$X$ is a random variable with density
$f(x,\theta,\lambda) = \lambda \theta^\lambda x^{-(\lambda+1)}$
where $x \ge \theta$ and $\lambda > 0$ and $\theta > 0$ and unknown parameters.
- given $\psi > 0$ retrieve the distribution of $X$ conditioned to the event $X > \psi$
Assuming $\lambda$ known:
- show if the statistic $S = \text{min}(X_1,\dots,X_n)$ is sufficient for $\theta$ and show if it is also minimal
- retrieve the estimator of $\theta$ with maximum likelihood method and establish whether it is unbiased
- retrieve the distribution of $S$ and establish whether there is an unbiased estimator for $\theta$ within the class of estimators of type $c \cdot s$ where $c$ is a constant
I proceed as follows:
$P(X > \psi) = \int_\psi^\infty \lambda \theta^\lambda x^{-(\lambda+1)} = -\theta^\lambda x^{-\lambda}\Big|_\psi^\infty = \theta^\lambda \psi^{-\lambda}$
$f(X,\theta,\lambda|X>\psi) = \frac{f(X,\theta,\lambda)}{P(X>\psi)} = \lambda \psi^\lambda X^{-(\lambda+1)}$
$L(X,\theta,\lambda) = \prod_{i=1}^n \lambda \theta^\lambda x_i^{-(\lambda+1)}[x_i\ge \theta] = \lambda^n \theta^{n\lambda}(\prod_{i=1}^n x_i)^{-(\lambda+1)}\prod_{i=1}^n[x_i\ge\theta]$
$\prod_{i=1}^n[x_i\ge\theta] = [\text{min}(x_i)\ge\theta]$
If the density $f_\theta(X)$ can be factored into a product of $h(X)$ and $g_\theta[T(X)]$, where $g_\theta[T(X)]$ depends on $X$ only through $T(X)$, $T(X)$ is a sufficient statistic.
In our case, $f_\theta(X) = h(X) g_\theta[T(X)]$ where $h(X) = \lambda^n(\prod_{i=1}^n x_i)^{-(\lambda+1)}$ and $g_\theta[T(X)] = \theta^{n\lambda}[T(X)\ge\theta]$ therefore $S(X) = \text{min}(x_i)$ is a sufficient statistic.
$S(X)$ is minimal because it can be represented as a function of any other sufficient statistic. There are $n$ sufficient statistics, one for each $j$: $S(X)_j = \prod_{i=1}^j[x_{(i)}\ge\theta]$ where $x_{(i)}$ are the $Xs$ in increasing order.
$L(X,\theta,\lambda) = \lambda^n \theta^{n\lambda}(\prod_{i=1}^n x_i)^{-(\lambda+1)}[\text{min}(x_i)\ge\theta]$
$\text{LogL}(X,\theta,\lambda) = n\text{log}\lambda + n\lambda\text{log}\theta - (\lambda+1)\sum_{i=1}^n \text{log}x_i$
Here I'm not able to represent the condition $[\text{min}(x_i)\ge\theta]$ with a log-transformation.
$\frac{\delta\text{LogL}(X,\theta,\lambda)}{\delta\theta} = \frac{n\lambda}{\theta} > 0$
In order to maximize LogL, since $x_i \ge \theta$ for each $i$, the maximum value of $\theta$ such that $x_i \ge \theta$ is min($x_i$). Therefore:
$\hat{\theta}_{ML} = \text{min}(x_i)$
$E(\hat{\theta}_{ML}) = ?$
