642-12465-5_1.

[15]

H. B. FangK. T. Fang and S. Kotz, The meta-elliptical distributions with given marginals, J. Multivariate Anal., 82 (2002), 1-16.  doi: 10.1006/jmva.2001.2017.

[16]

H. B. FangK. T. Fang and S. Kotz, Corrigendum to: "The meta-elliptical distributions with given marginals" [J. Multivariate Anal. 82: 1-16 (2002)], J. Multivariate Anal., 94 (2005), 222-223.  doi: 10.1016/j.jmva.2004.10.001.

[17]

D. Faraggi and E. L. Korn, Competing risks with frailty models when treatment affects only one failure type, Biometrika, 83 (1996), 467-471.  doi: 10.1093/biomet/83.2.467.

[18]

G. FrahmM. Junker and A. Szimayer, Elliptical copulas: Applicability and limitations, Statist. Probab. Lett., 63 (2003), 275-286.  doi: 10.1016/S0167-7152(03)00092-0.

[19]

M. J. Frank, On the simultaneous associativity of $F(x, y)$ and $x+y-F(x, y)$, Aequationes Math., 19 (1979), 194-226.  doi: 10.1007/BF02189866.

[20]

E. W. Frees and E. A. Valdez, Understanding relationships using copulas, N. Am. Actuar. J., 2 (1998), 1-25.  doi: 10.1080/10920277.1998.10595667.

[21]

C. Genest, Frank's family of bivariate distributions, Biometrika, 74 (1987), 549-555.  doi: 10.1093/biomet/74.3.549.

[22]

C. GenestB. Rémillard and D. Beaudoin, Goodness-of-fit tests for copulas: A review and a power study, Insurance Math. Econom., 44 (2009), 199-213.  doi: 10.1016/j.insmatheco.2007.10.005.

[23]

É. J. Gumbel, Distributions des valeurs extrêmes en plusieurs dimensions, Publ. Inst. Statist. Univ. Paris, 9 (1960), 171-173. 

[24]

R. C. GuptaM. E. Ghitany and D. K. Al-Mutairi, Estimation of reliability from a bivariate log-normal data, J. Stat. Comput. Simul., 83 (2013), 1068-1081.  doi: 10.1080/00949655.2011.649284.

[25]

R. C. Gupta and S. Subramanian, Estimation of reliability in a bivariate normal distribution with equal coefficients of variation, Commun. Stat. Simul. Comput., 27 (1998), 675-698. 

[26]

D. D. Hanagal, Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers, 38 (1997), 453-459.  doi: 10.1007/BF02926000.

[27]

P. Hougaard, A class of multivanate failure time distributions, Biometrika, 73 (1986), 671-678.  doi: 10.2307/2336531.

[28]

T. P. Hutchinson and C. D. Lai, The Engineering Statistician's Guide to Continuous Bivariate Distributions, Rumsby Scientific Publishing, Adelaide, 1991.

[29]

E. S. Jeevanand, Bayes estimation of $P(X_1 < X_2)$ for a bivariate Pareto distribution, Statistician, 46 (1997), 93-99. 

[30]

H. Joe, Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability, 73. Chapman & Hall, London, 1997. doi: 10.1201/b13150.

[31]

R. E. Keith and E. Merrill, The effects of vitamin C on maximum grip strength and muscular endurance, J. Sports Medicine and Physical Fitness, 23 (1983), 253-256. 

[32]

C. Kleiber, A guide to the Dagum distributions, In Modeling Income Distributions and Lorenz Curves, Springer, New York, (2008), 97–117.

[33]

C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2003. doi: 10.1002/0471457175.

[34]

S. Klugman and R. Parsa, Fitting bivariate loss distributions with copulas, Insurance Math. Econom., 24 (1999), 139-148.  doi: 10.1016/S0167-6687(98)00039-0.

[35]

S. Kotz, Y. Lumelskii and M. Pensky, The Stress-Strength Model and its Generalizations, Theory and applications. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564511.

[36]

E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153.  doi: 10.1214/aoms/1177699260.

[37]

Y. Malevergne and D. Sornette, Testing the Gaussian copula hypothesis for financial assets dependences, Quant. Finance, 3 (2003), 231-250.  doi: 10.1088/1469-7688/3/4/301.

[38] A. J. McNeilR. Frey and P. Embrechts, Quantitative Risk Management, Princeton University Press, Princeton, NJ, 2005. 
[39]

S. Nadarajah, Reliability for some bivariate beta distributions, Math. Problems Eng., 2005 (2005), 101-111.  doi: 10.1155/MPE.2005.101.

[40]

S. Nadarajah, Reliability for some bivariate gamma distributions, Math. Probl. Eng., 2005 (2005), 151-163.  doi: 10.1155/MPE.2005.151.

[41]

S. Nadarajah and S. Kotz, Reliability for some bivariate exponential distributions, Math. Probl. Eng., 2006 (2006), Art. ID 41652, 14 pp. doi: 10.1155/MPE/2006/41652.

[42]

R. B. Nelsen, Properties of a one-parameter family of bivariate distributions with specified marginals, Comm. Statist. A—Theory Methods, 15 (1986), 3277-3285.  doi: 10.1080/03610928608829309.

[43] R. B. Nelsen, An Introduction to Copulas, 2$^{nd}$ edition, Springer, New York, 2006. 
[44]

D. Oakes, Bivariate survival models induced by frailties, J. Amer. Statist. Assoc., 84 (1989), 487-493. 

[45]

D. Oakes, A model for association in bivariate survival data, J. Roy. Statist. Soc. Ser. B, 44 (1982), 414-422. 

[46]

R. L. Plackett, A class of bivariate distributions, J. Amer. Statist. Assoc., 60 (1965), 516-522. 

[47]

R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2021. Available from: https://www.R-project.org/.

[48] B. F. RyanB. L. Joiner and T. A. Ryan Jr, Minitab Handbook, 2$^{nd}$ edition, Duxbury Press, Boston, 1985. 
[49]

A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8 (1959), 229-231. 

[50]

W. Wang and M. T. Wells, Model selection and semiparametric inference for bivariate failure-time data, J. Amer. Statist. Assoc., 95 (2000), 62-72.  doi: 10.2307/2669523.

Figure 1.  Reliability measure $ R $ for Dagum margins with Gaussian copula
Figure 2.  Reliability measure $ R $ for Dagum margins with Student copula
Figure 3.  Reliability measure $ R $ for Dagum margins with Clayton copula
Figure 4.  Reliability measure $ R $ for Dagum margins with Frank copula
Figure 5.  Reliability measure $ R $ for Dagum margins with Gumbel copula
Figure 6.  Reliability measure $ R $ for Dagum margins with Plackett copula
Table 1.  Two data sets
Data set 1 Data set 2
Patient Day2 Day4 Subject Vitamin Placebo
1 270 218 1 145 417
2 236 234 2 185 279
3 210 214 3 387 678
4 142 116 4 593 636
5 280 200 5 248 170
6 272 276 6 245 699
7 160 146 7 349 372
8 220 182 8 902 582
9 226 238 9 159 363
10 242 288 10 122 258
11 186 190 11 264 288
12 266 236 12 1052 526
13 206 244 13 218 180
14 318 258 14 117 172
15 294 240 15 185 278
16 282 294
17 234 220
18 224 200
19 276 220
20 282 186
21 360 352
22 310 202
23 280 218
24 278 248
25 288 278
26 288 248
27 244 270
28 236 242
Data set 1 Data set 2
Patient Day2 Day4 Subject Vitamin Placebo
1 270 218 1 145 417
2 236 234 2 185 279
3 210 214 3 387 678
4 142 116 4 593 636
5 280 200 5 248 170
6 272 276 6 245 699
7 160 146 7 349 372
8 220 182 8 902 582
9 226 238 9 159 363
10 242 288 10 122 258
11 186 190 11 264 288
12 266 236 12 1052 526
13 206 244 13 218 180
14 318 258 14 117 172
15 294 240 15 185 278
16 282 294
17 234 220
18 224 200
19 276 220
20 282 186
21 360 352
22 310 202
23 280 218
24 278 248
25 288 278
26 288 248
27 244 270
28 236 242
Table 2.  Copula goodness of fit test for two data sets
Data set 1 Data set 2
Copula under $ H_0 $ Test p-value Parameters p-value Parameters
Estimation Estimation
Normal $ S_n^{(B)} $ 0.812 0.6882564 0.936 0.6180132
$ S_n^{(C)} $ 0.697 0.607
Student $ S_n^{(B)} $ 0.860 0.4343909 0.902 0.6179961
$ S_n^{(C)} $ 0.606 $ df=1.253 $ 0.622 df=10771
Clayton $ S_n^{(B)} $ 0.600 1.634037 0.829 1.146932
$ S_n^{(C)} $ 0.732 -
Frank $ S_n^{(B)} $ 0.869 4.166703 0.969 4.395282
$ S_n^{(C)} $ 0.798 0.895
Gumbel $ S_n^{(B)} $ 0.914 1.731146 0.773 1.530552
$ S_n^{(C)} $ 0.780 0.621
Plackett $ S_n $ 0.552 6.4489 0.755 6.3581
Data set 1 Data set 2
Copula under $ H_0 $ Test p-value Parameters p-value Parameters
Estimation Estimation
Normal $ S_n^{(B)} $ 0.812 0.6882564 0.936 0.6180132
$ S_n^{(C)} $ 0.697 0.607
Student $ S_n^{(B)} $ 0.860 0.4343909 0.902 0.6179961
$ S_n^{(C)} $ 0.606 $ df=1.253 $ 0.622 df=10771
Clayton $ S_n^{(B)} $ 0.600 1.634037 0.829 1.146932
$ S_n^{(C)} $ 0.732 -
Frank $ S_n^{(B)} $ 0.869 4.166703 0.969 4.395282
$ S_n^{(C)} $ 0.798 0.895
Gumbel $ S_n^{(B)} $ 0.914 1.731146 0.773 1.530552
$ S_n^{(C)} $ 0.780 0.621
Plackett $ S_n $ 0.552 6.4489 0.755 6.3581
Table 3.  Estimation of reliability measure R based on various copulas for two data sets
Reliability Measure $ R $
Copula Function Data set 1 Data set 2
Normal 0.7298 0.3063
Student 0.7422 0.3061
Clayton 0.7349 0.2973
Frank 0.7087 0.3002
Gumbel 0.7285 0.3248
Plackett 0.7218 0.2971
Reliability Measure $ R $
Copula Function Data set 1 Data set 2
Normal 0.7298 0.3063
Student 0.7422 0.3061
Clayton 0.7349 0.2973
Frank 0.7087 0.3002
Gumbel 0.7285 0.3248
Plackett 0.7218 0.2971
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