Find MLE and show that it is unbiased.

Question

I'm trying to solve a problem but not sure how to approach it because of the weird density function:

Would appreciate any constructive advice!

Answer 1

For these types of problems, the steps to follow are:

1. Calculate the likelihood function for identically distributed random variables $X_1$,$X_2$,..,$X_n$. Are these random variables assumed to be independent? If so, the likelihood will simply be the product of their densities.
2. Differentiate the likelihood function with respect to the parameter ($\theta$) you wish to estimate. Often, taking the natural logarithm of the likelihood function (known as the log likelihood), and differentiating this instead can help. As the logarithm is a monotonic function, the stationary points will coincide
3. Set the derivative of either the likelihood function or its logarithm to zero and solve for $\theta$, to obtain the MLE ($\hat{\theta}_{MLE}$), which will be a function of the random variable,.i.e. $\hat{\theta}_{MLE}=g(x)$
4. Calculate the expectation of $\hat{\theta}_{MLE}$ with the value you have obtained, and show that it equals parameter $\theta$, given that it is an unbiased estimator. i.e. show that $$\int_0^1\hat{\theta}_{MLE}\theta^{-1}x^{(1-\theta)/\theta}\ dx=\int_0^1g(x)\theta^{-1}x^{(1-\theta)/\theta}\ dx=\theta$$ Hint for this stage: use integration by parts