Question

If \( \hat{\beta} \) is a consistent estimator of a population parameter \( \beta \); \( \epsilon \) and \( \delta \) are very small quantities, then ..........

• A. \( \text{Prob} \left( \left| \hat{\beta} - \beta \right| \geq \epsilon \right) \geq 1 - \delta \)
• B. \( \text{Prob} \left( \left| \hat{\beta} - \beta \right|<\epsilon \right) = 0 \)
• C. \( \text{Prob} \left( \left| \hat{\beta} - \beta \right|<\epsilon \right) = \delta \)
• D. \( \text{Prob} \left( \left| \hat{\beta} - \beta \right|<\epsilon \right) = 1 + \delta \)

Hint:

Consistency of an estimator ensures that as the sample size increases, the probability of the estimate being within a small \( \epsilon \) of the true value increases.

Answer 1

The Correct Option is A

Step 1: Understand consistency of an estimator.
An estimator \( \hat{\beta} \) is consistent if, as the sample size tends to infinity, it converges in probability to the true value of the parameter \( \beta \). This means that for any small \( \epsilon>0 \), the probability that the estimate \( \hat{\beta} \) is within \( \epsilon \) of \( \beta \) approaches 1 as the sample size increases.
Step 2: Explanation of the options.
- Option (A) is correct. By the definition of consistency, the probability that the estimator \( \hat{\beta} \) is within an \( \epsilon \) distance from \( \beta \) increases as the sample size grows. In other words, for small \( \epsilon \), the probability that \( \hat{\beta} \) is farther than \( \epsilon \) from \( \beta \) becomes very small, i.e., \( \text{Prob} \left( \left| \hat{\beta} - \beta \right| \geq \epsilon \right) \leq \delta \). This implies that the probability of the estimator being close to \( \beta \) is at least \( 1 - \delta \).
- Option (B) is incorrect because, while \( \hat{\beta} \) is consistent, the probability of \( \hat{\beta} \) being exactly equal to \( \beta \) is 0, not necessarily 0 for small \( \epsilon \).
- Option (C) is incorrect because \( \delta \) is a very small quantity, and the probability that the estimator is within \( \epsilon \) of \( \beta \) should be close to 1, not equal to \( \delta \).
- Option (D) is incorrect because the probability cannot exceed 1, and \( 1 + \delta \) is not a valid probability.
Final Answer: \[ \boxed{\text{Prob} \left( \left| \hat{\beta} - \beta \right| \geq \epsilon \right) \geq 1 - \delta} \]