Consider a generalized Laplace density                             f(x; θ) = constant · exp{−|x − θ| α }, x ∈ R, where α ∈ (0, 1) and θ ∈ θ ⊂ R. Show that...

Consider a generalized Laplace density

                            f(x; θ) = constant · exp{−|x − θ| α }, x ∈ R,

where α ∈ (0, 1) and θ ∈ θ ⊂ R. Show that

                           ∂ ∂θ log f(x; θ) = −α|x − θ| α−1 sign(x − θ)

so that

                             Z R ¯ ¯ ¯ ∂ ∂θ log f(x; θ) ¯ ¯ ¯ dF(x; θ) <>∀α ∈ (0, 1),

but that for α ≤ 1/2,

                             Z R n ∂ ∂θ log f(x; θ) o2 dF(x; θ) 6<>

Comment on the asymptotic properties of the MLE of θ (when α is known and 0 <>