Let $(X_1,\dots,X_n)$ be a random sample from a population $X$ having probability density function $$f(x;\vartheta,\lambda)=\lambda\,\vartheta^\lambda\,x^{-(\lambda +1)}$$ $$\vartheta>0\qquad\lambda>0\qquad x\geq\theta$$ If $\lambda$ is known (hence fixed):
- Find $\hat{\vartheta}_n$, the MLE (Maximum Likelihood Estimator) of the parameter $\vartheta$ and verify if it is unbiased
- Find the distribution of $\hat{\vartheta}_n$
- If $\hat{\vartheta}_n$ is biased, propose an unbiased estimator in the class of the estimators of the type $c\cdot\hat{\vartheta}_n$, with $c$ an appropriate constant
While I'm pretty confident on how to solve the first and last point, I'd like some advice on the second one.
1.
To find the MLE $\hat{\vartheta}_n$, the likelihood function is calculated: $$\mathscr{L}(\underline{x};\vartheta) = \lambda^n\,\vartheta^{n\lambda}\,\left( x_1 \times\dots\times x_n \right)^{-(\lambda +1)} \qquad\qquad(\vartheta\leq x_i,\, \forall\,i=1,2,\dots,n)$$ Given the condition on $\vartheta$, the value that maximizes the likelihood function can be starightforwardly seen to be $\hat{\vartheta}_n = \min{\left(x_1,\dots,x_n\right)} = x_{(1)}$ where $\left(x_{(1)},\dots,x_{(n)} \right)$ is the sample re-arranged in non-decreasing order.
We have to verify if $\mathbb{E}\left(\hat{\vartheta}_n\right)=\vartheta$, so we calculate $$\mathbb{E}\left(\hat{\vartheta}_n\right)=\mathbb{E}\left(x_{(1)}\right)=\mathbb{E}\left(x\right)=\int_\vartheta^{+\infty} x\,\lambda\,\vartheta^\lambda\,x^{-(\lambda +1)}\stackrel{\lambda>1}{=}\frac{\lambda}{\lambda-1}\vartheta\neq\vartheta$$
So the MLE is biased.
2?
$\hat{\vartheta}_n = x_{(1)}$ has the same distribution of every $x$ random variable, that is $f(x;\vartheta,\lambda)$.
Am I right about this? No, I'm not!
3.
Given what was calculated at point 1, if $c=\frac{\lambda -1}{\lambda}$ we have that $c\cdot\hat{\vartheta}_n$ is an unbiased MLE.
Any help/correction would be greatly appreciated!