Consider the pdf given by
\(f(x \mid θ)=\frac{e^{(x-θ)}}{\left[1+\left.e^{(x-θ)}\right|^2\right.}\), -∞ < x < ∞, -∞ < θ < ∞

Based on one observation X with the above pdf, a UMP test of size α for testing H0 ∶ θ ≤ θ0 versus H1 ∶ θ > θ0 is

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. > k for some k such that α = \(P_{θ_0}\)[X > k]
  2. < k for some k such that α = \(P_{θ_0}\)[X > k]
  3. X > k for some k such that α = \(P_{θ_0}\)[X < k]
  4. X < k for some k such that α = \(P_{θ_0}\)[X < k]

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Option 1 : > k for some k such that α = \(P_{θ_0}\)[X > k]
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Based on one observation X with the given pdf, a UMP test of size α for testing H0 ∶ θ ≤ θ0 versus H1 ∶ θ > θ0 is

X > k for some k such that α = \(P_{θ_0}\)[X > k]

Option (1) is correct

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