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: Nov 25, 2025
- \(\frac{1}{2}(X_{(2)}+X_{(3)})\) 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
To solve this problem, we need to evaluate the given statements based on the properties of the probability distribution and order statistics of the random sample.
- Firstly, consider the probability density function (pdf) given as \(\frac{1}{2}e^{-|x-\theta|}\). This is the pdf of a Laplace distribution with mean \(\theta\) and scale parameter 1. The problem involves a random sample from this distribution and its order statistics.
- Now, we evaluate each statement one by one:
- Option 1: \(\frac{1}{2}(X_{(2)}+X_{(3)})\) is the unique maximum likelihood estimator of \(\theta\).
In the case of the Laplace distribution, the median is the maximum likelihood estimator of \(\theta\). For four samples, \(\frac{1}{2}(X_{(2)} + X_{(3)})\) may seem like a natural estimator, but it is not uniquely the MLE for \(\theta\). Hence, this statement is false. - Option 2: (\(X_{(1)}\), \(X_{(2)}\), \(X_{(3)}\), \(X_{(4)}\)) is a sufficient statistic for \(\theta\).
The order statistics constitute a sufficient statistic for location parameter \(\theta\) due to the invariance property of the sufficiency principle. Thus, this statement is true. - Option 3: \(\frac{1}{4}(X_{(2)}+X_{(3)})(X_{(2)}+X_{(3)}+2)\) is a maximum likelihood estimator of \(\theta(\theta + 1)\).
We verify this by examining the structure. The estimator seems formed to estimate a function of the form \(\theta(\theta + 1)\). Due to the nature of the Laplace distribution, and without exhaustive proof here, it matches the criteria for MLE of this function. The statement is true. - Option 4: (\(X_{1}X_{2}X_{3}\), \(X_{1}X_{2}X_{4}\)) is a complete statistic.
For continuous distributions like the Laplace, a complete statistic typically involves all sample components; having pairwise products only does not achieve completeness. Thus, this statement is false.
Thus, the correct statements are: (\(X_{(1)}\), \(X_{(2)}\), \(X_{(3)}\), \(X_{(4)}\)) is a sufficient statistic for \(\theta\), and \(\frac{1}{4}(X_{(2)}+X_{(3)})(X_{(2)}+X_{(3)}+2)\) is a maximum likelihood estimator of \(\theta(\theta + 1)\).
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