All Questions
Tagged with sufficient-statistics umvue
16
questions
2
votes
1
answer
31
views
Prove that $T$ is a complete statistic and find a UMVUE for $p$
While preparing for my prelims, I came across this problem:
Let $X_1, X_2,\cdots, X_n$ be a sequence of Bernoulli trials, $n \geq 4.$ It is given that, $X_1,X_2,X_3 \stackrel{\text{i.i.d.}}{\sim} Ber(\...
- 21
1
vote
0
answers
100
views
Finding UMVUE of a parameter in form of probability of discrete random variables
We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$.
Their pmf's are:
$f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$
$f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
- 37
0
votes
1
answer
294
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UMVUE for P(X > k) in exponential distribution [duplicate]
I have to find UMVUE for
$exp(-k*a)$ where X ~ Exponential(a); k is a positive real number.
I tried it using Lehmann-Scheffe theorem.
Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
- 107
0
votes
0
answers
338
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UMVUE of Bernoulli random variables
Let $X_1, X_2..... X_n$ be a random sample from a Bernoulli population with parameter $p$.
A sufficient statistic is $\sum_{i=1}^{n}X_i$. If we define
$$
U(X_1,X_2,\ldots,X_n)= \begin{cases}1/2n &\...
- 115
1
vote
1
answer
386
views
Unbiased Estimator based on Sufficient Statistic
suppose $X_1, ... , X_n$ are iid with pdf
$f(x|\beta) = e^{-(x-\beta))}I_{(\beta, \infty)}(x)$
and the pdf of ( the smallest order statistic) $X_{(1)}$ is given by
$f_{X_1}(x)$ = n $ *$ $e^{n(\...
- 213
2
votes
1
answer
187
views
Proving the MVUE is the following
I am stuck on the following question and I was wondering if can get some help.
Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
- 31
1
vote
0
answers
49
views
Sufficient statistic for the mean of a generic distribution?
Is there such thing as a "sufficient statistics for the expectation?"
From what I understand, a sufficient statistic is defined only when there is a family of distributions parametrized by $\theta$ ...
- 175
4
votes
0
answers
563
views
The relationship between UMVUE and complete sufficient statistic [duplicate]
Let $X_1,...X_n$ $U(-\theta , \theta)$
I want to find the UMVUE of $\theta$ if it is exists.
My answer is , there is no UMVUE in this case.
Because there is no complete sufficient statistic that ...
- 889
2
votes
2
answers
1k
views
UMVUE for $\theta$ where $X \sim Unif\{1 ,\ldots, \theta\}$
Say we have $X \sim Unif\{1, \ldots , \theta\}$ and we want to find the uniformly minimum variance unbiased estimator for $\theta$.
My first assumption was $X_{(n)}$. Which I managed to show is ...
- 770
1
vote
1
answer
4k
views
Finding sufficient statistic for Weibull density function
I am given the follow problem and am having trouble finding the sufficient statistic.
Suppose that Y$_1$, Y$_2$, ..., Y$_n$ denote a Weibull density function, given by:
f ( y | $\theta$ ) =
Let $...
- 389
1
vote
1
answer
99
views
Find the UMVUE of $U(n_1,θ)$ where $θ>n_1$
Let $X_1,X_2,\dots ,X_n$ follows $U(n_1,θ)$ where $θ>n_1$ and $n_1\le n$. Find the UMVUE of $θ$.
My answer is: $$\frac{(n+1)X_n}n - \frac{n_1}n$$
Is that correct?
- 141
4
votes
0
answers
120
views
Relating sufficient statistics to parameters
I'm studying sufficient statistics and I came across this problem:
A dataset consists of independent triples $(W_i,Y_i,Z_i)$ of independent random variables with distributions as follows,
$$ W_i \sim ...
- 676
2
votes
1
answer
508
views
UMVUE of a function of sufficient statistic
Let $X_1,\ldots, X_n$ be a random sample from the following distribution:
$$f(x\mid\theta)= \frac{4}{\theta}x^3e^{-\frac{x^4}{\theta}}$$
I know that $S = \sum X_i^4$ is a complete and sufficient ...
- 95
2
votes
1
answer
280
views
How does one can guarantee that any unbiased estimator is MVUE due to it containing a minimal sufficient statistic?
Sufficiency is okay. But I don't really get it why the fact guarantees it has minimal variance? Can anyone explain to me somewhat intuitively?
- 23
10
votes
2
answers
2k
views
Find the unique MVUE
This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388.
Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta<x<2\theta,$ ...
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