All Questions
Tagged with sufficient-statistics unbiased-estimator
18
questions
2
votes
1
answer
35
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
0
votes
1
answer
120
views
Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]
Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$.
$(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$?
To ...
- 345
5
votes
1
answer
320
views
How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?
Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family
$$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
- 2,096
4
votes
1
answer
66
views
Does an estimator need to be unbiased in order to be sufficient?
I am reviewing some theoretical statistics content, and I was wondering if an estimator need to be unbiased in order to be sufficient? Is there any way to prove this? Thanks!
- 504
2
votes
1
answer
164
views
For iid $X_1, \dots, X_n \sim N(0,\sigma^2)$, get sufficient statistic $T = \sum_{i=1}^nX_i^2$, how to find unbiased estimator of $\sigma^a$
For $X_1, \dots, X_n \sim N(0,\sigma^2)$, we define a sufficient statistic $T = \sum_{i=1}^nX_i^2$. There is a positive number $a$. My question is how to find unbiased estimator of $\sigma^a$ using ...
- 21
0
votes
0
answers
217
views
Rao blackwell theorem but the unbiased estimator is a function of the sufficient statistic
The Rao-Blackwell Theorem states the following:
Let $T(\mathbf X)$ be a sufficient statistic for the statistical model $(S, \{f_{\theta}: \theta \in \Theta\})$ and $\hat \theta(\mathbf X)$ be and ...
user255658
0
votes
1
answer
503
views
Finding the conditional distribution of single sample point given sample mean for $N(\mu, 1)$
Suppose that $X_1, \ldots, X_n$ are iid from $N(\mu, 1)$. Find the conditional distribution of $X_1$ given $\bar{X}_n = \frac{1}{n}\sum^n_{i=1} X_i$.
So I know that $\bar{X}_n$ is a sufficient ...
- 51
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
10
votes
1
answer
484
views
Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?
General description
Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$?
Say we compare the ...
- 82.2k
3
votes
1
answer
265
views
Comparing variances of two unbiased estimators
This question is from a Ph.D Qualifying Exam for Mathematical Statistics. Main reference is Casella & Berger's Statistical Inference.
Let $W_1$ and $W_2$ be unbiased estimators of a parameter $\...
- 413
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
3
votes
2
answers
2k
views
Best unbiased estimator for a location family
Mainly for pedagogical reasons, I'm considering the "simple" one dimensional model:
$$x=\theta+\epsilon$$
where $\epsilon$ has a known distribution $p$ that is independent of $\theta$. This ...
- 8,707
3
votes
1
answer
955
views
Are MVUEs and MLEs always functions of a minimal sufficient statistic?
Is it the case that both minimum variance unbiased estimators (MVUEs) and maximum likelihood estimators (MLEs) are always functions of a minimal sufficient statistic?
If so, how do we know? If not, ...
- 713
2
votes
1
answer
264
views
Unbiased estimator and sufficient statistics [closed]
Let $X_1,..,X_n$ be a random sample of $f(x;\theta)=\theta
x^{\theta-1}I_{[0,1]}(x)$
Find a sufficient statistic for $\theta$ and construct a unbiased
estimator for $\theta$ as a function of ...
user72621
3
votes
0
answers
377
views
Sufficient statistics and UMVUE for joint poisson, bernoulli
Given a pair $(X,Y)$ of r.v.s such that: $$X \sim
\text{Poisson}(\lambda)\quad \text{and}\quad Y \sim
B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a
one-dimensional sufficient ...
- 31