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

#1
\(

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
#2
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
#3
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
 
#4
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.
 
#6
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.

http://books.google.com/books?id=8E...eW3S6uAIIT6zASJ2shN&cd=1#v=onepage&q=&f=false

pg 32 or there abouts
Best book ever on linear models.

My money is on X_bar is umvue for u.



:shakehead:yup::mad:

t distribution is one of exponential family, isn't it? Since Cauchy distribution is absolutely not in exponential family, I'm not sure if it is correct!