I have a question regarding completeness of a statistic. So the problem is:

$n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and $b$ has been lost.

  • Find a minimal sufficient statistic for $(a, b)$ and check that it is complete.

I have found the minimal sufficient statistic $(\min X, \max X) \equiv (U,V)$ and their distributions, which are, respectively

$$ f_u = n f(x) [1-F(x)]^{n-1} = n \frac{1}{b-a} \left[ 1-\frac{x-a}{b-a} \right]^{n-1}, \\ f_v = n f(x) F(x)^{n-1} = n \frac{1}{b-a} \left[ \frac{x-a}{b-a} \right]^{n-1}, $$

exploiting the fact that the samples come from a uniform distribution. However, I can only show that $V$ is complete, not $U$... More specifically, for a function $g$, $\mathrm{E}[g(U)]=0$ for $x = b$. What am I doing wrong here?

  • 2
    $\begingroup$ What do you mean by $x=b$? You need to find $g$ such that $\mathbb{E}[g(U,V)]=0$ for all $(a,b)$'s or prove it cannot be anything but the zero function. You also need to derive the joint distribution of $(U,V)$ as the sufficient statistic is the pair $(U,V)$. $\endgroup$ – Xi'an Dec 9 '16 at 14:39
  • 1
    $\begingroup$ Ok, so is this the correct joint pdf? $$ f_u,v = \frac{n(n-1)}{(b-a)^2} \left( \frac{v-u}{b-a} \right)^{n-1} $$ $\endgroup$ – statstudent Dec 10 '16 at 11:43

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