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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 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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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 ...
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Sufficient statistic for linear regression model?

What is the sufficient statistic for the normal linear regression model? That is: $$y_i = x_i^T\beta + e_i,$$ where $e_i \sim N(0,\sigma^2)$. I know that the MLE is $\hat{\beta} = (X^TX)^{-1}Xy$ ...
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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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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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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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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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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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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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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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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 ...
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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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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}}...
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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 $f(x;\boldsymbol{\theta}) = a(\boldsymbol{\theta})g(x) \exp\{\sum_{i=1}^{k}b_(\boldsymbol{\theta}) R_i(x)\}$ ...
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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 ...
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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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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 $\...
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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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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,...
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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}\...
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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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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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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 ...
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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 ...
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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 ...
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1answer
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Sufficient statistic for sample without replacement

This is an exercise that I faced today: Let $\mathcal{M}$ be a population of ’individuals’ with identification numbers in $\mathbb{Z}$. We assume that the set of all identification numbers equals $\{...
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Minimal sufficient statistic for multivariate normal

I have the following iid. variables $X_1,..,X_n,Y_1,..,Y_m$ with distribution $X_i\sim N(\mu_1,\sigma_1^2), Y_j\sim N(\mu_2,\sigma_2^2)$. How do I find the minimal sufficient statistic for $(\mu_1,\...
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Proving that X is not complete

So, the question is the following: "Be X a single sample from $Beta(\theta,\theta )$, is X a sufficient statistic? Is it complete?" So, since it's the entire sample, it's easy to say that it is ...
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Maximum likelihood estimation for $\alpha$ with population pdf $f(x;\alpha)=\frac{2}{\alpha^2}(\alpha-x)I_{(0,\alpha)}(x)$

A sample of size two is taken from the distribution $ f(x;\alpha)=\frac{2}{\alpha^2}(\alpha-x)I_{(0,\alpha)}(x)$. We need to find the maximum likelihood estimator for $\alpha$. The likelihood ...
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question about correct interpretation of sufficiency [on hold]

I am currently trying to understand the concept of sufficiency and I am trying to create my own example. Assume that we have a statistic$$T:(\mathscr X, \mathscr A, \mathscr P) \to (\mathbb{R}^2, B^2)...