Let X1,...,Xn be i.i.d. random variables with density θ ∈ R. Note that the range depends on θ, violating the Cramer–Rao–Fr ´ echet regularity ´ conditions. Show that the smallest order statistic...

Let X₁,...,X_n
be i.i.d. random variables with density

θ ∈ R. Note that the range

depends on θ, violating the Cramer–Rao–Fr ´ echet regularity ´ conditions. Show that the smallest order statistic X_n:1
is the MLE and further, that n(X_n:1
− θ) has asymptotically Exp(1) distribution.