All Questions
Tagged with sufficient-statistics likelihood
25
questions
1
vote
1
answer
53
views
Birnbaum's Theorem: Strong belief in a model $\implies$ the likelihood function must be used as a data reduction device?
Working through understanding section 6.3.2 (pg. 292-294) in Casella and Berger's Statistical Inference (2nd-ed).
The following definitions and principles are given:
Definition (Experiment): An ...
2
votes
1
answer
41
views
Why does the sufficient statistic for the bivariate normal not imply a sufficient statistic for the correlation under bivariate normality?
This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven ...
- 642
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
5
votes
3
answers
165
views
Likelihood principle and inference
I've been reading Casella and Berger's Statistical Inference. In section 6.3 the author stated the likelihood principle: if the likelihood functions from two samples are proportional, then the ...
- 53
1
vote
0
answers
69
views
Likelihood ratio as minimal sufficient statistics in infinite parameter space
I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first:
Consider a family of density functions $f(x|\theta)$ where the ...
- 31
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
$$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
6
votes
1
answer
1k
views
What is the score function of two parameters?
According to this wikipedia article, score is the derivative of the log-likelihood function. However, I don't understand what if we have two parameters? For example, the logarithm of pdf has the ...
12
votes
1
answer
2k
views
What is "Likelihood Principle"?
While I was studying "Bayesian Inference", I happen to encounter the term, "Likelihood Principle" but I don't really get the meaning of it. I assume it is connected to "...
- 3,525
4
votes
1
answer
1k
views
Likelihood Ratio and Sufficient Statistics
I am not very experienced with statistics, so I apologize if this is an incredibly basic question. A book I am reading (Examples and Problems in Mathematical Statistics - Zacks) makes the following ...
- 141
1
vote
1
answer
1k
views
Poisson sufficient statistics problem
I have the following problem:
Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ distribution has the probability function ...
- 2,096
1
vote
1
answer
173
views
Definition of $k$-parameter exponential family
I am currently studying the concept of sufficient statistics in mathematical statistics. The following definition is presented:
Definition: $k$-parameter exponential family
Let $\mathbf{Y} \sim f_\...
- 2,096
0
votes
1
answer
35
views
Variance of sum calculation in example illustrating completeness for minimally sufficient statistic
I have an example where it is said that $$\sum_{i = 1}^n Y_i \sim N(n \mu, n a^2 \mu^2)$$ and
$$\begin{align} E \left[ \left( \sum_{i = 1}^n Y_i \right)^2 \right] &= \text{Var} \left( \sum_{i = 1}^...
- 2,096
0
votes
1
answer
294
views
Joint distribution simplification in minimal sufficient statistics proof
My notes introduce the concept of minimal sufficient statistics as follows:
Definition
A sufficient statistic $T(\mathbf{Y})$ is called a minimal sufficient statistic if it is a function of any other ...
- 2,096
3
votes
1
answer
8k
views
Sufficient statistics in the uniform distribution case
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
- 2,096
6
votes
1
answer
1k
views
Sufficient statistics are not unique?
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
- 2,096