Question

Let \( X_1, X_2, ...., X_n \) be a random sample of size \( n \) from a population having uniform distribution over the interval \( \left( \frac{1}{3}, \theta \right) \), where \( \theta>\frac{1}{3} \) is an unknown parameter. If \( Y = \max \{ X_1, X_2, ...., X_n \} \), then which one of the following statements is true?

• A. \( \left( \frac{n+1}{n} \right) (Y - \frac{1}{3}) + \frac{1}{3} \) is an unbiased estimator of \( \theta \)
• B. \( \left( \frac{n}{n+1} \right) (Y - \frac{1}{3}) + \frac{1}{3} \) is an unbiased estimator of \( \theta \)
• C. \( \left( \frac{n+1}{n} \right) (Y + \frac{1}{3}) - \frac{1}{3} \) is an unbiased estimator of \( \theta \)
• D. \( Y \) is an unbiased estimator of \( \theta \)

Hint:

- For a uniform distribution, the expected value of the maximum is given by \( E(Y) = a + \frac{n}{n+1} (b - a) \).
- Use the formula for \( E(Y) \) to find the unbiased estimator of the parameter.

Answer 1

The Correct Option is A

1) Understanding the problem:
The random variable \( X_1, X_2, ...., X_n \) follows a uniform distribution over the interval \( \left( \frac{1}{3}, \theta \right) \). The maximum \( Y = \max \{ X_1, X_2, ...., X_n \} \) is the largest of these random variables. We aim to find an unbiased estimator for \( \theta \) based on \( Y \).
2) The expected value of the maximum:
For a uniform distribution \( U(a, b) \), the expected value of the maximum of \( n \) random variables is given by: \[ E(Y) = a + \frac{n}{n+1} (b - a) \] For our distribution \( U\left( \frac{1}{3}, \theta \right) \), the expected value of the maximum is: \[ E(Y) = \frac{1}{3} + \frac{n}{n+1} \left( \theta - \frac{1}{3} \right) \] 3) Solving for an unbiased estimator:
An unbiased estimator \( \hat{\theta} \) is one for which the expected value is equal to \( \theta \). Setting \( E(Y) = \theta \), we get: \[ \frac{1}{3} + \frac{n}{n+1} \left( \theta - \frac{1}{3} \right) = \theta \] Solving for \( \theta \), we get: \[ \theta = \left( \frac{n+1}{n} \right) \left( Y - \frac{1}{3} \right) + \frac{1}{3} \] Thus, the unbiased estimator for \( \theta \) is \( \left( \frac{n+1}{n} \right) (Y - \frac{1}{3}) + \frac{1}{3} \), which corresponds to option (A).