How to check whether an MLE always yields an over-estimated values?

Question

I have a log-likelihood function with two parameters: $\mathcal{l}(\theta_1,\theta_2)$.

When I run a simulation, the maximum likelihood estimator $\widehat{\theta}=[\widehat{\theta}_1,\widehat{\theta}_2]'$ yields a reasonable value for $\theta_1$ and an over-estimated value for $\theta_2$.

For example, if the true $\theta_1$ and $\theta_2$ are 10 and 15, respectively, then $\widehat{\theta}_1=9.97$ and $\widehat{\theta}_2=19.5$.

In my setup, the incorrect estimate is an expected event. That is, theoretically the estimator $\widehat{\theta}_2$ is inconsistent.

However, I want to know whether the MLE always produces an over-estimated value.

Hence, my question is that Is it possible to know whether the MLE over-estimates the parameters always?

Answer 1

``Is it possible to know whether the MLE over-estimates the parameters always?''

Yes, for specific distributions, this is possible to know.

---

An example is the MLE for a uniform distribution $$x_i \sim U(\theta,1)$$ which has as MLE the minimum of the sample which always overestimates the true parameter

$$\hat\theta = \min(x_1, x_2, \dots, x_n) > \theta $$

Another example is the MLE for the mean of a normal distribution, where we know that it is unbiased, and hence we know whether it overestimates (namely it doesn't overestimate).

---

Aside from those examples, more generally you will need to consider the distribution everytime again. I don't believe that there is an easy formula that anders you yes or no on the question whether the MLE always overestimates.

I imagine that it is possible that it is too difficult to compute/analyse the distributions, but one can always use some simulations and see whether the MLE is, for those simulations, always an overestimation of the true value.

I find it hard to come up with other situations than such cases as the uniform distribution whose domain is bounded and determined by parameters, so I guess that in practice, this method of simulations should handle most cases.