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, minimal, complete

Are all complete statistics functions of each other? For example if I have T and S complete statistics Can you always write T in terms of S and S in terms of T?
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sufficient statistics for bernoulli distribution

Let Y1, . . . , Yn be a random sample of size n where each Yi ~ Bernoulli(p), and let Y = $\sum$ Yi for i = 1, . . . , n. The estimator is W= (Y+1)/(n+2) Is the estimator a sufficient statistics for ...
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Is this a sufficient statistic for variance?

I have $X_1,\dots,X_n,X_{n+1}\overset{iid}{\sim}F_X(x)$, where $F_X$ has a finite mean $\mu$ and variance $\sigma^2$. If I calculate $\bar X_n = \dfrac{1}{n}\sum_{i=1}^n$ and $S^2_n = \dfrac{1}{n-1}\...
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How is $\text{Cov}(\bar{Y}, Y_i - \bar{Y}) = \dfrac{1}{n^2} \text{Cov} \left( \sum_{j = 1}^n Y_j, nY_i - \sum_{j = 1}^n Y_j \right)$?

I have this example of sufficiency: Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence $$\...
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How does the result $\dfrac{1}{n^T} \dfrac{T!}{\prod_{i = 1}^n Y_i!}$ tell us what distribution $T(\mathbf{Y})$ is?

This follows on from my question here. I have the following problem: Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ ...
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Is there a standard measure of the sufficiency of a statistic?

Given a parametrical model $f_\theta$ and a random sample $X = (X_1, \cdots, X_n)$ from this model, a statistic $T(X)$ is sufficient if the distribution of $X$ given $T(X)$ doesn't depend on $\theta$. ...
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Minimal sufficient statistic with parameter-dependent support on density function

I'm having trouble finding a minimal sufficient statistic for this particular population. It has the pdf defined as $$f_\theta (x) = 4\theta^4 x^{-5}$$ if $\theta \leq x$, and 0 otherwise. We take $n$ ...
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Intuition for the perfect correlation between sufficient statistic and score function

I have recently learned that: Cramer Raw Lower Bound is achieved when there is perfect correlation between T(X) statistic and U(theta, X). Is there any intuition behind this? I cant quite understand ...
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Rao Cramèr Lower Bound problem

Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
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Does this distribution belong to the exponential family? [duplicate]

I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
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Is the ordering of a sufficient statistic meaningful (soft question, maybe)?

Suppose we have distribution with a vector of parameters, and a random sample of size n from said distribution. Is the writing order of a sufficient statistic meaningful? For a normal distribution, we ...
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Sufficiency for Truncated Geometric

Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics: Let $$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
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Bayesian definition of sufficient statistics

Some time ago I wrote a question about what I think/thought (up to my understanding) is an ambiguity of the common definition of sufficient statistics : Conditioning in the definition of sufficient ...
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Rao-Blackwellisation using non-sufficient statistics

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1") Now, I do understand that the ...
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Sufficient Statistic for $f(x,\theta)=\dfrac{2}{\theta^{2}} (\theta-x) \cdot 1_{(0,\theta)}(x), \;\forall \theta \in (0,\theta) $

Let $X_{1},\ldots, X_{n}$ be random variables independent and identically distributed; show that the following density function is in the exponential family and find the sufficient statistic for $\...
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Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution

Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function: $$ f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R} $$ Find the joint ...
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What are the "Dangers" of using "Non-Sufficient" Statistics?

I was reading one of the answers listed on this previous Stackoverflow question about the importance of sufficient statistics (Generalized Linear Models - What's special about the exponential ...
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Distribution of unbiased estimator given the sufficient statistic

Let T be a sufficient statistic for parameter a, and W be an unbiased estimator of a, then will the distribution of W|T always be independent of parameter a? I understand that T being sufficient for ...
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Subscripts for Expectations and variances in for estimators [duplicate]

Is there any significance for subscripts to E and Var? For example, the risk function of an estimator $\delta(\mathbf x)$ of $\theta$ in my book is: $$ R(\theta,\delta)=E_\theta[L(\theta,\delta(\...
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UMVUE for P(X > k) in exponential distribution [duplicate]

I have to find UMVUE for $exp(-k*a)$ where X ~ Exponential(a); k is a positive real number. I tried it using Lehmann-Scheffe theorem. Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
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What is the space that a class of probability distributions spans when T is a complete sufficient statistic?

There are a few good posts/notes (see here, and here) giving high level geometric intuition of a complete statistic ($E_{T}[g(T); \theta] = 0 \Rightarrow P(g(T)=0; \theta) = 1 \text{ almost everywhere}...
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Order statistics is minimal sufficient statistics for unknow density function

I'm trying to prove the problem, but there is a problem on definition of term. The theorem that I use to prove it is However, what exact meaning of "family of densities ~ all have common support&...
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Logistic distribution's minimal sufficient statistics

Distribution function(pdf) is If x is sample from population, f(x|θ) is I calculated f(x|theta)/f(y|theta), but I don't know how can I find minimal sufficient statistics.
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What is the goal of sufficient dimension reduction? Under what circumstances can it be achieved?

I have recently heard the term "sufficient dimension reduction" tossed around, although I have struggled to find material on the concept that I fully understand or that clearly explains why ...
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Understanding the Importance of "Sufficiency" within Statistics

I am trying to better understand what it means to be a "sufficient statistic". "In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown ...
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Sufficient Statistic for Absolutely Continuous Distribution [duplicate]

The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes. Let $x_{1}, x_{2},...,x_{M}$ be i.i.d. samples from the absolute continuous ...
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Practical Value of Sufficient Statistics [duplicate]

Can anyone provide some comments about the practical value of sufficient statistics research? Nearly all sufficiency results, such as the Fisher-Neyman factorization rule, build upon parametric models ...
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Likelihood and 2-dimensional Sufficient statistic

I was asked to find the likelihood of a function and calculate a two-dimensional sufficient statistic for (a, θ) for the function $$ \frac {(a+1)(θ^{a+1})}{y^{(a+2)}}$$ I've calculated the likelihood ...
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Are These Conjectures Regarding Sufficient Statistics True?

I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true ---...
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Is This A Counter-Example To The Theorem by Barndorff-Nielsen-Pedersen (1968)?

In the textbook "Theory of Point Estimation" 2nd Ed. by Lehmann and Casella, Theorem 6.18 states: Suppose $X_1, ..., X_n$ are real-valued IID according to a distribution with density $f_\...
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Sufficient statistic and complete sufficient statistic [duplicate]

I'm trying self-study some inference and now I'm trying to understand how to solve some problems on this topic but I found this basic problem that I'm not being able to solve. Problem: Let $X_{1},...,...
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How can a statistician who has the data for a non-normal distribution guess better than one who only has the mean?

Let's say we have a game with two players. Both of them know that five samples are drawn from some distribution (not normal). None of them know the parameters of the distribution used to generate the ...
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Mean is not a sufficient statistic for the normal distribution when variance is not known?

According to the PDF here: https://www.math.arizona.edu/~tgk/466/sufficient.pdf, the sum of a sample of data is not a sufficient statistic for the normal distribution when the variance is unknown. ...
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How is a sufficient complete estimator minimal sufficient [duplicate]

the question is quite theoretical But I am finding it extremely hard to understand it without cramming the derivation ,although I understand the basic concept of it . Can someone explain how a ...
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Sufficient statistics for a non exponential

I think this is not an exponential family but does it mean that we can't find a sufficient statistic for $\theta$ if $X_1, X_2,..., X_n$ are a random sample from this density? $$ f_{\theta} (x) = \...
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exponential sufficient statistics [duplicate]

A family of pdfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ and the statistic $T$ is sufficient iff $f(x;\theta) = h(x)...
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Technical term for natural statistics comined with log-partition

One way to find the posterior of an exponential family distribution with a conjugate prior is to use the natural reparametrization of the likelihood and prior and combining the sufficient statistics ...
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Sufficient statistics and randomized estimator in TPE

I've been reading Lehmann and Casella's Theory of Point Estimation 2nd Edition (TPE). In Chapter 1 Section 6 (pp.32-33), they introduce the idea "randomized estimator". Their explanation is, ...
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How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?

Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family $$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
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Does an estimator need to be unbiased in order to be sufficient?

I am reviewing some theoretical statistics content, and I was wondering if an estimator need to be unbiased in order to be sufficient? Is there any way to prove this? Thanks!
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Rao–Blackwellization of Metropolis–Hastings

I am trying to achieve a Rao–Blackwellization of Metropolis–Hastings algorithm. In the paper by Robert et al. 2018, the following is given. \begin{align} ℑ=&\frac{1}{T}\sum_{t=1}^Th(\theta^{(t)})=\...
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Showing that $f_\varphi(x)$ is a member of the one-parameter exponential family and $\sum_{i = 1}^n - \log(X_i)$ is sufficient for $\varphi$

Let $X_1, \dots, X_n$ denote a random sample from the PDF $$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{...
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How do we conclude that a statistic is sufficient but not minimal sufficient? #2

This is related to a question I recently asked. I want to show that the statistic $\left(\sum_{i = 1}^n Y_i, \sum_{i = 1}^n Y_i^2 \right)$ is sufficient for $\mu$ but not minimal sufficient where $(...
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Did I show sufficiency and minimal sufficiency correctly?

I am currently trying to show that the statistic $\sum\limits_{y = 1}^n Y_i^2$ is minimal sufficient for $\mu$ where $Y_1, \dots, Y_n$ is a random sample from $N(\mu,\mu)$ for $\mu > 0$. The ...
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Why does this theorem for minimal sufficient from the "All of Statistics" textbook by Wasserman have these exponents of $n$?

In the textbook All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman, the definition of minimal sufficient is given as follows: 9.35 Definition. A statistic $T$ is minimal ...
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How do I proceed from here for factorization of likelihood / joint normal density for finding sufficient statistics?

I'm trying to show that the statistic $\left(\sum_{i = 1}^n Y_i, \sum_{i = 1}^n Y_i^2 \right)$ is sufficient for $\mu$ where $(Y_1, \dots, Y_n)$ is a random sample from $N(\mu, \mu)$ for $\mu > 0$. ...
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How do we conclude that a statistic is sufficient but not minimal sufficient?

I want to show that the statistic $\left(\sum_{i = 1}^n Y_i, \sum_{i = 1}^n Y_i^2 \right)$ is sufficient for $\mu$ but not minimal sufficient where $(Y_1, \dots, Y_n)$ is a random sample from $N(\mu, \...
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What is the difference between using a probability distribution and using the mean of observations only as a measure to describe the data?

I am wondering using the mean only without a specific probability distribution to predict or infer an appropriate measure for the average height of female students. I ask this question because I ...
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Completeness calculation: Where does the $\sum_{i = 0}^n g(i)$ in $E_p[g(T)] = \sum_{i = 0}^n g(i) {n\choose{i}} p^i (1 - p)^{n - i}$ come from?

I have the following definition of completeness: Definition Let $Y_1, \dots, Y_n$ be an i.i.d. $f_\theta (y)$, where $\theta \in \Theta$. A statistic $T(\mathbf{Y})$ is complete if no function $g(T)$ ...
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How do $g(S(\mathbf{y}_1), \theta)$ and $g(S(\mathbf{y}_2), \theta)$ cancel?

My notes introduce the concept of minimal sufficient statistics as follows: Definition A sufficient statistic $T(\mathbf{Y})$ is called a minimal sufficient statistic if it is a function of any other ...
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