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 statistic vector of single parameter?

Can the sufficient statistic for a single parameter be a vector? In my case, I am finding the sufficient statistics for the Poisson parameter in a HMM mixture. The parameter enters my log likelihood ...
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How does one can guarantee that any unbiased estimator is MVUE due to it containing a minimal sufficient statistic?

Sufficiency is okay. But I don't really get it why the fact guarantees it has minimal variance? Can anyone explain to me somewhat intuitively?
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complete sufficient statistic exercise

I have to find complete sufficient statistic of the following pdf $$f(x|\theta)=\frac{\theta}{(1+x)^{(1+\theta)}},\quad 0<x<\infty,\theta>0.$$ My Attempt: The joint density $$f(\mathbf x|\...
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Composite random variables and sufficiency

I wish to find a sufficient statistic for a composite random variable ... Suppose $Z$ is Bernoulli$(p)$ and let $$X|Z=z\sim\begin{cases} N(0,1) & \text{if }z=1\\ E(\lambda)& \...
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Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
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Sufficiency - Knowns and Unknowns

I have a self-study question that goes as follows: Let $X$ be one observation from a $N\sim(0, \sigma^2)$ population. Is $|X|$ a sufficient statistic? My question is - since you KNOW the $\mu$ ...
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975 views

Basic intuition about minimal sufficient statistic

As stated by Wikipedia: A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, $S(X)$ is minimal sufficient if and ...
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Puzzling Sufficient Statistic

If $X_1\sim U(0,\theta)$ then $X_1$ is a sufficient statistic for $\theta$. Also when $X_2\sim U(0,\theta + 1)$ then $X_2$ is a sufficient statistic for $\theta$. Is that right? Now if $X_1, X_2$ ...
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Find the unique MVUE

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388. Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta<x<2\theta,$ ...
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Applying Lehmann-Scheffe Theorem to an example

Let me state the theorem first: Let $T$ be a sufficient and complete statistic for the statistical model $\mathcal{P}$ and let $\tilde{\gamma}_1$ be an unbiased estimator for the parameter $\gamma ...
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940 views

A sufficient statistic for Laplace distribution

Suppose we have p dimensional vector of $X =[X_1 \dots X_n]$ where X is Laplace distributed. What will be a sufficient statistics for estimating covariance of $X$? Would it be the sample covariance,...
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Finding a sufficient statistic for a rectangular distribution

Iam trying to use this theorem for the problem below: a statistic is sufficient for $\mathcal{P} = \{P_{\theta}: \theta \in \Theta \}$ iff there exist nonnegative functions $g(\cdot; \theta)$ and $...
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Understanding a characterization of minimal sufficient statistics

I have some questions regarding the proof of the theorem below. First we need a definition: A statistic $T$ is minimal sufficient iff $T$ is a function of any other sufficient statistic. That is, ...
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Distribution of sum of order statistics

The question is from a problem I am trying to solve in Robert Hogg's introduction to Mathematical Statistics 6th version problem 7.2.9 in page 380. The problem is: We consider a random sample $X_1,...
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$T$ is sufficient for $\mathcal{P}$, sufficient for $\theta$ , sufficient for $F$

The definition which is given in my book about sufficiency is: A Statistic $T$ is said to be sufficient for the statistical model $ \mathcal{P}= \{P_{\theta} : \theta \in \Theta \}$ of $\boldsymbol{...
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Minimum dimension of sufficient statistics

Suppose that we have a parameter of $k$-dimensions. Say, for example, for $N(u,\theta)$ both unknown then the parameter is of two dimensions, and $n$ i.i.d. observations. Is it possible to find a ...
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Sufficient Statistic for non-exponential family distribution

Question: Let $X_1,X_2,\ldots,X_n$ be an iid sample from $N(\theta , 4 \theta^2 )$. I want to show that this model is not a member of the exponential family and to find a sufficient statistic for $\...
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Conditional independence conjecture

Suppose there are four random variables (events), $A$, $B$, $C$ and $X$. If we have $X\perp A|B$, saying $X$ and $A$ are conditional independent given $B$, then I want to ask that whether we have $X\...
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Unbiased estimator and sufficient statistics [closed]

Let $X_1,..,X_n$ be a random sample of $f(x;\theta)=\theta x^{\theta-1}I_{[0,1]}(x)$ Find a sufficient statistic for $\theta$ and construct a unbiased estimator for $\theta$ as a function of ...
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Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional sufficient ...
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Find a complete sufficient statistic,

Let $X_1,...,X_n$ be iid observations.Find a complete sufficient statistics for i)$f(x|\theta)=\frac{\theta}{(1+x)^{1+\theta}}I_{[0\infty)}(x), \theta>0$ What I did $$\frac{\theta}{(1+x)^{1+\...
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Foundational sufficient statistics

I've been reading through Casella and Berger's Statistical Infererence and have am having a little trouble understanding something in their explanation of sufficient statistics. Here is the passage ...
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231 views

Minimal sufficiency and UMVUE in a pseudo-Normal distribution

I already asked a (stupid) question about this problem here thinking I wouldn't have problems to continue it but I was pretty wrong. I'm finding several more problems trying to solve it. I'll try to ...
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In which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all $x$'s are not zero ($0$)?

Say $X$ is a random variable and $x$'s are realizations of $X$ . Say , $\mathbb E[X]=\sum _ix_i P[x_i]=0$ . But I do not understand in which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all ...
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Is the posterior a sufficient statistic when observations are conditionally independent?

Suppose there are two random variables, $X_1$ and $X_2$, and we're trying to infer $\theta$. If $X_1$ and $X_2$ are conditionally independent, then is $f(\theta|X_1)$ a sufficient statistic for $X_1$?...
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Summary of estimator properties (consistency, bias, sufficiency, etc.)

I've read about various properties of estimators, but I'm wondering if there's some source with a summary (maybe a list, table, or graphic) of the properties for different kinds of estimators. ...
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Proof that n-order statistics are sufficient for a sample of size n

This is problem 1.5.8 in Mathematical Statistics by Bickel and Doksum. It seems straightforward, but I am not sure if my proof is lacking in some way. It doesn't seem quite correct. Question Let $...
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Apparent inconsistency arising from showing that $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{(0, \theta)}$

The problem is to show that the largest order statistic $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{x \in (0, \theta)}$ is a uniform distribution. I believe I have ...
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Why are the fixed effects of a panel probit regression inconsistent?

I was taught that a probit with fixed effects would not be consistent because the estimates of a non-linear model with a link function other than the canonical (in this case the logit) are not ...
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Sufficient Statistic for inverse Gaussian Distribution

Let $X_1,...,X_n$ be a random sample from population with the pdf of the inverse Gaussian distribution $$f(x|\theta,\beta)=(\frac{\beta}{2 \pi x^3})^\frac{1}{2}e^{-\frac{\beta(x-\theta)^2}{2\...
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Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic

Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$ a)Show that $U_2=\mathbb{E}(...
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Are complete sufficient statistics unique?

I'm under the impression that up to a one-to-one function complete sufficient statistics are unique. How can I show this?
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Are sufficient statistics for regression equivalent in the frequentist and Bayesian cases? [duplicate]

If I have a Poisson regression such that $\lambda = \alpha + \beta t$, $\alpha + \beta t \geq 0$ $\forall t, \alpha, \beta$ and $Y_t \sim \textrm{Poisson}(\lambda_t)$ for which I have 10 observations ...
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Univariate random variables with a multiparameter exponential family

I have this expression of a posterior distribution $$h(\Delta|z_1,\ldots,z_m)=\exp\left\{\frac{(z_1,\ldots,z_m)^T(\Delta,\ldots,\Delta)}{2}+\frac{(z_1,\ldots,z_m)^T(z_1,\ldots,z_m)}{4}-\sum_{j=1}^{...
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Sufficient statistic for a Gamma distribution

I am confused about the steps I need in order to solve the equation below. I must use conditional distribution (and NOT the factorization theorem). Q: $X_1, . . . , X_n$ is a random sample from a ...
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Are the order statistics minimal sufficient for a location-scale family?

Suppose we have a location-scale family with pdf $$\frac{1}{\sigma}f(\frac{x - \mu}{\sigma}).$$ Let $X_1$, $X_2$, $X_3$,..., $X_n$ be a random sample from the location-scale family. Is the statistics $...
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Showing that sum of samples is sufficient statistic

If $X_i$ where $i=1,2,\dots,n$ is a random independent sample from $\textrm{Beta(1,0)}$ having the pdf $f_X(x \mid \theta) = \theta x^{\theta-1}, 0 < x < 1$ and transform this random variable as ...
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Why a sufficient statistic contains all the information needed to compute any estimate of the parameter?

I've just started studying statistics and I can't get an intuitive understanding of sufficiency. To be more precise I can't understand how to show that the following two paragraphs are equivalent: ...
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Exponential family distribution with high-order statistics

An exponential family distribution in its simplest form is given by $p(x|\theta) = \exp(\theta^\top T(x) - A(\theta))$ where $T(x)$ is a vector of sufficient statistics, $\theta$ is its natural ...
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Sufficiency or Insufficiency

Consider a random sample $\{X_1,X_2,X_3\}$ where $X_i$ are i.i.d. $Bernoulli(p)$ random variables where $p\in(0,1)$. Check if $T(X)=X_1+2X_2+X_3$ is a sufficient statistic for $p$. Firstly, how ...
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How to prove that the permutation of the points are the minimal sufficient statistics for Cauchy distribution?

I see this everywhere that the permutation of the samples $X_{(1)}, ..., X_{(n)}$ is the minimal sufficient statistic for the Cauchy distribution [1]. It is clear that it is a sufficient statistic,but ...
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Complete sufficient statistic for $f(x|\theta) = \frac{1}{6\theta^4}x^3e^{-x/\theta}$ and the UMVUE for $\theta$

Looking to find a complete sufficient statistic for the following pdf. $X_1, ..., X_n$ from a random sample with distribution: $$f(x|\theta) = \frac{1}{6\theta^4}x^3e^{-x/\theta}$$ $$ 0 \lt x \lt \...
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269 views

Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
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Intuitive and Formulaic Justification for the Rao-Blackwell Theorem

I've tried looking online and I can't seem to grasp the Rao-Blackwell theorem. Could someone please give an intuitive explanation backed up by formulae.
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Is there a way to visualize what “Minimal sufficiency” and “Completeness” of a statistic means? [duplicate]

As defined (for example, in Wikipedia): Completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of ...
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When if ever is a median statistic a sufficient statistic?

I came across a remark on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic but, besides the obvious case of one or two observations where it equals the ...
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Jointly sufficient statistic?

A random sample $X_{1},...,X_{n}$ is pulled from a gamma distribution. Are there jointly sufficient statistics based on these observations for the two unknown parameters? The definition of a gamma ...
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685 views

Sufficient and complete statistic

Let $ X_1, ... , X_n $ be i.i.d random variables with pdf given by $$f(x;\theta) = \exp(-(x-\theta))I_{(\theta, \infty)}(x)$$ It is asked to find a sufficient statistics for $ \theta $ and to verify ...
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Exponential Family: Observed vs. Expected Sufficient Statistics

My question arises from reading reading Minka's "Estimating a Dirichlet Distribution", which states the following without proof in the context of deriving a maximum-likelihood estimator for a ...
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How to find conditional distribution for Rao-Blackwellizing an estimator?

Let's say I have an unbiased estimator $u(\underline x)$ for function $v(\theta)$ where $\theta$ is a parameter of the distribution of $x$, and $T(\underline x)$ which is a sufficient statistic for $\...