Question

Let \( x_1 = 2, x_2 = 1, x_3 = \sqrt{5}, x_4 = \sqrt{2} \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{2\theta}, & \text{if } -\theta \leq x \leq \theta \\ 0, & \text{otherwise}, \quad \theta > 0 \end{cases} \] Then the method of moments estimate of \( \theta \) is

• A. 1
• B. 3
• C. 2
• D. 4

Hint:

In the method of moments, the population moments (e.g., mean) are matched with the sample moments to estimate the parameters of the distribution.

Answer 1

The Correct Option is B

Step 1: Recall the method of moments formula.
For a uniform distribution U(-θ, θ), the mean is:
E[X] = 0 (since symmetric about 0).
The second moment (about origin) is:
E[X^2] = θ^2 / 3.

Step 2: Compute the sample second moment.
The observed values are x1 = 2, x2 = 1, x3 = √5, x4 = √2.
The sample second moment is:
M2 = (1/4)(2^2 + 1^2 + (√5)^2 + (√2)^2)
= (1/4)(4 + 1 + 5 + 2) = (1/4)(12) = 3.

Step 3: Equate sample second moment to theoretical second moment.
θ^2 / 3 = M2 = 3
θ^2 = 9
θ = 3 (since θ > 0).

Final Answer: 3