Mathematics · Statistical Inference
Let X be a random sample of size one from the probability density function f(x|θ) = θe-θ(x-1) if x > 1, and 0 otherwise, where θ (> 0) is the unknown parameter. Suppose we want to test the null hypothesis H0: θ = 1 against the alternative hypothesis H1: θ ≠ 1, based on the observed value x of X. Then, which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The likelihood ratio test for testing H0 against H1 rejects H0 if x > c, for some c > 1
- 2The likelihood ratio test for testing H0 against H1 rejects H0 if (x - 1)e-(x-1) 0✓
- 3The maximum value of the likelihood function is e-1x-1
- 4The likelihood function is maximized at θ = 1 for all x
Correct answer
The likelihood ratio test for testing H0 against H1 rejects H0 if (x - 1)e-(x-1) 0
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