Question

Let X1, X2, X3, X4 be a random sample from a continuous distribution with the probability density function f(x) = $\frac{1}{2}e^{-|x-\theta|}$, x ∈ $\R$, where θ ∈ $\R$ is unknown. Let the corresponding order statistics be denoted by X(1) < X(2) < X(3) < X(4). Then which of the following statements is/are true ?

• A. $\frac{1}{2}(X_{(2)}+X_{(3)})$ is the unique maximum likelihood estimator of θ
• B. (X(1) , X(2) , X(3) , X(4)) is a sufficient statistic for θ
• C. $\frac{1}{4}(X_{(2)}+X_{(3)})(X_{(2)}+X_{(3)}+2)$ is a maximum likelihood estimator of θ(θ + 1)
• D. (X1X2X3, X1X2X4) is a complete statistic

Answer 1

Answer: (B), (C)

(X(1) , X(2) , X(3) , X(4)) is a sufficient statistic for θ

Answer 2

The correct option is (B) : (X(1) , X(2) , X(3) , X(4)) is a sufficient statistic for θ and (C) : $\frac{1}{4}(X_{(2)}+X_{(3)})(X_{(2)}+X_{(3)}+2)$ is a maximum likelihood estimator of θ(θ + 1).