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

The Correct Option is B, C

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.

1. 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.
2. Now, we evaluate each statement one by one:
3. 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.
4. 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.
5. 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.
6. 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 \(\theta(\theta + 1)\).