Questions tagged [sufficient-statistics]
A sufficient statistic is a lower dimensional function of the data which contains all relevant information about a certain parameter in itself.
466
questions
33
votes
2
answers
2k
views
When if ever is a median statistic a sufficient statistic?
I came across a casual remark on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic but, besides the obvious case of one or two observations where it ...
- 107k
32
votes
3
answers
3k
views
How can a statistician who has the data for a non-normal distribution guess better than one who only has the mean?
Let's say we have a game with two players. Both of them know that five samples are drawn from some distribution (not normal). None of them know the parameters of the distribution used to generate the ...
- 2,600
31
votes
3
answers
4k
views
Sufficient statistics for layman
Can someone please explain sufficient statistics in very basic terms? I come from an engineering background, and I have gone through a lot of stuff but failed to find an intuitive explanation.
- 1,351
30
votes
2
answers
3k
views
Why do we not care about completeness, sufficiency of an estimator as much anymore?
When we begin to learn Statistics, we learn about seemingly important class of estimators that satisfy the properties sufficiency and completeness. However, when I read recent articles in Statistics I ...
- 413
23
votes
7
answers
3k
views
Why does a sufficient statistic contain all the information needed to compute any estimate of the parameter?
I've just started studying statistics and I can't get an intuitive understanding of sufficiency. To be more precise I can't understand how to show that the following two paragraphs are equivalent:
...
- 381
22
votes
3
answers
7k
views
Jointly Complete Sufficient Statistics for Uniform$(a, b)$ Distributions
Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that ...
- 221
16
votes
1
answer
2k
views
Sufficient statistic, specifics/intuition problems
I'm teaching myself some statistics for fun and I have some confusion regarding sufficient statistics. I'll write out my confusions in list format:
If a distribution has $n$ parameters then will it ...
- 161
14
votes
1
answer
1k
views
Intuitive understanding of the Halmos-Savage theorem
The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...
- 3,104
13
votes
2
answers
1k
views
How does Bayesian Sufficiency relate to Frequentist Sufficiency?
The simplest definition of a sufficient statistics in the frequentist perspective is given here in Wikipedia. However, I recently came across in a Bayesian book, with the definition $P(\theta|x,t)=P(\...
- 5,822
12
votes
1
answer
2k
views
What is "Likelihood Principle"?
While I was studying "Bayesian Inference", I happen to encounter the term, "Likelihood Principle" but I don't really get the meaning of it. I assume it is connected to "...
- 3,525
12
votes
3
answers
4k
views
Solution to German Tank Problem
Is there a formal mathematical proof that the solution to the German Tank Problem is a function of only the parameters k (number of observed samples) and m (maximum value among observed samples)? In ...
- 295
12
votes
1
answer
510
views
Sufficiency or Insufficiency
Consider a random sample $\{X_1,X_2,X_3\}$ where $X_i$ are i.i.d.
$Bernoulli(p)$ random variables where $p\in(0,1)$. Check if
$T(X)=X_1+2X_2+X_3$ is a sufficient statistic for $p$.
Firstly, how ...
- 1,565
12
votes
1
answer
1k
views
Do the mean and the variance always exist for exponential family distributions?
Assume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f.
$$ f_X(x|\boldsymbol \theta) = h(x) \exp\left(\sum_{i=1}^s \eta_i({\boldsymbol \theta}) T_i(x) - A({\...
- 345
11
votes
1
answer
643
views
Is there a standard measure of the sufficiency of a statistic?
Given a parametrical model $f_\theta$ and a random sample $X = (X_1, \cdots, X_n)$ from this model,
a statistic $T(X)$ is sufficient if the distribution of $X$ given $T(X)$ doesn't depend on $\theta$.
...
- 2,628
11
votes
1
answer
2k
views
Proof of Pitman–Koopman–Darmois theorem
Where can I find a proof of Pitman–Koopman–Darmois theorem? I have googled for quite some time. Strangely, many notes mention this theorem yet none of them present the proof.
- 1,716