Probability density function is given and we have to find a consistent estimator for $\theta$

Question

Let X1,X2,......Xn b a random sample from population with probability density function

$$f(X)=\frac {4}{\theta}{x^3}{e^\frac {-x^4}{\theta}}$$

Where $\theta$ is unknown. Then consistent estimator of $\theta$ is Options are:

A. $\frac1n\sum_1^n X^4$

B.$[{\sum_1^n\frac{X}{n}}]^4$

C.$\bar X$

D.$\prod_1^n X^3$

I tried doing this by checking the options one by one. For an estimator to be consistent its estimate should be equal to $\theta$. But I can't integrate with $\sum_1^n X^4$.

Answer 1

Hint: Mle's are always consistent.

I am suspecting this question came in jam 2015. So how did you do in your exam?