consistency of maximum likelihood estimator

Question

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.

Please help me to do that.

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.

J. Delaney · Accepted Answer · 2022-02-05 18:58:49Z

-1

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)

answered Feb 5, 2022 at 18:58

J. Delaney

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$\begingroup$ 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. $\endgroup$
– Dave
Commented Feb 5, 2022 at 19:05
1

$\begingroup$ 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. $\endgroup$
– J. Delaney
Commented Feb 5, 2022 at 19:18

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