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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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 ...
0
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0answers
12 views
Distribution of General Pivotal Quantity [on hold]
Let $f(x; \theta) = g(\theta)h(x)$ for $ a(\theta) \leq x \leq b(\theta)$ where $ a(\theta)$ decreases and $b(\theta)$ increases with $\theta$.
I'm trying to show that
\begin{equation}
P(S ;\theta) ...
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43 views
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
42 views
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
53 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
56 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
50 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 ...
2
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1answer
16 views
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 ...
0
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0answers
29 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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1answer
233 views
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
40 views
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 ...
4
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1answer
292 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
171 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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1answer
121 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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0answers
20 views
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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0answers
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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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3answers
105 views
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{...
3
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1answer
95 views
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 ...
1
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2answers
58 views
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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0answers
32 views
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
26 views
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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0answers
34 views
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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0answers
183 views
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
242 views
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
42 views
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 ...
4
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2answers
74 views
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 ...
5
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1answer
69 views
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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0answers
39 views
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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0answers
79 views
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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0answers
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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
44 views
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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0answers
63 views
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 ...
5
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2answers
247 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
46 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
109 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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1answer
82 views
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
33 views
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), $$ when $x>i\theta $ and $x=0$ otherwise. Let $T$ be the ...
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0answers
102 views
Complete and sufficient statistics of Laplace Distribution [duplicate]
Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,
$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}...
5
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1answer
117 views
Is the negative exponential distribution a member of the exponential family?
Please correct me if I am wrong.
The general form of k-parameter exponential family is
$f(x;\boldsymbol{\theta}) = a(\boldsymbol{\theta})g(x) \exp\{\sum_{i=1}^{k}b_(\boldsymbol{\theta}) R_i(x)\}$
...
4
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1answer
283 views
Sufficient statistics for Uniform (-θ,θ)
So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter θ. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the paramether θ?
In other words, can a ...
0
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1answer
50 views
Why isn't the binomial distribution completely defined by its mean and variance?
I'm currently reading Gelman and Hill (ARM) chapter 7 on simulation of probability models and statistical inferences. To motivate simulation of probability models, the authors explain that "if the ...
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1answer
61 views
Comparing variances of two unbiased estimators
This question is from a Ph.D Qualifying Exam for Mathematical Statistics. Main reference is Casella & Berger's Statistical Inference.
Let $W_1$ and $W_2$ be unbiased estimators of a parameter $\...
4
votes
1answer
49 views
Finding a function of sufficient statistic with the same expected value with $\psi(X)$
This problem is from a qualifying exam of mathematical statistics of Department of Mathematics, POSTECH, and I'm trying to solve this problem on reference of Casella & Berger's Statistical ...
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0answers
80 views
Minimal Sufficient Statistic for Gaussians with different means
I have the following problems on my Statistics course (using Casella and Berger's book) problem set:
1) Let $Y_{i} = X_{i}'\theta + U_{i}$ where $\theta \in \mathbb{R}^k$
and $U_{i}$ are iid $N(0,...
1
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1answer
532 views
Sufficient and complete statistic function for $\theta$ of geometric distribution [duplicate]
I am trying to find a sufficient and complete statistics function for $0<\theta<1$ of a sample $X = X_1, \dots, X_n$ from the Geometric Distribution.
We have $f(x;\theta) = (1-\theta)^{x-1}\...
2
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0answers
67 views
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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0answers
231 views
The relationship between UMVUE and complete sufficient statistic
Let $X_1,...X_n$ $U(-\theta , \theta)$
I want to find the UMVUE of $\theta$ if it is exists.
My answer is , there is no UMVUE in this case.
Because there is no complete sufficient statistic that ...
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0answers
35 views
Minimal sufficiency of smallest order statistic
I am attempting to show that the smallest order statistic T is minimally sufficient for the mean of a distribution when the variance is known. In particular, iid random variables $X_1,\ldots,X_n$ have ...
1
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0answers
49 views
Sufficient statistic function for $f(x) = \theta x^{-2}, \; \; 0 < \theta \leq x < \infty$ [duplicate]
Exercise :
Let $X_1, \dots, X_n$ be a random sample from a distribution with probability density function $f(x) = \theta x^{-2}, \; \; 0 < \theta \leq x < \infty$, where $\theta$ unknown ...
4
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1answer
140 views
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 ...
