Let X1, , Xn be i.i.d. with density f(x, ) = [1 + cos(x)]/2, where

Let X1, ··· , Xn be i.i.d. with density f(x, θ) = [1 + θ cos(x)]/2π, where the parameter θ satisfies |θ| < 1 and x ranges between 0 and 2π. (The observations Xi may be interpreted as directional data. The case θ = 0 corresponds to the uniform distribution on the circle.) Construct an efficient likelihood estimator of θ, as explicitly as possible.