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Is there a case when uniform distribution posses unbiased and umvue estimate of $\theta$?
Suppose $X_1, X_2, \dots, X_{45}$ ~ uniform on interval $[\theta-1/2\ , \theta+1/2]$

My views: I know $\max(X_1, X_2, \dots, X_{45})$ and $\min(X_1, X_2, \dots, X_{45})$ are mle.
2 sample mean is unbiased in case when lower limit is $0$ for uniform distribution.
What gave me doubt is.. this is a symmetric distribution, so what property does sample mean and median has for this distribution over this limits?

Edit : Adding more details, My problem is I want to know if sample mean and sample median are unbiased for this particular distribution.? My intuition says yes, but I couldn’t connect it with the theory.

User0405
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  • The distribution is symmetric about $\theta$, which is both the population mean and median. Consequently sample median is unbiased for $\theta$ here (sample mean is of course unbiased for population mean). But there is no UMVUE of $\theta$. – StubbornAtom Feb 25 '21 at 18:18
  • Yes.. agree but how did you reach to that conclusion, can you suggest some readings on it? – User0405 Feb 26 '21 at 00:48

1 Answers1

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Actually the likelihood is

$$L(\theta)=\mathbb{1}_{[x_{(n)}-0.5;x_{(1)}0.5]}(\theta)$$

Thus any value in the interval is MLE. As an example you can take the midrange

$$T_1=\frac{x_{(1)}+x_{(n)}}{2}$$

as a MLE.

Also $T_2=(X_{(1)},X_{(n)})$ is sufficient, minimal but the problem is that this estimator is NOT complete.

In fact the given density belongs to the Location Family and $(x_{(n)}-x_{(1)})$ is location invariant too, thus it is also ancillary and thus

$$\mathbb{E}[X_{(n)}-X_{(1)}]=C$$

where C does not depend on $\theta$ which implies that, $\forall \theta$

$$\mathbb{E}[X_{(n)}-X_{(1)}-C]=0$$

and clearly, $\forall \theta$

$$\mathbb{P}[X_{(n)}-X_{(1)}-C=0] \neq 1$$

With these preliminary statements, perhaps it is better that you add all the details of your problem.

tommik
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  • This answer is helpful, further I have added details to the questions as asked – User0405 Feb 25 '21 at 13:24
  • @User0405 Sample mean is obviously unbiased for $\theta$. As sample median is concerned I think it is not difficult using Order Statistics. Have you already tried in this way? – tommik Feb 25 '21 at 13:30
  • In what all cases median will be unbiased and when is it not? Is it only when distribution is symmetric, for uniform cases only? – User0405 Feb 26 '21 at 00:50