Abstract
When data generating mechanism generates two correlated data sets both defined on the unit interval, a bivariate probabilistic distribution defined on the unit square is needed for modelling the data. For this purpose, we give a Marshall-Olkin type bivariate extension of an omega distribution in this paper. This is in fact a bivariate unit-exponentiated-half-logistic distribution. We study its mathematical properties in detail. The distribution contains neither an exponential term nor any special function which complicates the computations. Maximum likelihood estimation method and its large sample inference are considered for model parameters. Alternatively, we also propose an expectation- maximization algorithm to compute the estimates. To see the performances of the proposed estimators and validate the theoretical results obtained for estimation, we present the results of a simulation study. Data fitting demonstrations show its applicability in modelling random proportions.
Acknowledgement
The authors would like to thank the Area Editor (Gejza Wimmer, Ph.D.) and the two anonymous referees for their comments.
(Communicated by Gejza Wimmer )
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Appendix A. Proofs of Theorems
Proof
Proof of Theorem 2.1.
□
Proof
Proof of Theorem 2.3. We have FY1,Y2 (y1, y2) = SY1, Y2(y1, y2)+FY1(y1)+FY2(y)−1, where FY1 and FY2 denote the marginal CDF’s of Y1 and Y2. Using this formula, one can obtain the result in the theorem.□
Proof
Proof of Theorem 2.4. For the absolutely continuous part of the joint PDF, we take the second order partial derivatives of the joint CDF. For the singular part, we use the following fact:
Since
we have
Since
we have
□
Proof
Proof of Theorem 2.5. If y1 < y2, then
The other two cases can be shown similarly.□
Proof
Proof of Theorem 2.6. If y1 < y2, then
The other two cases can be shown similarly.□
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