# how to find the UMVUE of the location parameter under t distribution

#### egpivo

##### New Member
$$f(x;\theta)={{\Gamma({{v+1}\over2})}\over{\Gamma({v\over2})\sqrt{\pi v}}}\Big(1+\frac{(x-\theta)^2}{v}\Big)^{-\frac{v+1}{2}} , v>=3$$

i have no idea how to create the UMVUE of $$\theta$$, and please help me , give me some tips.. thx

#### WeeG

##### TS Contributor
hard one.....
I suppose you can't assume that v is known...
I would start by finding a sufficient and complete statistic for $$\theta$$ , but I haven't got a clue right now what it is. Do you ?

#### mp83

##### TS Contributor
Is this the case?

Let (X1, ...,Xn) be a random sample from P∈P containing all symmetric distributions with ﬁnite means and with Lebesgue densities on R.
(i) When n = 1, show that X1 is the UMVUE of µ.
(ii) When n> 1, show that there is no UMVUE of µ = EX .

From Shao, Ex 3.22

#### egpivo

##### New Member
hard one.....
I suppose you can't assume that v is known...
I would start by finding a sufficient and complete statistic for $$\theta$$ , but I haven't got a clue right now what it is. Do you ?
No, I only used the factorization theorem to find a sufficient statistic but haven't got a clue to prove completeness.

#### egpivo

##### New Member
No, I know for normal distribution X_bar and var_hat = stdev^2
are umvue.

there is some theories out there that describe umvue estimation in
exponential family, which includes T distribution as I recall.