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 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Problem Deriving Expected Sufficient Statistic

Any exponential family distribution can be expressed as $p(x|\theta) = g(\theta) f(x) e^{\phi(\theta)^T T(x)} = f(x) e^{\phi(\theta)^T T(x) - A(\theta)}$ where $A(\theta) = -\log{g(\theta)}$. We ...
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Reducing dimension $P(\theta|y) = P(\theta|s)$ in the posterior distribution

Given a sample of $n$ independent observations $\boldsymbol{y}$. Let $S(\boldsymbol{y})$ be a sufficient statistic for the underlying parameter $\boldsymbol{\theta}$ so that the density can be ...
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“Magical” variance reduction problem

I recently came across this toy problem: You have two sticks of unknown lengths $a>b$ and a measuring device with constant variance $1$ that you can only use twice. How can you construct ...
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Problem on sufficient statistics

Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ...
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Solution to German Tank Problem

Is there a formal mathematical proof that the solution to the German Tank Problem is a function of only the parameters k (number of observed samples) and m (maximum value among observed samples)? In ...
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Weighted average Sufficient Statistic

I have the following question about sufficient statistic: suppose that you have N agents indexed by $i$ and $j$ each starting with a noisy signal $x_i^0 = \theta + \epsilon_i$ where $\theta$ is a ...
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Sufficient Statistic of Uniform $(-\theta,0)$

Let $X_1, ... , X_n$ be i.i.d random variables Uniform $(-\theta,0)$ , with $\theta > 0$ parameter \begin{align}f_{\theta}(x_1,x_2,\cdots,x_n)&=\prod_{i=1}^nf(x_i;\theta) \\&=\frac{1}{(\...
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What does the 'sufficient statistic' on unknown parameter mean? How is it related to conjugate priors? [duplicate]

By definition, it's stated that statistic said to be sufficient for θ if the conditional distribution of X1, X2, ..., Xn, given the statistic Y, does not depend on the parameter θ. There is an ...
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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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Sufficient statistic for Gaussian $AR(1)$

Question Does the Gaussian $AR(1)$ model, with a fixed sample size $T$, have nontrivial sufficient statistics? The model is given by $$ y_t = \rho y_{t-1}, \, t = 1, \cdots, T, \; \epsilon_i \...
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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 $\...
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If $T$ is a sufficient statistic for $\theta$, is $H(\theta\mid x) = H(\theta\mid T(x))$?

I was trying to prove that sufficient statistics attain equality in the data processing inequality by a slightly different route than I usually see, and came across an odd expression. (I care more ...
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Sufficient statistics from exponential distributions with different means [closed]

If $X$ and $Y$ are independent exponential random variables with means $\theta$ and $2\theta$ respectively, then show that $X + 2Y$ is sufficient for $\theta$. I know how to find sufficient ...
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The minimal sufficient statistic of $f(x) = e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$

The Casella Berger (2002) solutions manual says that the minimal sufficient statistic for $$f(x) = e^{-(x-\theta)}e^{-e^{-(x-\theta)}}, \qquad x\in \mathbb{R}$$ are the order statistics $(X_{(1)},\...
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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 ...
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Sufficient Statistic and Unbiased Estimate in Exponential Family

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he deals with sufficient statistics in exponential distributions ...
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Proof of Rao Blackwellization

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he introduces the idea of minimizing the variance of an unbiased ...
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Minimal sufficient statistics: how should we define and interpret it? [duplicate]

Through my studies of statistics inference, I came into the concept of minimal sufficient statistics. However, I find it a little bit cumbersome. Could someone provide me its definition and how should ...
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How to prove or disprove that $T(X_{1},X_{2}) = X_{1} + X_{2}$ is a sufficient statistic

Let $X_{1},X_{2},\ldots,X_{n}$ be random sample from a population whose distribution is given by $X\sim\text{Bernoulli}(\theta)$, $0 < \theta < 1$. a. Show that $T(x) = \displaystyle\sum_{i=1}^{...
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Unbiased Estimator based on Sufficient Statistic

suppose $X_1, ... , X_n$ are iid with pdf $f(x|\beta) = e^{-(x-\beta))}I_{(\beta, \infty)}(x)$ and the pdf of ( the smallest order statistic) $X_{(1)}$ is given by $f_{X_1}(x)$ = n $ *$ $e^{n(\...
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Showing the sample mean is a sufficient statistics from an exponential distribution

Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\theta$), where $\theta>0$ is unknown. If we observe $\overline x = 5.2$ for a sample of $20$ light ...
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Conditional Testing of Two-way Contingency Independence

I am reading about testing independence in two-way contingency tables from Mood Graybill and Boes's Introduction to the Theory of Statistics and is confused about testing independence. We have a two-...
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Expectation of Sufficient Statistic for Beta Distribution

I am looking at question 1b of the following notes: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1/asst1.pdf In 1a, I have shown that the Beta distribution has a density that can be written in the ...
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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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Binomial distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be random sample from $X\sim\text{Binomial}(2,\theta)$. (a) Find the least variance from the set of all unbiased estimators of $\theta$. (b) Find a sufficient ...
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Prove that the order statistics are minimal sufficient for a random sample from an unknown density $f$ [duplicate]

This is Exercise 6.29 from Casella and Berger's Statistical Inference, so I'll just post the question in full, and I'll also post the answer included in the solutions manual. I'll make the part that I ...
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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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Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem

Question: Given, $X\sim N(0,\sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $\sigma^2$. My Attempt: This problem is very easy if we use Fisher–Neyman ...
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obtain (minimal) sufficient statistic for $\gamma$ knowing the canonical statistic $\theta(\gamma)$

A statistical model for a data set y is an exponential family , with canonical parameter vector $\theta= (\theta_1,\theta_2,..\theta_k)$ and canonical statistic $t(y) = (t_1(y),t_2(y),..t_k(y))$ if ...
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Checking if a minimal sufficient statistic is complete

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[-\theta, 2\theta]$ with $\theta \in \mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete. I found ...
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Sufficient Statistic and MLE

Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
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Gamma Distribution Sufficient Statistics

I've been asked to show the gamma distribution can be written in the form $p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$ where $T(x)$ is a sufficient statistic. .... I have ...