  An empirical best linear unbiased prediction (EBLUP) estimator is utilized
for efficient inference in small-area estimation. To measure its uncertainty,
we need to estimate its mean squared error (MSE) since the true MSE cannot
generally be derived in a closed form. The "naive MSE estimator", one of the
estimators available for small-area inference, is unlikely to be chosen, since
it does not achieve the desired asymptotic property, namely second-order
unbiasedness, although it maintains strict positivity and tractability.
Therefore, users tend to choose the second-order unbiased MSE estimator. In
this paper, we seek a new adjusted maximum-likelihood method to obtain a naive
MSE estimator that achieves the required asymptotic property. To obtain the
result, we also reveal the relationship between the general adjusted
maximum-likelihood method for the model variance parameter and the general
functional form of the second-order unbiased, and strictly positive, MSE
estimator. We also compare the performance of the new method with that of the
existing naive estimator through a Monte Carlo simulation study. The results
show that the new method remedies the underestimation associated with the
existing naive estimator.
