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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Sufficient Statistics - Relating the Intuition with the Mathematical Definition

I believe the heuristic definition of a Sufficient Statistic makes sense to me - when you take a sample in order to make an inference about the parameter related to the probability distribution, and ...
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When a function of sufficient statistic is itself sufficient?

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
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Statistics Theory Question

Casella& Berger Theorem 6.2.28: If a minimal sufficient statistics exists, any complete statistics is minimal sufficient. So let's suppose $X_1...X_n$ are iid $Bernoulli(p)$ $p\in (0,1)$, then $\...
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How to identify one-one correspondance in Sufficient Statistics?

The correct answer to the given question is (1),(3) and (4). I understood how 3 and 4 are correct but I could not understand how (1) is also a correct answer. I know that here $\sum_i X_i$ is a ...
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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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Rao blackwell theorem but the unbiased estimator is a function of the sufficient statistic

The Rao-Blackwell Theorem states the following: Let $T(\mathbf X)$ be a sufficient statistic for the statistical model $(S, \{f_{\theta}: \theta \in \Theta\})$ and $\hat \theta(\mathbf X)$ be and ...
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468 views

Is Complete Statistic Uncorrelated with Ancillary Statistic

By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be ...
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How can I find a complete, minimal sufficient statistic from a $Beta(\sigma,\sigma)$ distribution?

Let $X_1,\cdots,X_n$ be a random sample from $Beta(\sigma,\sigma)$, where $\sigma > 0$ is unknown. Is the minimal sufficient statistic for $\sigma$ complete? My work I found the minimal ...
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71 views

Show that $T(\mathbf{X})=(\sum X_i, \sum X_i^2)$ is not complete

Let $X_1, \cdots X_n \stackrel{\text{iid}}{\sim} N(\alpha \sigma, \sigma^2)$, where $\alpha$ is known, and $\sigma > 0$ is unknown. Show that the family of distributions of $$T(\mathbf{X})=(\sum ...
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How can I show that $\sum X_i$ is not a sufficient statistic for $\theta$?

Let $X_1,\ldots, X_n \sim f(x\mid \theta)=\frac{x}{\theta}e^{-x^2/(2\theta)}, x > 0$ independently. $\theta > 0$ is unknown. How can I show that $\sum X_i$ is not a sufficient statistic for $\...
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Does the UMVUE have to be a minimal sufficient statistic?

I'm studying point estimation and I have found this question that seems pretty tricky to me. If $T$ is a minimal sufficient statistic for $\theta$ with $E(T) = \tau(\theta)$, can you say that $T$ ...
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Is it correct to write $\Bbb E[X]$ or $\Bbb E_{\theta}[X]$?

Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known ...
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Is it ok to write $\Bbb E[X|T(X)]$? [duplicate]

Suppose that we observe the discrete random variable $X = (X_1, . . . , X_n)$ with state space $S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known ...
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1answer
82 views

What is the difference between $T(x)$ and $T(X)$?

Suppose that we observe the discrete random variable $X = (X_1, . . . , X_n) $ with state space $S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known ...
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1answer
978 views

Basic intuition about minimal sufficient statistic

As stated by Wikipedia: A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, $S(X)$ is minimal sufficient if and ...
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Proving that $T(X)$ is a sufficient statistic for $\theta$

I have that $X=(X_1,...,X_n)$ is a rv consisting of $n$ iid exponential rv's where $\theta$ is the parameter (and thus mean $1\over \theta$) . I have to prove that $T(X)=\sum^n_{i=1}X_i$ is a ...
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Is it true that Fisher information for a statistic and the sample are equal if and only if the statistic is sufficient?

According to https://en.wikipedia.org/wiki/Fisher_information#Sufficient_statistic we have if and only if, but according to https://projecteuclid.org/download/pdfview_1/euclid.imsc/1362751193 we don'...
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Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
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How to derive a pdf of Complete Sufficient Statistic of exponential family

While studying Mathematical statistics through "Introduction to Mathematical Statistics 7th" (by Hogg and Craig), I've been stuck in the Theorem above. The answer of the exercise 7.5.8 is not given in ...
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replication of minimal sufficient statistic

Suppose we have a minimal sufficient statistic for observations $X_1, ...,X_n$ that are i.i.d from distribution $f(X|\theta)$, namely $T(X) = (T_1,...,T_k)$ which is a $k$ dimensional statistics. Now ...
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Dimension of minimal sufficient statistic

Is this true that "dimension of every minimal sufficient statistic is less than any sufficient statistic (minimal or not)"?
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Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
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Sufficient statistic for p in a binom(1,p)

I don't really understand why the statistic $T = \sum_{i=1}^{n} X_{i}$ is a sufficient statistic for p , for $X_{1}, ... X_{n}\sim^{iid} Binom(1,p)$. Shouldn't it be $\bar{X}$, i.e. $\frac{T}{n}$?
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Finding complete sufficient statistic

Let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such. I know that $T=(X_{(1)}, X_{(n)} )$ is a ...
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Is the sample mean complete sufficient for the Expectation when $X \sim F$ where $F$ is some symmetric distribution?

Let $X_1,X_2... X_n$ be iid $\sim F$ where $F$ is any symmetric continuous distribution and let $\mid E(X)\mid<\infty$. Is $\bar{x}$ complete sufficient for $E(x)=\int{xf(x)dx}$? Assume that all ...
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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.
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Conditioning in the definition of sufficient statistics

Let $X_1,...,X_n$ be an i.i.d. sample with parameter $\theta$ and $T$ a statistics. The statistics is called sufficient if, given a value $t$, the distribution $P_{\theta}(X_1,..,X_n|T=t)$ does not ...
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What is the broadest context where $\bar{X}$ is complete sufficient for $\mathbb E(X)$?

$\bar{X}$ is complete sufficient for $\mathbb E(X)$ if $X$ is Normal with known standard deviation $\sigma$. Are there broader contexts? Like for exponential families in general or more general than ...
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Can the Fisher factorization theorem be understood as a product of densities?

Let $T$ be some random variable on a probability space $\Omega$. Then we have, for $x\in\Omega$: $$P(x) = P(x|T=T(x))P(T = T(x))$$ This equation is nonsense in an arbitrary probability space but ...
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What is exponential family criterion to test the sufficiency and completeness of an estimator?

I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families: Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...
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nonexistence of a sufficient statistic

Let $X_1,X_2,\dots,X_n$ be a random sample from a $\Gamma(\theta,\theta)$ distribution. Then $$ \prod_{i=1}^n f(x_i;\theta) = \frac{1}{\Gamma(\theta)^n\theta^n}(\prod_{i=1}^n x_i)^{\theta-1}e^{-\frac{...
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Compute the information matries related to normal distribution

This is a problem that I have trouble with. Suppose that we have $X_{1}, \ldots, X_{m}$ are iid $N\left(\mu, \sigma^{2}\right), Y_{1}, \ldots, Y_{n}$ are iid $N\left(0, \sigma^{2}\right),$ the $X$...
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309 views

When is further reduction than the order statistics not possible?

Let $X_1, ..., X_n$ be a random sample from a population with location pdf $f(x-\theta)$. Show that the order statistics $T(X_1, ..., X_n) = (X_{(1)}, ..., X_{(n)})$ are a sufficient statistic for $\...
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What is the intuition behind the factorization theorem? (Sufficient statistics)

By the Fisher's factorization theorem, a statistics is a sufficient statistic if (and only if) the joint density, $$ f(x_1, x_2, x_3, \dots x_n; \theta) $$ can be factorized into two functions, $ g(...
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Relationship between completeness and sufficiency

hopefully this isn't a duplicate of another question (at least I didn't find one). Here is a question I have about completeness and sufficiency: Problem: Suppose $T(x)$ is complete sufficient for $\...
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Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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Finding the conditional distribution of single sample point given sample mean for $N(\mu, 1)$

Suppose that $X_1, \ldots, X_n$ are iid from $N(\mu, 1)$. Find the conditional distribution of $X_1$ given $\bar{X}_n = \frac{1}{n}\sum^n_{i=1} X_i$. So I know that $\bar{X}_n$ is a 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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Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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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 (...
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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 ...
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Proving Order Statistics are Minimal Sufficient

I am given continuous i.i.d. random variables $X_1,\dots,X_n$ having an unknown p.d.f. $f$ and am trying to show that the vector of order statistics $(X_{(1)}, \dots, X_{(n)})$ is minimal sufficient ...
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Does Fisher's factorization theorem provide the pdf of the sufficient statistic?

From Wikipedia Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...
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Intutitive meaning behind the formal definition of sufficient statistic?

According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter. What I am ...
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253 views

Showing sufficiency using the Fisher-Neyman factorization theorem

I have derived a likelihood function for $\theta$ as follows: $$L(\theta)=(2\pi\theta)^{-n/2} \exp\left(\frac{ns}{2\theta}\right)$$ Where $\theta$ is an unknown parameter, $n$ is the sample size, ...
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Why do we use $S^2$ while estimating the variance?

Sorry the title is a bit silly, but I currently confront a problem related to Fisher's information. Let $X_1, X_2, \cdots, X_n$ be of $N(\mu , \sigma^2 )$ distribution where $\mu$ is known, $U^2 := ...
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1answer
315 views

UMVUE of a function of sufficient statistic

Let $X_1,\ldots, X_n$ be a random sample from the following distribution: $$f(x\mid\theta)= \frac{4}{\theta}x^3e^{-\frac{x^4}{\theta}}$$ I know that $S = \sum X_i^4$ is a complete and sufficient ...
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556 views

UMVUE for $\theta$ where $X \sim Unif\{1 ,\ldots, \theta\}$

Say we have $X \sim Unif\{1, \ldots , \theta\}$ and we want to find the uniformly minimum variance unbiased estimator for $\theta$. My first assumption was $X_{(n)}$. Which I managed to show is ...
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Obtaining the expected value

Suppose we have $X_1,\dots, X_n \overset{iid}{\sim} N(\mu = 0, \sigma^2 = 1)$, for a known $n$. And we want to calculate $E[X_{(1)} | \bar{X} = c]$, where $c \in \mathbb{R}$ is known, $X_{(1)}$ is the ...

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