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 ?
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 ?
Updated On: Oct 1, 2024
- \(\frac{1}{2}(X_{(2)}+X_{(3)})\) is the unique maximum likelihood estimator of θ
- (X(1) , X(2) , X(3) , X(4)) is a sufficient statistic for θ
- \(\frac{1}{4}(X_{(2)}+X_{(3)})(X_{(2)}+X_{(3)}+2)\) is a maximum likelihood estimator of θ(θ + 1)
- (X1X2X3, X1X2X4) is a complete statistic
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The Correct Option is B, C
Solution and Explanation
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).
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