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.
98
questions with no upvoted or accepted answers
9
votes
0
answers
501
views
How do sufficiency statistics help in the interpretation of regression results?
One of the results why canonical link functions are widely used in GLMs is the existence of sufficiency statistics for the regression parameters, which in turn allow for:
... minimal
sufficient ...
- 4,452
6
votes
0
answers
215
views
Does $f : p_\theta\mapsto p_{T\,\mid\,\theta}$ being injective imply statistic $T $ is sufficient?
Wikipedia says
... consider the map $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$. The ...
- 284
5
votes
0
answers
696
views
Intuitive understanding of the Aldous-Hoover representation theorem for row-column exchangeable arrays
I would like to ask a couple of questions about the Aldous-Hoover theorem for the representation of probability distributions over (2D) arrays with exchangeable rows and columns. I'd be happy about ...
- 1,306
4
votes
0
answers
223
views
How to prove or disprove that a complete sufficient statistic exists?
We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
- 545
4
votes
0
answers
75
views
If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?
I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" ...
- 2,564
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,
$$ W_i \sim ...
- 676
4
votes
0
answers
166
views
Sufficient statistics of posterior (with Poisson data)
Suppose that, for year $t$, the data $y$ is Poisson with mean $a + bt$. Assume also a uniform prior on $(a,b)$. If we have $n$ years of data then I think the posterior for $(a,b)$ will be
\begin{...
- 41
3
votes
0
answers
188
views
Minimal sufficient statistic: a measurability issue in a well-known theorem
Given a statistical model $\{\mathbb{P}_\theta\,|\,\theta\in\Theta\}$ on $(\Omega,\mathscr{F})$, and given a real-valued random variable $X$, we say a real-valued random variable $T=T(X)$ is a ...
- 202
3
votes
0
answers
96
views
Given a single sample X from $N(0, \theta)$ is $|X|$ a sufficient statistic for $\theta$?
My first idea on how to proceed was to treat $|X|$ as piecewise where $X=$ $X$ for $X \in [0, \infty)$ and $-X$ for $X \in (-\infty, 0)$, then use the conditional probability definition for a ...
- 31
3
votes
0
answers
377
views
Sufficient statistics and UMVUE for joint poisson, bernoulli
Given a pair $(X,Y)$ of r.v.s such that: $$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
3
votes
0
answers
97
views
Is the posterior a sufficient statistic when observations are conditionally independent?
Suppose there are two random variables, $X_1$ and $X_2$, and we're trying to infer $\theta$.
If $X_1$ and $X_2$ are conditionally independent, then is $f(\theta|X_1)$ a sufficient statistic for $X_1$?...
3
votes
0
answers
4k
views
Why are the fixed effects of a panel probit regression inconsistent?
I was taught that a probit with fixed effects would not be consistent because the estimates of a non-linear model with a link function other than the canonical (in this case the logit) are not ...
- 858
3
votes
0
answers
146
views
Kolmogorov's paper defining Bayesian sufficiency
I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper
Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
- 1,443
2
votes
1
answer
25
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
2
votes
0
answers
42
views
Reference request for the existence of minimal sufficient statistics
I'd like a recent paper or book that shows in what conditions we can guarantee the existence of a minimal sufficient statistic.
I know the paper "Sufficiency and Statistical Decision Functions&...
- 133