A proposal for modeling and simulating correlated discrete Weibull variables

Document Type : Article

Author

Department of Economics, Management and Quantitative Methods, 4 Universit`a degli Studi di Milano, via Conservatorio 7, 20122 Milan, Italy

Abstract

Researchers in applied sciences are often concerned with multivariate random vari9
ables. In particular, multivariate discrete data often arise in many fields (statistical
10 quality control, biostatistics, failure and reliability analysis, etc.) and modeling such
11 data is a relevant task, as well as simulating correlated discrete data satisfying some spe12
cific constraints. Here we consider the discrete Weibull distribution as an alternative to
13 the popular Poisson random variable and propose a procedure for simulating correlated
14 discrete Weibull random variables, with marginal distributions and correlation matrix as15
signed by the user. The procedure indeed relies upon the Gaussian copula model and an
16 iterative algorithm for recovering the proper correlation matrix for the copula ensuring
17 the desired correlation matrix on the discrete margins. A simulation study is presented,
18 which empirically assesses the performance of the procedure in terms of accuracy and
19 computational burden, also in relation to the necessary (but temporary) truncation of
20 the support of the discrete Weibull random variable. Inferential issues for the proposed
21 model are also discussed and are eventually applied to a dataset taken from the literature,
22 which shows that the proposed multivariate model can satisfactorily fit real-life correlated
23 counts even better than the most popular or recent existing ones.

Keywords

Main Subjects


References

1. Nakagawa, T. and Osaki, S. The discrete Weibull distribution", IEEE Transactions on Reliability, 24(5), pp. 300-301 (1975).
2. Stein, W.E. and Dattero, R. A new discrete Weibull distribution", IEEE Transactions on Reliability, 33(2),
pp. 196-197 (1984).
3. Padgett, W.J. and Spurrier, J.D. Discrete failure
models", IEEE Transactions on Reliability, 34(3), pp.
253-256 (1985).
4. Englehardt, J.D. and Li, R.C. The discrete Weibull
distribution: An alternative for correlated counts with
con rmation for microbial counts in water", Risk
Analysis, 31(3), pp. 370-381 (2011).
5. Khan, M.S.A., Khalique, A. and Abouammoh, A.M.
On estimating parameters in a discrete Weibull distribution",
IEEE Transactions on Reliability, 38(3),
pp. 348-350 (1989).
6. Kulasekera, K.B. Approximate MLE's of the parameters
of a discrete Weibull distribution with type I
censored data", Microelectronics Reliability, 34(7), pp.
1185-1188 (1984).
7. Barbiero, A. A comparison of methods for estimating
parameters of the type I discreteWeibull distribution",
Statistics and Its Interface, 9(2), pp. 203-212 (2016).
8. Bebbington, M., Lai, C.D., Wellington, M. and Zitikis,
R. The discrete additive Weibull distribution: A
bathtub-shaped hazard for discontinuous failure data",
Reliability Engineering & System Safety, 106, pp. 37-
44 (2012).
9. Lai, C.D. Discrete Weibull Distributions and Their
Generalizations", In Generalized Weibull Distributions,
Springer Berlin Heidelberg, pp. 97-113 (2014).
A. Barbiero/Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 386{397 397
10. R Core Team. R: A language and environment for statistical
computing", R Foundation for Statistical Computing,
Vienna, Austria (2016). URL https://www.Rproject.
org/
11. Barbiero, A. DiscreteWeibull: Discrete Weibull
distributions (Type 1 and 3)", R package version 1.1.
http://CRAN.R-project.org/package=Discrete
Weibull (2015).
12. Sarabia, J.M. and Gomez-Deniz, E. Construction of
multivariate distributions: A review of some recent
results", SORT, 32(1), pp. 3-36 (2008).
13. Nelsen, R.B., An Introduction to Copulas, Springer,
New York (1999).
14. Nikoloulopoulos, A.K. Copula-based models for multivariate
discrete response data", Copulae in Mathematical
and Quantitative Finance, Lecture Notes in
Statistics, In P. Jaworski et al., Eds., pp. 231-249
(2013).
15. Lai, C.D. Constructions of discrete bivariate distributions",
Advances in Distribution Theory, Order
Statistics, and Inference, in N. Balakrishnan et al.,
Eds., pp. 29-58 (2006). Birkhauser Boston.
16. Englehardt, J. Distributions of Autocorrelated First-
Order Kinetic Outcomes: Illness Severity", PLoS
ONE, 10(6) (2015). DOI: 10.1371/journal.pone.
0129042
17. Barbiero, A. and Ferrari, P.A. GenOrd: Simulation
of ordinal and discrete variables with given correlation
matrix and marginal distributions", R package version
1.4.0. http://CRAN.R-project.org/package=GenOrd
(2015).
18. Ferrari, P.A. and Barbiero, A., Simulating ordinal
data", Multivariate Behavioral Research, 47(4), pp.
566-589 (2012).
19. Barbiero, A. and Ferrari, P.A. Simulating correlated
ordinal and discrete variables with assigned marginal
distributions", in V. Melas, S. Mignani, P. Monari,
L. Salmaso, Eds., Topics in Statistical Simulation.
Research from the 7th International Workshop on Statistical
Simulation, Springer-Verlag, New York (2014).
20. Cario, M.C. and Nelson, B.L. Modeling and generating
random vectors with arbitrary marginal distributions
and correlation matrix", Technical Report, Department
of Industrial Engineering and Management
Sciences, Northwestern University, Evanston, Illinois
(1997).
21. Nelson, B.L. Foundations and Methods of Stochastic
Simulation: A First Course, Springer, New York
(2013).
22. Madsen, L. and Dalthorp, D. Simulating correlated
count data", Environmental and Ecological Statistics,
14(2), pp. 129-148 (2007).
23. Madsen, L. and Birkes, D. Simulating dependent
discrete data", Journal of Statistical Computation and
Simulation, 83(4), pp. 677-691 (2013).
24. Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F.,
Scheipl, F. and Hothorn, T. mvtnorm: Multivariate
normal and t distributions", R package version
1.0-2. http://CRAN.R-project.org/package=mvtnorm
(2014).
25. Barbiero, A. and Ferrari, P.A. Simulating correlated
Poisson variables", Applied Stochastic Models in Business
and Industry, 31, pp. 669-680 (2015).
26. Demirtas, H. and Hedeker, D. A practical way for
computing approximate lower and upper correlation
bounds", The American Statistician, 65, pp. 104-109
(2011).
27. Barbiero, A. Simulating correlated discrete Weibull
variables: a proposal and an implementation in the R
environment", International Conference of Computational
Methods in Science and Engineering, Athens,
20-23 March 2015. AIP Conference Proceedings,
1702(190017) (2015).
28. Joe, H. Asymptotic eciency of the two-stage estimation
method for copula-based models", Journal of
Multivariate Analysis, 94(2), pp. 401-419 (2005).
29. Mitchell, C.R. and Paulson, A.S. A new bivariate negative
binomial distribution", Naval Research Logistics
Quarterly, 28, pp. 359-374 (1981).
30. Famoye, F. A new bivariate generalized Poisson
distribution", Statistica Neerlandica, 64(1), pp. 112-
124 (2010).