The Pareto pdf with scale parameter $α > 0$ and shape parameter $β > 0$ is $f(y;α,β) = βα^βy^{−(β+1)}I_{(α,∞)}(y) = βα^{β}y^{−(β+1)}$ for $y > α.$
Is the MLE for α unbiased for $α$? If it is not unbiased, will it overestimate or underestimate $α$? (no explicit calculations are necessary, just argue from the type of the estimator.)
I have found $α_\text{MLE}= \min_i(y_i)$.
Could you please answer/explain the question?
Kind regards.
Lets take an analogical slightly "cleaner" problem. Let $X_1,....,X_n \sim \mathrm{U}[0,\theta]$. Clearly, the MLE for $\theta$ is $\max\{X_1,...,X_n\}=X_{(n)}$. Note that $X_{(n)} < \theta$ w.p $1$. Technically, you can show that $X_{(n)}$ has continuous distribution hence $P(X_{(n)}=c)=0$ for all $c\in [0,\theta]$. However, asymptotically $\lim_{n\to \infty} P(X_{(n)}<\theta - \epsilon)=0$. As such $X_{(n)}$ approaches $\theta$ "from beneath", so it underestimates $\theta$ for any finite $n$. Applying the same logic for you case, note that $X_{(1)} > \alpha$ for all finite $n$, hence the minimum overestimates $\alpha$ for any finite sample size.