An asymptotic relation arising in the decomposition of the likelihood of order statistics

In the decomposition of the score function based on all the order statistics, partial derivative/partial derivative theta log L(1...n) = partial derivative/partial derivative theta log L(1...n) + partial derivative/partial derivative theta log L(r + 1...n/r), we study the asymptotic behaviour of the maximum conditional likelihood estimator, based on the second term on the right, and suggest an indirect way to study that of the maximum likelihood estimator based on the first term. We also give an asymptotic relation between the maximum likelihood estimators in each part.