Bivariate Birnbaum-Saunders distribution and associated inference. (English) Zbl 1177.62073
Summary: The univariate Z.W. Birnbaum and S.C. Saunders distribution [J. Appl. Probab. 6, 319–327 (1969; Zbl 0209.49801)] has been used quite effectively to model positively skewed data, especially life time data and crack growth data. We introduce a bivariate Birnbaum-Saunders distribution which is an absolutely continuous distribution whose marginals are univariate Birnbaum-Saunders distributions. Different properties of this bivariate Birnbaum-Saunders distribution are then discussed. This new family has five unknown parameters and it is shown that the maximum likelihood estimators can be obtained by solving two nonlinear equations. We also propose simple modified moment estimators for the unknown parameters which are explicit and can therefore be used effectively as an initial guess for the computation of the maximum likelihood estimators. We then present the asymptotic distributions of the maximum likelihood estimators and use them to construct confidence intervals for the parameters. We also discuss likelihood ratio tests for some hypotheses of interest. Monte Carlo simulations are then carried out to examine the performance of the proposed estimators. Finally, a numerical data analysis is performed in order to illustrate all the methods of inference here discussed.
MSC:
62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
62H12 | Estimation in multivariate analysis |
62E20 | Asymptotic distribution theory in statistics |
62F25 | Parametric tolerance and confidence regions |
65C05 | Monte Carlo methods |
62F03 | Parametric hypothesis testing |
62E15 | Exact distribution theory in statistics |
62H10 | Multivariate distribution of statistics |
Keywords:
maximum likelihood estimators; modified moment estimators; Fisher information matrix; asymptotic distribution; likelihood ratio testCitations:
Zbl 0209.49801References:
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