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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.

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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 ...
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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 ...
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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" ...
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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 ...
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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{...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $X_1$?...
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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 ...
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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. ...
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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&...
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Sufficient Statistic for a family of distributions consisting of Poisson family and Bernoulli family

Suppose $(X_1, . . . ,X_n)$ is an i.i.d. sample from the distribution $f_{\theta,k}(x)$, where $\theta \in (0, 1)$ and $k = 1, 2$. Assume that $$f_{\theta, k}(x)=\begin{cases} \text{Poisson($\theta)$},...
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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 ...
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Sufficiency and completeness of truncated distribution

[From Theory of Point Estimation (Lehmann and Casella, 1999, Exercise 6.37)] Let $P=\{P_\theta:\theta \in \Theta\}$ be a family of probability distributions and assume that $P_\theta$ has pdf $p_\...
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What is the space that a class of probability distributions spans when T is a complete sufficient statistic?

There are a few good posts/notes (see here, and here) giving high level geometric intuition of a complete statistic ($E_{T}[g(T); \theta] = 0 \Rightarrow P(g(T)=0; \theta) = 1 \text{ almost everywhere}...
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Obtaining the expected value $E[X_{(1)} \mid\overline X = c]$

Suppose we have $X_1,\dots, X_n \overset{\text{iid}}{\sim} N(\mu = 0, \sigma^2 = 1)$, for a known $n$. And we want to calculate $E[X_{(1)} \mid \overline X = c]$, where $c \in \mathbb{R}$ is known, $...
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Does family of sufficient $\sigma$-subalgebras depend on the reference measure?

Let $\{ P_{\gamma} \}$ be a parametric family of probability measures on $(\Omega, \mathcal{F})$, such that $P_{\gamma} \ll \mu$ for all $\gamma$, for some $\sigma$-finite $\mu$. Consider the Radon-...
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Does any choice of sufficient statistic and natural parameter create valid exponential family pdfs?

So exponential families can be parameterized as $f(x|\theta) = h(x)e^{T(x)n(\theta)-A(n)}$. I'm trying to understand what the conditions are on $h(x)$, $T(x)$, and $n(\theta)$ are for $f(x|\theta)$ ...
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Sufficiency and completeness of distribution

Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf $p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\ ...
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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 ...
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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 ...
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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 ...
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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?
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Complete sufficient statistic for $f(x|\theta) = \frac{1}{6\theta^4}x^3e^{-x/\theta}$ and the UMVUE for $\theta$

Looking to find a complete sufficient statistic for the following pdf. $X_1, ..., X_n$ from a random sample with distribution: $$f(x|\theta) = \frac{1}{6\theta^4}x^3e^{-x/\theta}$$ $$ 0 \lt x \lt \...
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How to find conditional distribution for Rao-Blackwellizing an estimator?

Let's say I have an unbiased estimator $u(\underline x)$ for function $v(\theta)$ where $\theta$ is a parameter of the distribution of $x$, and $T(\underline x)$ which is a sufficient statistic for $\...
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Sufficient Statistic for the beta distribution with one fixed shape parameter

How can I show that the geometric mean $( \prod_{i=1}^{n} x_i )^{1/n}$ of a random sample of size $n$ from a distribution with pdf $f(x;\theta)=\theta x^{\theta-1},0<x<1,$, zero elsewhere, and $...
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3 parameter Exponential Family and sufficient statistics

This is a homework problem. I've derived the following distribution from an earlier part in the problem $$ f_{X_1,X_2}(x_1,x_2) = \dfrac{\Gamma(x_1+x_2+r)\alpha_1^{x_1}\alpha_2^{x_2}\theta^r}{\Gamma(r)...
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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 ...
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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 $\bar{x}$ ...
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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 ...
Sextus Empiricus's user avatar
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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 ...
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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 ...
absolutelyzeroEQ's user avatar
1 vote
1 answer
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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 ...
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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 ...
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How do I know which statistic is for which parameter when calculating joint sufficient statistics using factorization criteria?

For the normal distribution for example, after factorization we get $\mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{n\mu^2}{2\sigma^2}\right) \exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=...
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Sufficient Statistic for a finite family of Normal distributions

Suppose we have a finite family of normal distributions $P=\{N(0, 1), N(0, 2), N(1, 2), N(2, 2)\}$ and we want to find a sufficient statistic for this family. Intuitively it is clear that as the means ...
user671269's user avatar
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Finding UMVUE of a parameter in form of probability of discrete random variables

We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$. Their pmf's are: $f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$ $f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
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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 ...
Christian Chapman's user avatar
1 vote
2 answers
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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 $\text{Pois}(\lambda)$ ...
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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 ...
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Is the Pearson Product Moment Correlation Coefficient (PPMCC) a sufficient statistic for a bivariate normal distribution?

The PPMCC can be defined as the following: $$\rho_{X,Y} = \frac{\mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]}{\sqrt{\mathbb{E}[X^2] - (\mathbb{E}[X])^2}\sqrt{\mathbb{E}[Y^2] - (\mathbb{E}[Y])^2}}...
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Sufficient Statistic for Variance (Normal Distribution)

Suppose $X_1, \dots, X_n \sim N(\mu, \sigma^2)$. If $\mu$ is known and $\sigma^2$ is unknown, prove that $S^2$ is a sufficient statistics for $\sigma^2$. Likelihood: $$ L = (2\pi\sigma^2)^{-n/2} \cdot ...
James Hampshire 's user avatar
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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 ...
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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 $\theta$....
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Minimal sufficient statistics

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 ...
User197307's user avatar
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1 answer
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Gaussian sufficient statistic calculation

Consider the Gaussian model $$ Y_i = \beta + \epsilon_i,\, i = 1, \cdots, n,\; \mbox{where}\; \epsilon_i \stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^2), $$ parametrized by $\beta$, with known $\...
Michael's user avatar
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Prove that the Pitman estimator is itself complete sufficient

It's a homework question, but I just have no idea about it ... Let be random variables according to a distribution having joint density ,where is a location parameter. Assume that there exists a ...
Tsy's user avatar
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Sufficient statistic for the mean of a generic distribution?

Is there such thing as a "sufficient statistics for the expectation?" From what I understand, a sufficient statistic is defined only when there is a family of distributions parametrized by $\theta$ ...
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