Mathematics · Mathematical Statistics
Let X 1 , ..., X n (n ≥ 3) be a random sample from a distribution having probability density function: f(x|θ) = θe -θx for x > 0, and 0 otherwise (where θ > 0 is an unknown parameter). Let T n = 1 n Σ i=1 n X i . Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1(1 - e -1/T n ) is a consistent estimator of P θ (X 1 ≤ 1)
- 2The uniformly minimum variance unbiased estimator of θ attains the Cramer-Rao lower bound
- 3The Cramer-Rao lower bound for the variance of any unbiased estimator of θ is θ 2 n✓
- 4The uniformly minimum variance unbiased estimator (UMVUE) of θ is n - 1 n T n
Correct answer
The Cramer-Rao lower bound for the variance of any unbiased estimator of θ is θ 2 n
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