Question

Let 𝑋₁,𝑋₂, … , 𝑋_𝑛 be a random sample from a \(u(θ +\frac{σ}{\sqrt3},θ+\sqrt3σ)\) distribution, where 𝜃 ∈ ℝ and 𝜎>0 are unknown parameters. Let 𝑋̅ =\(\frac{ 1}{ 𝑛} ∑^n _{i=1}X_i\) and \(𝑆 =\sqrt \frac{1}{n} ∑^n_{i=1}(X_i-\overline{X})^2\) Let \(\^θ\) and \(\^σ\) be the method of moment estimators of 𝜃 and 𝜎 ,respectively. Then, which one of the following statements is FALSE?

• A. \(\^σ+\sqrt3\^θ=\sqrt3\overline{X}-3s\)
• B. \(2\sqrt3\^σ+\^θ=\overline{X}-4\sqrt3S\)
• C. \(\sqrt3\^σ+\^θ=\overline{X}+\sqrt3\,S\)
• D. \(\^σ-\sqrt3\,\^θ=9\,S-\sqrt3\,\overline{X}\)

Answer 1

The Correct Option is B

This problem involves estimating unknown parameters of a uniform distribution using the Method of Moments. Let's break down the steps to identify the FALSE statement among the given options. 

The given distribution is \( u\left(\theta + \frac{\sigma}{\sqrt{3}}, \theta + \sqrt{3}\sigma\right) \). The fundamental properties of a uniform distribution help us estimate the parameters:

• The mean \( \mu \) of a uniform distribution \( U(a, b) \) is given by \( \frac{a + b}{2} \).
• The variance \( \sigma^2 \) of a uniform distribution \( U(a, b) \) is given by \( \frac{(b - a)^2}{12} \).

In this context, we have:

• \(\theta + \frac{\sigma}{\sqrt{3}} = a\), and \(\theta + \sqrt{3}\sigma = b\).

Therefore, the mean and variance of the uniform distribution can be expressed as:

• Mean: \[ \overline{X} = \frac{a + b}{2} = \frac{\theta + \frac{\sigma}{\sqrt{3}} + \theta + \sqrt{3}\sigma}{2} = \theta + \frac{(\frac{\sigma}{\sqrt{3}} + \sqrt{3}\sigma)}{2} = \theta + \sigma. \]
• Variance: \[ S^2 = \frac{(b - a)^2}{12} = \frac{\left[(\theta + \sqrt{3}\sigma) - (\theta + \frac{\sigma}{\sqrt{3}})\right]^2}{12} = \frac{\left[(\sqrt{3} - \frac{1}{\sqrt{3}})\sigma\right]^2}{12} = \frac{\left(\frac{2\sigma}{\sqrt{3}}\right)^2}{12} = \frac{4\sigma^2}{12} = \frac{\sigma^2}{3}. \]

Using the method of moments, we equate the sample mean \( \overline{X} \) to the distribution mean \( \theta + \sigma \) and the sample variance \( S^2 \) to \( \frac{\sigma^2}{3} \). Solving these equations gives the method of moments estimators \( \hat{\theta} \) and \( \hat{\sigma} \).

• \( \overline{X} = \theta + \sigma \Rightarrow \theta = \overline{X} - \sigma \).
• \( S^2 = \frac{\sigma^2}{3} \Rightarrow \sigma = \sqrt{3}S \).

Now, substitute \( \sigma = \sqrt{3}S \) back into the equation for \( \theta \):

• \[ \theta = \overline{X} - \sqrt{3}S. \]

Based on substitutions, the estimators are:

• \( \hat{\theta} = \overline{X} - \sqrt{3}S \).
• \( \hat{\sigma} = \sqrt{3}S \).

Substituting these in the given options, validate each statement:

• \[ \hat{\sigma} + \sqrt{3}\hat{\theta} = \sqrt{3}\overline{X} - 3S: \quad \sqrt{3}S + \sqrt{3}(\overline{X} - \sqrt{3}S) = \sqrt{3}\overline{X} - 3S. \]
• \[ 2\sqrt{3}\hat{\sigma} + \hat{\theta} = \overline{X} - 4\sqrt{3}S: \quad 2\sqrt{3}(\sqrt{3}S) + (\overline{X} - \sqrt{3}S) \neq \overline{X} - 4\sqrt{3}S. \]
• \[ \sqrt{3}\hat{\sigma} + \hat{\theta} = \overline{X} + \sqrt{3}S: \quad \sqrt{3}(\sqrt{3}S) + (\overline{X} - \sqrt{3}S) = \overline{X} + \sqrt{3}S. \]
• \[ \hat{\sigma} - \sqrt{3} \hat{\theta} = 9S - \sqrt{3} \overline{X}: \quad \sqrt{3}S - \sqrt{3}(\overline{X} - \sqrt{3}S) = 9S - \sqrt{3}\overline{X}. \]

The second option does not satisfy the derived formulas: \[ 2\sqrt{3}\hat{\sigma} + \hat{\theta} = \overline{X} - 4\sqrt{3}S \] is FALSE.