All Questions
Tagged with sufficient-statistics factorisation-theorem
17
questions
4
votes
2
answers
500
views
Why is the weak likelihood principle not a theorem?
The weak likelihood principle (WLP) has been summarized as: If a sufficient statistic computed on two different samples has the same value on each sample, then the two samples contain the same ...
- 642
3
votes
1
answer
115
views
Sufficiency for Truncated Geometric
Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics:
Let
$$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
0
votes
0
answers
72
views
How do I proceed from here for factorization of likelihood / joint normal density for finding sufficient statistics?
I'm trying to show that the statistic $\left(\sum_{i = 1}^n Y_i, \sum_{i = 1}^n Y_i^2 \right)$ is sufficient for $\mu$ where $(Y_1, \dots, Y_n)$ is a random sample from $N(\mu, \mu)$ for $\mu > 0$. ...
- 2,096
0
votes
0
answers
56
views
Order Statistics - simple random sampling without replacement
I am studying the Watson-Royall theorem and I have a question in item 2:
"under simple random sampling, the vector of order statistics is sufficient ..."
I wish to use the factorisation ...
5
votes
1
answer
976
views
What is the intuition behind the factorization theorem? (Sufficient statistics)
By the Fisher's factorization theorem, a statistics is a sufficient statistic if (and only if) the joint density,
$$ f(x_1, x_2, x_3, \dots x_n; \theta) $$
can be factorized into two functions, $ g(s; ...
- 777
2
votes
1
answer
155
views
Problem on sufficient statistics
Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known.
Show that there is ...
3
votes
1
answer
815
views
Showing the sample mean is a sufficient statistics from an exponential distribution
Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\theta$), where $\theta>0$ is unknown. If we observe $\overline x = 5.2$ for a sample of $20$ light ...
- 447
0
votes
0
answers
88
views
Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem [duplicate]
Question:
Given, $X\sim N(0,\sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $\sigma^2$.
My Attempt:
This problem is very easy if we use Fisher–Neyman ...
1
vote
1
answer
2k
views
When a function of sufficient statistic is itself sufficient?
I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
- 571
8
votes
2
answers
2k
views
Puzzled by the definition of sufficient statistics in Mood, Graybill, and Boes
I am learning about sufficient statistic from Mood, Graybill, and Boes's Introduction to the Theory of Statistics. I am slightly confused by the book's definition of a sufficient statistic for ...
-1
votes
1
answer
53
views
Proving sufficiency by showing ratio of statistic pdf to sample pdf is independent of unknown parameter
Let $X_1,...,X_n$ be iid random variables with densities given by $$ f_{x_i}(x|\theta)=e^{i\theta - x}\mathbb{I}_{(i\theta,\infty)}(x), $$ when $x>i\theta $ and $x=0$ otherwise. Let $T$ be the ...
- 1,276
1
vote
0
answers
226
views
Minimal Sufficient Statistic for Gaussians with different means
I have the following problems on my Statistics course (using Casella and Berger's book) problem set:
1) Let $Y_{i} = X_{i}'\theta + U_{i}$ where $\theta \in \mathbb{R}^k$
and $U_{i}$ are iid $N(0,...
1
vote
1
answer
1k
views
Showing sufficiency using the Fisher-Neyman factorization theorem
I have derived a likelihood function for $\theta$ as follows:
$$L(\theta)=(2\pi\theta)^{-n/2} \exp\left(\frac{ns}{2\theta}\right)$$
Where $\theta$ is an unknown parameter, $n$ is the sample size, ...
- 1,627
0
votes
0
answers
97
views
Is an injective statistic always sufficient?
New to the concept of sufficient statistics. Does it follow, in general, from the factorization theorem that any injective statistic is necessarily a sufficient statistic? I cannot find anything ...
- 229
2
votes
2
answers
510
views
Can the Fisher factorization theorem be understood as a product of densities?
Let $T$ be some random variable on a probability space $\Omega$. Then we have, for $x\in\Omega$:
$$P(x) = P(x|T=T(x))P(T = T(x))$$
This equation is nonsense in an arbitrary probability space but ...
- 439