Statistical inference procedure for a bivariate exponential distribution (Q1327825)

Let us consider the bivariate exponential distribution \(BVE(\lambda_ 1, \lambda_ 2, \lambda_ 3)\) with distribution function \[ F(x,y)= P(X>x, Y>y)= \begin{cases} \exp\{| -\lambda_ 1 x-\lambda_ 2 x-\lambda_{12} \max (x,y)|\}, \quad &x\geq 0,\;y\geq 0,\\ 0 &\text{otherwise} \end{cases}. \] The author assumes a two-component series system with the joint distribution of component lifetimes BVE which follow type II censoring. More precisely: a life test with \(n\) prototypes of the series system is conducted and we observe the first \(r_ s\) failure times \(T_{(1)}\leq T_{(2)}\leq \dots\leq T_{(r_ s)}\) as well as the corresponding failure indicators \(I_{| j|}\), \(j=1,\dots, r_ s\). If the \(j\)-th system failure is caused by the failure of component \(C_ 1\) \((C_ 2)\), \(I_{| j|}= 0(1)\), and if the \(j\)-th system failure is caused by the failure of components \(C_ 1\) and \(C_ 2\) simultaneously, then \(I_{| j|} =0\) and 1, both with probability \(1/2\). Besides system testing the author also considers a life test of \(m_ 1\) units of component \(C_ 1\) and \(m_ 2\) units of component \(C_ 2\), where he observes the first \(r_ 1\) and \(r_ 2\) failure times, respectively. Aside summarization of known results the author derived maximum likelihood estimators \(\widehat {\lambda}^*\) and \(\widehat {\lambda}^*_{12}\) of the parameters \(\lambda_ 1= \lambda_ 2= \lambda\) and \(\lambda_{12}\) under the condition of identical marginals and obtained strong consistency and the limiting distribution of \(\widehat {\lambda}^*\), \(\widehat {\lambda}^*_{12}\). Besides, he established the uniform most powerful unbiased (UMPU) test for \(\lambda\) and \(\lambda_{12}\), including the UMPU test of independence, i.e. \(\lambda_{12} =0\). Moreover, without the restriction of identical marginals, the moment type estimators \(\widetilde {\lambda}^*_ 1\), \(\widetilde {\lambda}^*_ 2\), \(\widetilde {\lambda}^*_{12}\) of parameters \(\lambda_ 1\), \(\lambda_ 2= \lambda_{12}\) are derived and strong consistency and the limiting distribution of them is obtained. Finally, the author develops a UMPU test of equal marginals, i.e. \(\lambda_ 1 =\lambda_ 2\).