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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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 ...
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Invariance property

I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators. As far as I know, Invariance property of ...
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Sufficient statistic for Poisson in wiki?

In Wikipedia: https://en.wikipedia.org/wiki/Sufficient_statistic#Poisson_distribution it says that $X_1+\cdots+X_n$ is a sufficient statistic for the parameter of the Poisson distribution and its ...
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Minimal sufficient statistic for location exponential family

Let $X_1,\dots,X_n$ iid with pdf $$f(x|\theta)=e^{-(x-\theta)},\,\,\,\theta<x<\infty,\,\,\,-\infty<\theta<\infty.$$ Part (b) of Problem 6.9 in Casella and Berger asks to find a minimal ...
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Proving completeness of highest-order statistic using Leibnitz' Rule

Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$ I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
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Dirichlet expected sufficient statistics

If $\theta$ is a ($k$-dimensional) Dirichlet distribution, the sufficient statistics are $\log\theta_i, i = 1,\ldots, k$. It can be shown that if the Dirichlet has parameter $\alpha = (\alpha_1, \...
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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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Minimal sufficient statistics of increasing dimensionality (not equal to the number of observations)

Restricting the attention to the case of fixed parameters support, it's my understanding that (minimal) sufficient statistics of fixed dimensionality, i.e. a fixed number of of them, exists in, and ...
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Sufficient Statistic for $\beta$ in OLS

I have the classical regression model $$y = \beta X + \epsilon$$ $$\epsilon \sim N(0, \sigma^2)$$ where $X$ is taken to be fixed (not random), and $\hat\beta$ is the OLS estimate for $\beta$. It is ...
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Efficient Estimator from Insufficient Statistic

Suppose that I have a statistic $T(X)$, and I know for sure that it is not sufficient to estimate a parameter $\theta$. Is it still possible to have an estimator $\hat\theta(T(X))$ that is efficient (...
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Likelihood Function is Minimal Sufficient

What does it mean to say that "Likelihood Function is Minimal Sufficient"? Is this a general statement, or does it apply to only exponential family of distributions? I think I understand the concept ...
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Order statistics are minimal sufficient for the set of all continuous distributions

Previous question exists here, but no answer has been posted. I don't believe this is a duplicate of existing questions concerning minimal sufficiency of order statistics that have an answer. Problem:...
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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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Likelihood function when $X\sim U(0,\theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(0,\theta)$. Derive the likelihood function given the sample $x_1, ..., x_n$. Answer The likelihood function is: \begin{...
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1answer
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Sufficient statistic when $X\sim U(\theta,2 \theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(\theta,2 \theta)$. Find a sufficient statistic for $\theta$, and compute $\widehat{\theta}_{MLE}$. Answer The joint ...
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Probability conditional on a parameter?

This is a definition of the sufficient statistic from Wikipedia. A statistic $t = T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $...
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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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How to prove this Corollary regarding ratios of densities being sufficient

The following Corollary is used in "Theory of Point Estimation" by Lehmann to prove a theorem. However I'm unsure how to prove this Corollary (it's left as a problem, so proof is omitted). The ...
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reA function of sufficient statistic

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 ...