Suppose that*X*_{1},*X*_{2}, ...,*X*_{n}, ... is a sequence of i.i.d. random variables with a density*f(x*, θ). Let*c*_{n} be a maximum order of consistency. We consider a solution
$$\hat \theta _n $$
of the discretized likelihood equation
$$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$
where*a*_{n}(θ,*r*) is chosen so that
$$\hat \theta _n $$
is asymptotically median unbiased (AMU). Then the solution
$$\hat \theta _n $$
is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.