# 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.

97 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
499 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 ...
• 4,442
686 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
74 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" ...
• 2,554
120 views

• 207
133 views

### Solving the Neyman-Scott problem via Conditional MLE

In section 2.4 of the book Essential Statistical Inference by Boos and Stefanski, the authors discuss the idea conditional likelihoods and illustrate their usefulness by describing how they can be ...
• 1,501
90 views

126 views

• 903
73 views

### Parametric family problems

I came across such a problem that I cannot solve: Let $\mathcal{P} = \{\mathbb{P}_\theta : \theta \in \mathbb{R}\}$ be a parametric family over $\{0,1\} \times \mathbb{R}$ defined in the following ...
1k views

### Minimal sufficient statistic for two normal distributions

Let $X_1, . . . , X_m, Y_1, . . . , Y_n$ be independent with $X_i ∼ N(ξ, σ^2)$ and $Y_j ∼ N(η, τ^2).$ What is the minimal sufficient statistic for $(ξ,η,σ^2)$ where $σ^2 = τ^2$? I've seen MSS ...
• 903
258 views

### Sufficient statistics in multiparameter exponential family

I'm trying to work through a theorem in the Lehmann statistical inference book and I'm confused about a proof. They are proving that a set of tests are UMP unbiased level-alpha tests for a series of ...
• 229
1k views

### Are complete sufficient statistics unique?

I'm under the impression that up to a one-to-one function complete sufficient statistics are unique. How can I show this?
• 95
1k views

• 10.1k
143 views

### Finding a sufficient statistic

Consider an i.i.d. sample $(X_{1},\ldots, X_{n})$ where the $X_{i}$ have density $f(x) = k \cdot \exp(−(x − θ)^4)$ with $x$ and $\theta$ real, obtain the sufficient statistic and its dimension. What ...
• 49
189 views

### Likelihood estimation using iid normal samples

Given an i.i.d. sample $X = (x_{1}, \dots, x_{n}) \sim N(\mu, 1)$. I have been asked to show that the likelihood of $\mu$ based on the whole sample is proportional to the likelihood based on $\bar{x}$ ...
• 161
1 vote
10 views

### Sufficient statistic as iso-surfaces in the distribution density. Is it possible to generalise to multiple parameters?

For continuous distributions, there is a geometric intuition behind sufficient statistics that regards a multivariate probability density as several iso-surfaces. This works at least for cases where a ...
• 81.5k
1 vote
47 views

### Find minimal sufficient statistic of this random sample with cursed support

Suppose $X_1,X_2,...,X_n$ is a i.i.d random sample with probability mass function $p(x_i,\theta)$ where $x_i \in \{\theta,\theta+1,\theta+2,...\}$ and $\theta \in \mathbb{R}$. I claim that minimal ...
1 vote
38 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 ...
1 vote
62 views

### Find a two dimensional sufficient statistic for $\theta$

Let $\{X_i\}_{i=1}^n$ be conditional independent given $\theta$ with distribution $$p_{X_i | \theta} (x |\theta) = \frac{1}{2i\theta}, \ -i\theta<x<i\theta.$$ Find a two dimensional sufficient ...
• 255
1 vote
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 ...
1 vote
31 views

1 vote
100 views

### Are there any (exponential) families without a minimal sufficient statistic?

Bahadur's theorem says that if a minimal sufficient statistic exists, then a complete sufficient statistic is also minimal sufficient. Are there any (homogenous, identifiable) families with a complete ...
1 vote
56 views

### How does the result $\dfrac{1}{n^T} \dfrac{T!}{\prod_{i = 1}^n Y_i!}$ tell us what distribution $T(\mathbf{Y})$ is?

This follows on from my question here. I have the following problem: Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ ...
• 2,022
1 vote
734 views

### minimal sufficient statistics of 1-parameter Gamma distribution

If $x_i \sim Gamma(\alpha, \alpha)$, are the minimal sufficient statistics still $\Pi_i x_i$ and $\sum_i x_i$ (same as when $x_i \sim Gamma(\alpha, \theta)$ where $\alpha \neq \theta$)? My reasoning ...
• 231
1 vote
157 views

1 vote
29 views

### How to determine data size is statistically efficient?

I have a question about the data size for probability of default model. For each consumer, I have a binary bit to indicate whether the client goes default or not (1 is default and 0 is current). And I ...
• 11
1 vote
48 views

### What's wrong with this proof that the sample sum is sufficient for $\theta$ in $U(0,\theta)$?

So let's say $X_i ~ U(0, \theta)$, and let's consider the two-sample sample sum, $t = \bar{X_2} = (X_1 + X_2)/2$. So we want to show that $p(x|t) = p(x,t)/p(t) = p(x)/p(t)$ is independent of $\theta$....
• 457
1 vote
Suppose we have data $X = X_1,\ldots,X_n$, $Y = Y_1,\ldots,Y_n$ that is i.i.d. generated by a distribution $\mathbb{P}_\theta$. Let $T$ be a test statistic such that that $T(X) = T(Y)$ if and only ...