The asymptotics of maximum-likelihood estimates of parameters based on a data type where the failure and the censoring time are dependent

This article derives the asymptotic results of the maximum-likelihood estimates of the parameters in the general bivariate continuous distribution for the data type, in which the failure time and the censoring variables are dependent. This data type is motivated from life-testing two-component parallel systems. Because the duration of collecting all system failure times can be too long to justify its experimental cost, the testing experiment is terminated at the rth smallest failure time, X(r), of one component. The resulting bivariate censored data are (X*(1), Y*[i]), where X*(i) = X(i) if i [less-than-or-equals, slant] r; X*(i) = X(r) if i> r, Y*[i] = Y[i] if Y[i] [less-than-or-equals, slant] X(r); Y*[i] = X(r), if Y[i] [less-than-or-equals, slant] X(r), and Y[i]'s are the concomitants of the ordered statistics X(i)'s. The failure time Y[i] is dependent on X(r) due to the induced dependence in concomitant ordered statistics. Our procedures can be applied to derive asymptotics in other complicated data types, where the life-testing experiment is stopped at the rth smallest failure time of max(X, Y) or min(X, Y).