# consistency of maximum likelihood estimator

For population with n size and following density function

$$f(y, a)= (1/6a^4)y^3e^{-y/a}$$

For that, I have found the maximum likelihood estimator of a which is $$\hat{a}= \bar{y}/4$$

I have also shon that this is unbiased estimator of a.

But I cannot show whether this MLE estimator is consistent.

I know that if $$lim(|\hat{a}-a|>c)=0$$ then MLE is consistent estimator.

I do not know how I can show this limit.

An unbiased estimator is by definition also consistent, so you have actually already shown that.

To see it directly, you just have to show that $$a=E[y]/4$$, where $$E[y] = \int yf(y,a)$$ .

(This is because $$\lim_{n \to \infty} \bar y = E[y]$$, due to the law of large numbers)

• An unbiased estimator does not have to be consistent. For instance, let $X_1,\cdots,X_n\overset{\sim}{iid}N(\mu,1)$. Then $X_1$ is unbiased for $\mu$ but not consistent.
– Dave
Feb 5 at 19:05
• An unbiased estimator is consistent as long as it converges. So strictly speaking yes you also have to show that it converges, which is trivial in this case. Feb 5 at 19:18