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I was looking at this question on Sufficient Statistics and the Uniform Distribution: https://math.stackexchange.com/questions/1359183/why-should-we-care-about-sufficient-statistics

In this question, the following example is provided about the Uniform Distribution:

For example: if you are trying to estimate θ in Unif(0,θ), you might consider three candidate estimators: (a) double the sample mean, (b) double the sample median, and (c) (n+1)/n times the maximum observation among n. All three are unbiased. Only (c) is sufficient and it has the smallest variance of the three. Indeed, (c) has the smallest possible variance among unbiased estimators.

Specifically, when n=5 and θ=10, the standard deviations of these estimators are 1.69 for (c), compared with 2.58 for (a) and 3.78 for (b). So there is a considerable difference in the variability of these estimates.

My Question : Can someone please explain how the standard deviations of (a), (b) and (c) are calculated? Where do the numbers 1.69 2.58 and 3.78 come from?

I tried to work out the standard deviation for (c):

  • Standard Form: y = a*x

  • var(y)= (a^2) * var(x)

Using the estimator in (c):

  • thetha = ((n+1)/n) * max(x)

  • var(thetha) = ((n + 1)/n)^2 * var(max(x))

Since n = 5

  • sd(thetha) = sqrt( (6/5)^2 * var(max(x))

But I don't know how to proceed further.

Can someone please show me how the standard deviations of (a), (b) and (c) were calculated?

Thanks!

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  • $\begingroup$ Gave you some clues at the linked page. $\endgroup$
    – BruceET
    Nov 17 at 5:52
  • $\begingroup$ For max see, but note that max is biased downward, hence $(n+1)/n$ for unbiased estimator. $\endgroup$
    – BruceET
    Nov 17 at 6:02
  • $\begingroup$ More generally, for order statistics (including median) perhaps see $\endgroup$
    – BruceET
    Nov 17 at 6:06
  • 1
    $\begingroup$ Try with order statistics (the Wikipedia page is enough to solve this homework). $\endgroup$
    – Xi'an
    Nov 17 at 7:12

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