Mathematics · Statistical Inference
Consider the probability density function (pdf) given by: f(x | θ) = e (x-θ) [1 + e (x-θ) ] 2 , where -∞ < x < ∞ and -∞ < θ < ∞. Based on a single observation X with the above pdf, a Uniformly Most Powerful (UMP) test of size α for testing H 0 : θ ≤ θ 0 versus H 1 : θ > θ 0 is given by:
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- 1X > k for some k such that α = P θ 0 [X > k]✓
- 2X < k for some k such that α = P θ 0 [X > k]
- 3X > k for some k such that α = P θ 0 [X < k]
- 4X < k for some k such that α = P θ 0 [X < k]
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X > k for some k such that α = P θ 0 [X > k]
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