Part CCSIR NET June 2025the-likelihood-ratio-is-unimodal-so-small-values-occur-at-both-ends

The likelihood ratio is unimodal so small values occur at both ends

Let X be a random sample of size one from the probability density function f(x| $θ) =$ $θ e^{- θ (x - 1)}$ if x > 1, and 0 otherwise, where $θ (> 0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_{0} : θ = 1$ against the alternative hypothesis $H_{1} : θ \neq = 1$ , based on the observed value x of X. Then, which of the following statements are true?

A.The likelihood function is maximized at $θ = 1/ (x - 1)$ ✓
B.The maximum value of the likelihood function is $e^{- 1} / (x$ − 1)✓
C.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if (x − $1) e^{- x}$ < k, for some k > 0✓
D.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if x > c, for some c > 1

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests what the inference means.

See pricing

50 are analysed free — try those first.

The trap it tests

What the inference means

p-values, size, power and error types say specific things. This was not one of them.

Drill statements like this

The likelihood ratio is unimodal so small values occur at both ends

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