All Questions
Tagged with sufficient-statistics exponential-family
57
questions
6
votes
3
answers
135
views
Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?
I am interested in the Pitman-Koopman-Darmois theorem.
I'm having a hard time finding a simple rigorous version of this theorem as I struggle finding sources.
This helpful post provides three sources ...
- 2,628
1
vote
0
answers
43
views
How does reparametrization of the Fisher information matrix change the variance expression for the sufficient statistics?
If I have an exponential family distribution of the form $$p_{\theta}(x) = e^{\theta^T\cdot t(x) - \psi(\theta)},$$ where $\theta$ is a vector of parameters, $t(x)$ is a vector of sufficient ...
- 111
5
votes
1
answer
188
views
A lemma concerning the distribution of sufficient statistic from exponential family
I understand Lemma 8 in Chapter 1 from Lehmann's Testing Statistical Hypotheses [or Lemma 2.7.2 in Lehmann and Romano] as follows:
If the pdf of an exponential family is $$p_{\theta}(x)=\exp\bigg\{\...
- 65
1
vote
1
answer
38
views
Prove covariance between sufficient statistic and logarithm of base measure in exponential family is equal to zero
Exponential family form is
$$f_X(x) = h(x)\exp(\eta(\theta)\cdot T(x) - A(\theta))$$
I know
$$\operatorname{Cov}(T(x), \log(h(x)) = 0.$$
But how can I prove it?
- 11
1
vote
0
answers
54
views
Does this distribution belong to the exponential family? [duplicate]
I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
0
votes
1
answer
76
views
Sufficient Statistic for Absolutely Continuous Distribution [duplicate]
The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes.
Let $x_{1}, x_{2},...,x_{M}$ be i.i.d. samples from the absolute continuous ...
- 101
6
votes
2
answers
123
views
Are These Conjectures Regarding Sufficient Statistics True?
I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true ---...
- 491
2
votes
1
answer
69
views
Is This A Counter-Example To The Theorem by Barndorff-Nielsen-Pedersen (1968)?
In the textbook "Theory of Point Estimation" 2nd Ed. by Lehmann and Casella, Theorem 6.18 states:
Suppose $X_1, ..., X_n$ are real-valued IID according to a distribution with density $f_\...
- 491
0
votes
1
answer
108
views
Sufficient statistics for a non exponential
I think this is not an exponential family but does it mean that we can't find a sufficient statistic for $\theta$ if $X_1, X_2,..., X_n$ are a random sample from this density?
$$ f_{\theta} (x) = \...
- 11
1
vote
1
answer
69
views
exponential sufficient statistics [duplicate]
A family of pdfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ and the statistic $T$ is sufficient iff $f(x;\theta) = h(x)...
- 11
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
$$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
- 2,096
1
vote
2
answers
428
views
Showing that $f_\varphi(x)$ is a member of the one-parameter exponential family and $\sum_{i = 1}^n - \log(X_i)$ is sufficient for $\varphi$
Let $X_1, \dots, X_n$ denote a random sample from the PDF
$$f_{\varphi}(x)=
\begin{cases}
\varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\
0 &\text{otherwise}
\end{...
- 2,096
0
votes
0
answers
167
views
Sufficient statistic for a given distribution from exponential form
Given a particular form, i can verify whether it is sufficient statistic or not using $\frac{p_\theta(x_1,x_2...x_n)}{p_\theta(T(x_1,x_2...x_n))}$ is independendent of $\theta$ then i can say $T(\bar ...
10
votes
1
answer
597
views
Does a sufficient statistic imply the existence of a conjugate prior?
In the comments on this answer, user Scortchi asks:
So iff there's a sufficient statistic of constant dimension, there's a conjugate prior?
As far as I know this didn't get a complete answer, so I'm ...
- 425
9
votes
1
answer
417
views
Why is the EM algorithm well suited for exponential families?
I've been brushing up on the EM algorithm, and while I feel like I understand the basics, I keep seeing the claim made (e.g. here, here, among several others) that EM works particularly well for ...
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