The Maximum Likelihood Estimation with the Order Statistics

In this paper,a class of parameter space restricted by the sample maximum likelihood estimation problem is discussed.The corresponding relationship is considered between non-zero region in distribution of a random variable and the domain of likelihood function.The paper proposed that no matter the likelihood equation is solvable or not,if the distribution of non-zero region affected by parameter restrictions,the maximum likelihood estimation of parameters related to the order statistics X_((n)) or X_((1)) inevitably.This maximum likelihood estimation has three cases:equal,not equal or possible equal to the order statistics,and gives the corresponding Criterion.Finally,it is obtained that under the possible equal case,the likelihood estimation is X_((n)) or X_((1)) depends on the sample observations.