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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Minimal sufficient statistics: how should we define and interpret it?
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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1answer
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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 $ *$ $e^{n(\...
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Maximum Likelihood of Mean Parameter
I am very confused by the language of Questions 1&2 in the following link:
http://www.gatsby.ucl.ac.uk/teaching/courses/ml1/asst1.pdf
My first question:
I'm self teaching myself this material ...
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1answer
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Showing the sample mean is a sufficient statistics from an exponential distribution
Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\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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1answer
107 views
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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1answer
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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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45 views
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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1answer
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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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1answer
125 views
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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2answers
126 views
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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1answer
118 views
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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1answer
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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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39 views
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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1answer
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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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439 views
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 ...
7
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1answer
181 views
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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2answers
239 views
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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2answers
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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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1answer
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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 ...
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Explain sufficient statistic for Poisson distribution [duplicate]
The Wikipedia entry on this topic is, to me, very confusing. It states that:
If X1, ...., Xn are independent and have a Poisson distribution with parameter λ, then the sum T(X) = X1 + ... + Xn is a ...
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2answers
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Is there a difference between Bayesian and Classical sufficiency?
The title pretty much says it all. I wonder whether there is any difference in the way Bayesians understand sufficiency vs. the way orthodox statistics understands sufficiency, or are they equivalent? ...
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1answer
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Question about sufficiency
I learned in my (classical) statistics class that (if we have densities) $T(X)$ is sufficient iff $$f(x)= g(T(x))h(x)$$
I am reading "the Bayesian Choice" and there the factorization-lemma is quoted ...
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2answers
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MCMC combined with numerical integration towards more efficient Bayesian inference
I am quite new to Bayesian statistics so the question can be a bit naive.
My question is on how to deal with a model with individual coefficients. Simple versions of a task and a model I deal with is ...
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1answer
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Jointly sufficient statistics of a multi-parameter exponential family
Let $f_X$ be a joint density function that comes from an $s$-parameter exponential family with sufficient statistics $(T_1, T_2, \dots, T_s)$ so that the density $f_X$ can be expressed as
$$f_{X|\...
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Is an unbiased estimator based off multiple complete sufficient statistics also UMVUE?
If $T(X)$ is a complete sufficient statistic such that $ET(X) = \sigma^2$, then $T(X)$ is the UMVUE estimator of $\sigma^2$.
My question is, suppose $\tau (T(X),W(X))$ is an unbiased estimator of $\...
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Properties of MLE (minimal sufficient and complete) on a restricted parameter space [closed]
So, I bumped into this question of my Statistics Course:
Consider a random sample $X_1, X_2, ..., X_n$ from a one dimensional density $f_\theta$ with $\theta > 0$, or $\Theta = (0, \infty)$. The ...
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Sufficient statistics and wrong model assumption
Given any model for the underlying probability distribution $f(\theta)$, sufficient statistics provides us a way to estimate the model parameter $\theta$ with confidence without wasting the sample ...
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1answer
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Sufficient statistic for p in a binom(1,p)
this might be a stupid question but 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 ...
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Equivalence of sufficient statistics definitions [duplicate]
I'm reading about sufficient statistics, and have come across two definitions which seems unrelated, and I'm trying to understand their connection.
The first definition is from Wikipedia
A ...
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2answers
293 views
Puzzled by definition of sufficient statistics
I am learning about sufficient statistic from Mood, Graybill, and Boes's Introduction to the Theory of Statistics. I am slightly confused by the book's definition of a sufficient statistic for ...
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1answer
48 views
Minimal sufficient statistic whose dimension is less than dimension of parameter
Consider following example:
Suppose $ X\sim N(0, \sigma^2) $,
consider a random sample of size one from this population.
Clearly $X$ is sufficient statistic but $ |X| $ is minimal sufficient ...
2
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1answer
118 views
Conditional distribution of $(X_1,\cdots,X_n)\mid X_{(n)}$ where $X_i$'s are i.i.d $\mathcal U(0,\theta)$ variables
Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from $\mathcal U(0,\theta)$ distribution. It is a common exercise to prove that the maximum order statistic $X_{(n)}$ is sufficient for $\theta$ as ...
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Verification of sufficiency of a linear combination of the sample $(X_i)_{i\ge1}$ where $X_i\stackrel{\text{i.i.d}}\sim\text{Ber}(\theta)$
This question is in regards to this post where it asks if a certain statistic is sufficient for the parameter or not. My query is specifically with this problem:
Let $X_1,X_2,X_3$ be i.i.d ...
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1answer
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Proving sufficiency by showing ratio of statistic pdf to sample pdf is independent of unknown parameter
Let $X_1,...,X_n$ be iid random variables with densities given by $$ f_{x_i}(x|\theta)=e^{i\theta - x}\mathbb{I}_{(i\theta,\infty)}(x), $$ when $x>i\theta $ and $x=0$ otherwise. Let $T$ be the ...
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Complete and sufficient statistics of Laplace Distribution [duplicate]
Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,
$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
