A bivariate conditional Weibull distribution with application

Abstract

A three-parameter bivariate distribution is derived from the marginal and conditional Weibull distributions. Its joint properties are derived and method of estimation of its parameters discussed. It is generalised for the multivariate case when the assumptions of piecewise conditional independence of the variables is tenable. It is applied to a bivariate data set of wind speed and solar radiation, where the conditional dependence of the two variables involved is suspected. The result shows that the model is compatible with the data. It is proposed for application when there is need to allow for such conditional structure in a bivariate model.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Balakrishnan, N., Lai, C.D.: Continuous Bivariate Distributions, 2nd edn. Springer, New York (2009)

    Google Scholar 

  2. 2.

    Burrows, R., Salih, B.A.: Statistical modelling of long-term wave climate. In: Edge, I. (ed.) Twentieth Coastal Engineering Conference Proceedings, vol. 1 B, pp. 42–56. American Society of Civil Engineers, New York (1987)

    Google Scholar 

  3. 3.

    Constantinides, A.: Applied Numerical Methods and Applied Computers. McGraw-Hill, New York (1987)

    Google Scholar 

  4. 4.

    Dubey, S.D.: Compound gamma, beta and F distributions. Metrika 16, 27–31 (1970)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Genton, M.G.: Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Edited Volume, p. 416. Chapman & Hall/CRC, Boca Raton (2004)

    Google Scholar 

  6. 6.

    Gongsin, I.E., Saporu, F.W.O.: On the estimation of parameters in a Weibull wind model and its application to wind speed data from Maiduguri, Borno State Nigeria. Math. Model. Theory 6(7), 62–76 (2016)

    Google Scholar 

  7. 7.

    Gongsin, I.E., Saporu, F.W.O.: Solar energy potential in Yola, Adamawa State, Nigeria. Int. J. Renew. Energy Sources 4, 48–55 (2019). (ISSN: 2367-9123)

    Google Scholar 

  8. 8.

    Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 1, 2nd edn. Wiley, New York (1994)

    Google Scholar 

  9. 9.

    Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  10. 10.

    Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Multivariate Distributions, vol. 1, 2nd edn. Wiley, New York (2000)

    Google Scholar 

  11. 11.

    Kaminsky, F.C., Kirchoff, R.H.: Bivariate probability models for the description of average wind speeds at two heights. Sol. Energy 40, 49–56 (1988)

    Article  Google Scholar 

  12. 12.

    Krogstad, H.E.: Height and period predictions of extreme waves. Appl. Ocean Res. 7, 158–165 (1985)

    Article  Google Scholar 

  13. 13.

    Lee, C.K., Wen, M.J. A multivariate Weibull distribution. http://www.arxiv.org/pdf/math/0609585. Accessed 13 Feb 2019

  14. 14.

    Mihram, G.A., Hultquist, A.R.: A bivariate warning-time/failure-time distribution. J. Am. Stat. Assoc. 62, 589–599 (1967)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Myrhaug, D., Kjeldsen, S.F.: Parametric modelling of joint probability distributions for steepness and asymmetry in deep water waves. Appl. Ocean Res. 6, 207–220 (1984)

    Article  Google Scholar 

  16. 16.

    Ododo, J.C. (1994). New models for prediction of solar radiation in Nigeria. Paper Presented at the 2nd OAU/STRC Conference on New Renewable Energies at Bamako, Mali, May, 16–20

  17. 17.

    Proma, A.K., Pobitra, K.H., Sabbir, R.: Wind energy potential estimation for different regions of Bangladesh. Int. J. Renew. Sustain. Energy 3(3), 47–52 (2014)

    Article  Google Scholar 

  18. 18.

    Sarabia, M.J., Emilio, G.D.: Construction of multivariate distributions: a review of some recent results. SORT 32(1), 3–36 (2008)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Sens, S., Lamichhane, R., Diawara, N.: A bivariate distribution with conditional gamma and its multivariate form. J. Mod. Appl. Stat. Methods 13(2), 169–184 (2014). https://doi.org/10.22237/jmasm/1414814880. (article 9)

    Article  Google Scholar 

  20. 20.

    Vaidyanathan, V.S., Sharon, V.A.: Morgenstern type bivariate Lindley distribution. Stat. Opt. Inf. Comput. 4, 132–146 (2016). https://doi.org/10.19139/soic.v4i2.183

    MathSciNet  Article  Google Scholar 

  21. 21.

    Villanueva, D., Feijoo, A., Pazos, J.L.: Multivariate Weibull distribution for wind speed and wind power behavior assessment. Resources 2, 370–384 (2013). https://doi.org/10.3390/resources2030370

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to I. E. Gongsin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Tables 3 and 4.

Table 3 Daily solar radiation (MJ/m2) at Yola in 2015
Table 4 Daily wind speed (m/s) at Yola in 2015

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gongsin, I.E., Saporu, F.W.O. A bivariate conditional Weibull distribution with application. Afr. Mat. 31, 565–583 (2020). https://doi.org/10.1007/s13370-019-00742-8

Download citation

Keywords

  • Bivariate conditional Weibull distribution
  • Newton–Raphson method
  • Joint moments
  • Hessian matrix

Mathematics Subject Classification

  • 742