Let Y₁ < Y₂ < · · · < Y_n be the order statistics of a random sample of size n from the uniform distribution over the closed interval [−θ, θ] having pdf f(x; θ) = (1/2θ)I_[−θ,θ](x).
(a) Show that Y₁ and Y_n are joint sufficient statistics for θ.
(b) Argue that the mle of θ is ˆθ = max(−Y₁, Y_n).
(c) Demonstrate that the mle ˆθ is a sufficient statistic for θ and thus is a minimal sufficient statistic for θ.