Part BCSIR NET June 2025a-heavier-tail-parameter-pushes-the-sample-down-so-the-ump-test-rejects-small-values

A heavier tail parameter pushes the sample down so the ump test rejects small values

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a continuous distribution with the common probability density function f(x| $\theta) = 2^\theta \theta /$ $x^{\theta+1}$ if x > 2, and 0 otherwise, where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y > \chi^{2}_{m,\beta})$ $= \beta$ , where Y ~ $\chi^{2}_m$ . For testing $H_{0} : \theta = 1$ against $H_{1} : \theta > 1, a$ uniformly most powerful test of size $\alpha, 0 < \alpha < 1$ , will reject $H_{0}$ if

A. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} > (1/2)\chi^{2}_{2n,\alpha}$ + n ln 2
B. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} < (1/2)\chi^{2}_{2n,1-\alpha}$ + n ln 2✓ Correct
C. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} > \chi^{2}_{n,\alpha}$ + n ln 2
D. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} < \chi^{2}_{n,1-\alpha}$ + n ln 2

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

Related counterexample: A UMP test exists for every testing problem

A heavier tail parameter pushes the sample down so the ump test rejects small values

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