# Proving the nonexistence of UMVUE for $\text{Unif}\{\theta-1, \theta, \theta+1\}$

I am trying to prove that

There is no UMVUE for $$\theta$$ for the distribution $$\text{Unif}\{\theta-1, \theta, \theta+1\}$$, $$\theta$$ is an integer.

Here is what I have attempted.

I am trying to use the theorem 1.7 from Lehmann and Casella's book but I got stuck.

The theorem says

Let $$X$$ have a distribution $$\mathbb{P}_\theta$$, $$\theta\in\Theta$$, let $$\delta$$ be an estimator so that $$\mathbb{E}_\theta\delta^2<\infty$$, and let $$\mathcal{U}$$ denote the set of all unbiased estimator of zero, which also satisfy $$\mathbb{E}_\theta U^2<\infty,\forall U\in\mathcal{U}$$. Then $$\delta$$ is a UMVUE for its expectation $$g(\theta)=\mathbb{E}(\delta)$$ if and only if $$\mathbb{E}_\theta(\delta U)=0,\forall U\in\mathcal{U}\text{ and }\theta\in\Theta.$$

Suppose $$\{X\}_{i=1}^n\overset{i.i.d}{\sim}\text{Unif}\{\theta-1, \theta, \theta+1\}$$. Thus, $$\mathbb{P}(X=\theta-1)=\mathbb{P}(X=\theta)=\mathbb{P}(X=\theta+1)=1/3$$.

All the unbiased estimator of $$0$$ is of the form $$\mathcal{U}=\{a\mathbb{1}_{(X=\theta-1)}+b\mathbb{1}_{(X=\theta)}+c\mathbb{1}_{(X=\theta+1)},\text{ }\forall a,b,c\in\mathbb{R},a+b+c=1\}$$ This is because $$\forall U\in\mathcal{U}$$, we have $$\mathbb{E}(U)=1/3(a+b+c)=0.$$

Also, notice $$\mathbb{E}(X)=\theta$$

I was trying to use the theorem, but I do not know how to proceed further.

• Your set $\mathcal U$ is not made of estimators since the expressions involve $\theta$. Jan 25 '21 at 19:01
• Ha, that's right. I probably need $X_{(n)}-X_{(1)}$ to get the unbiased estimator of zero. Thanks.
– Tan
Jan 25 '21 at 20:11
• Not sure if this helps but there is no complete sufficient statistic in this model: stats.stackexchange.com/q/61990/119261. May 5 '21 at 15:55

Let's first assume there exist a UMVUE $$\hat\theta$$ for $$\theta$$. Then, by the theorem, any zero estimator $$\eta$$ will have:
1. $$E_{\theta}\eta = 0$$, for any $$𝜃∈Θ$$.
2. $$E_𝜃[\eta \hat \theta]=0$$
Let's start with the first term: $$E_𝜃[\eta ]=\frac{1}{3}[\eta(\theta-1) + \eta(\theta) + \eta(\theta+1)] = 0$$ Then any function that satisfies $$\eta(x-1) + \eta(x) + \eta(x+1) = 0$$ will meet the requirement. This surely can not derive $$E_{\theta}(\eta \hat\theta)= 0$$, so in this case there is no UMVUE.