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Let $Y_1, \ldots, Y_n$ be a random sample from a distribution with the density function

$$f_θ(y) = \frac{3θ^3}{y^4} \text{ for } y ≥ θ > 0.$$

Is there a UMP test at level α for testing $H_0 : θ ≤ θ_0$ vs $H_1 : θ > θ_0$?

I've begun this question by looking for an MLR, but I'm already stuck. I've got the likelihood ratio as just $(θ_2/θ_1)^{3n}$, which means it is non-decreasing, but it's not a function of a statistic, just the parameter. Does this mean

a) there is no MLR for this distribution and hence, no UMP?

b) there is no MLR for this distribution, but this doesn't necessarily mean no UMP?

or c) I have misunderstood how to calculate an MLR and there is one.

Thanks!

Michael Hardy
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Brandon Barry
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  • Did you take into account the piecewise nature of the definition of the density? – Michael Hardy Oct 30 '17 at 16:14
  • Wouldn't the other half of the piecewise function just be that the density is 0 for any other values (that don't satisfy y≥θ>0). How would that help? – Brandon Barry Oct 31 '17 at 04:34
  • See https://math.stackexchange.com/questions/2767098/ump-level-alpha-test-with-distribution-fx-mid-lambda-lambda-x-2?rq=1 and https://en.wikipedia.org/wiki/Uniformly_most_powerful_test#The_Karlin%E2%80%93Rubin_theorem. – StubbornAtom Jun 21 '19 at 10:52

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