I've recently started studying statistical inference. I've been working through various problems and this one has me completely stumped.

Let $X_1,\dots,X_n$ be a random sample from a discrete distribution which assigns with probability $\frac{1}{3}$ the values $\theta-1,\space\theta,\space\text{or}\space\theta+1$, where $\theta$ is an integer. Show that there does not exist a complete sufficient statistic.

Any ideas?

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    $\begingroup$ What do you have so far? $\endgroup$ – gung - Reinstate Monica Jun 18 '13 at 3:09
  • $\begingroup$ I can write the likelihood as: $(\frac{1}{3})^n$ times the product of the indicator functions that each observation is equal to either $\theta-1,\space\theta,\space\text{or}\space\theta+1$. From this it looks like the sufficient statistic is the order statistics. I've been thinking about this for days, it's like nothing I've seen before. $\endgroup$ – Tony Jun 18 '13 at 3:21
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    $\begingroup$ What do you know about completeness? $\endgroup$ – Glen_b Jun 18 '13 at 4:59
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    $\begingroup$ A statistic, $T$, is complete if it satisfies the condition that, for some function $g(T)$, if $E[g(T)]=0$, then $g(T)=0$ $a.e.$ $\endgroup$ – Tony Jun 18 '13 at 5:06
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    $\begingroup$ So you need to find a counter-example ... what clearly ancillary statistic can you find from the sample minimum & maximum? $\endgroup$ – Scortchi - Reinstate Monica Jun 18 '13 at 12:56

(1) Show that for a sample size $n$, $T=\left(X_{(1)}, X_{(n)}\right)$, where $X_{(1)}$ is the sample minimum & $X_{(n)}$ the sample maximum, is minimal sufficient.

(2) Find the sampling distribution of the range $R=X_{(n)}-X_{(1)}$ & hence its expectation $\newcommand{\E}{\operatorname{E}}\E R$. It will be a function of $n$ only, not of $\theta$ (which is the important thing, & which you can perhaps show without specifying it exactly).

(3) Then simply let $g(T)=R-\E R$. It's not a function of $\theta$, & its expectation is zero; yet it's not certainly equal to zero: therefore $T$ is not complete. As $T$ is minimal sufficent, it follows from Bahadur's theorem that no sufficient statistic is complete.

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  • $\begingroup$ Could you give a reference for Bahadur' theorem where it states that if a minimal sufficient statistic is not complete, then a complete sufficient statistic does not exist? I was searching for this result, but could not find it anywhere. $\endgroup$ – StubbornAtom Jul 24 '18 at 10:00
  • $\begingroup$ @StubbornAtom: Bahadur's theorem states that if a statistic's complete it's minimal sufficient (providing a minimal sufficient statistic exists at all). So once you show that a minimal sufficient statistic exists & is incomplete, you don't need to worry about the possibility of complete non-minimal sufficient statistics. (Or of course about the possibility of other complete minimal sufficient statistics - they're all one-to-one functions of each other.) $\endgroup$ – Scortchi - Reinstate Monica Jul 24 '18 at 20:10
  • $\begingroup$ Thinking about it, it would've been simpler just to say that $T$, being minimal sufficient, is some function $f(\cdot)$ of any sufficient statistic $S$, & therefore $g(T)=g(f(S))$ also goes to show the incompleteness of $S$. $\endgroup$ – Scortchi - Reinstate Monica Jul 24 '18 at 20:44

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