Determine what is wrong with the following argument: Suppose that the random variable X has the uniform distribution on the interval [0, θ], where the value of θ is unknown (θ > 0).Then f (x|θ) = 1/θ, λ(x|θ)= −log θ and λ'(x|θ)=−(1/θ). Therefore,
I(θ) = Eθ{[λ'(X|θ)]2} = 1/θ2.
Since 2X is an unbiased estimator of θ, the information inequality states that
Var(2X) ≥ 1/I(θ) = θ2.
But
Var(2X) = 4 Var(X) = 4 . θ2/12 = θ2/3 < θ2.
Hence, the information inequality is not correct.