Bayesian analysis for two-components systems and a conjugate family of bivariate densities (Q2640301)

Let \(\Theta\) be a vector of unknown parameters. Let \(X_ 1,X_ 2,..\). be a sequence of real nonnegative random variables and let \(Y_ 1,Y_ 2,..\). be another sequence of real nonnegative random variables. Suppose that, given \(\Theta\), the \(X_ i's\) are i.i.d. and also the \(Y_ i's\) are i.i.d. Furthermore assume that, given \(\Theta\), the \(X_ i's\) and the \(Y_ i's\) are independent. Set \(U_ h\equiv \min (X_ h,Y_ h)\), \(h=1,2,...\). Denote by D(x,y,u;a,b) an event of the form \[ X_ 1=x_ 1,...,X_ a=x_ a,\quad X_{a+1}>x_{a+1},...,X_{\ell}\geq x_{\ell}, \] \[ Y_ 1=y_ 1,...,Y_ b=y_ b,\quad Y_{b+1}>y_{b+1},...,Y_ m>y_ m, \] \[ U_{h_ 1}=u_ 1,...,U_{h_ n}=u_ n,\quad \max (\ell,m)<h_ 1<...<h_ n. \] Such events arise naturally in reliability theory when the observer has some past data consisting of lifetimes \(U_{h_ 1},...,U_{h_ n}\), of two-components series systems, and also some more recent data of such systems in which the component which caused the system failure can be identified, and, on top of that, the observer also observes some such systems which are still functioning. The author derives general formulas for the posterior density of \(\Theta\) given D(x,y,u;a,b) and also for some `predictive reliability functions'. When the \(X_ i's\) and the \(Y_ i's\) have proportional hazards, with the coordinates of \(\Theta\) being the coefficients of proportionality, the author identifies a sufficient statistic and a natural conjugate family of densities. The conjugate family of densities is analyzed. The case of parallel systems is also considered.