Questions tagged [sufficient-statistics]

For questions about sufficient statistics. A statistic is sufficient for a parametric model if the distribution of the data conditioned on the statistic is parameter-free. For more general questions about statistics and estimators, please use "statistical-inference".

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Exponential Family with Complete Sufficient Statistic

Suppose that $X$ is in an exponential family taking values in $\sigma$-finite space $(\mathcal{X}, F_{\mathcal{X}}, \nu)$ probability density function $f_{\theta}(x)=h(x) \exp \{\eta(\theta)^T T(x)-\...
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Strongest Result on Existence of Minimal Sufficient Statistic

Let $X$ be a random variable taking values in a measurable space $(\mathcal{X}, F_{\mathcal{X}})$ whose distribution $P_{\theta}$ is chosen from a parametric family of probability measures $\mathcal{P}...
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$\mathbb E(X_1 X_2|Y)$ when $Y = X_1 + \cdots + X_n$

Find $\mathbb E(X_1 X_2|Y)$ when $Y = X_1 + \cdots + X_n$ for a Bernoulli distribution for coin flipping with $n$ flips. Here heads = $1$, and tails = $0$. I understand that $X_1 X_2$ equals $1$ if ...
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Basu's theorem and completeness

Recently, I was reading up on the Basu's theorem and what i gathered of it was that if a statistic $T$ is complete and minimal sufficient then it is free from Ancillary statistics. My question is why ...
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Proving that $T_{n}(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$ given a sample of i.i.d random variables

I am asked to show that the statistic $T(X):=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$, where $X_i\sim Geom(p)$ are i.i.d random variables. Given a sample $x=(x_1,x_2,\dots,x_n)$ I have to ...
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Minimum variance unbiased estimator for $\mu$ in Normal location model with known but random variance

Consider observing $X \mid \sigma \sim N(\mu, \sigma^2)$ and $\sigma \sim F$ for some known distribution $F$ supported on the positive reals. We observe a single draw $(X, \sigma)$. An estimator $T(X, ...
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Calculate the expected value of $S(X)=\sum_{i=1}^{n}x_i$

Let $x=(x_1,x_2,\dots, x_n)$, be observations of i.i.d. random variables with probability function $$\mathbb{P}(X_i=x_i)=p(1-p)^{x_i-1}$$ where $x_i\in\mathbb{N}$ and $p\in(0,1)$. I am asked to ...
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Let $Y_1, \dots, Y_n \sim \; \textrm{iid}$ with pdf $f_Y(y)$. Show that the UMVUE of $\theta$ is given by $\frac{n-1}{\sum_{i=1}^n Y_i}$ [duplicate]

I'm having a difficult time figuring out where to go here. Question: Let $Y_1,\dots, Y_n$ be iid random variables with pdf $f_Y(y) = \theta e^{-\theta y} \;,\; y >0\;,\;\theta >0.$ Show that the ...
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Does this show $\bar{Y}$ is a sufficient statistic?

Question If $Y_1,\dots, Y_n \sim \; \textrm{iid geometric(p)}$, show that $\bar{Y}$ is a sufficient statistic for p. My work Factorization Theorem If $Y_1, \dots, Y_n \sim \; \textrm{iid}$ then U is a ...
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If $T_i$ is sufficient for $\theta_i$ for $i=1,2$, then $(T_1,T_2)$ is sufficient for $(\theta_1,\theta_2)$.

I am trying to solve the following problem from a book on mathematical statistics: Suppose $\mathcal{P}=\{f(x|\theta_1,\theta_2)|\theta=(\theta_1,\theta_2)\in\Theta_1\times\Theta_2\}$ is a family of ...
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How do I apply the Rao-Blackwell Theorem to find MVUE of parameter theta?

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables having the same population distribution with density: f(y; θ) = ( θ(3^θ)/y^(θ+1) , y ⩾ 3; 0, elsewhere.) where θ is a ...
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Proving Independence of Complete and Sufficient Statistic

Let $X_1, \ldots, X_n$ be i.i.d r.v. with the p.d.f. $$ f(x; \theta_1, \theta_2) = \begin{cases} \frac{1}{\theta_2} \exp\left(-\frac{x - \theta_1}{\theta_2}\right), & \text{if } x > \theta_1, \...
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Links between sufficient statistics and Chentsov's characterization of Fisher metric

I've been self-studying "Information Geometry" by Ay et al. fascinated by the connection between Geometry, Probability and even Statistics. The proofs are clear to me, nonetheless, even in ...
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Joint sufficient statistics for normal distribution (denominator n - 1)

Example 24-6 here, for i.i.d. $X_i$ from a normal distribution $(\theta_1, \theta_2)$, expresses the joint density $$ f(\textbf{x}; \theta_1, \theta_2) = \exp\bigg[\frac{-1}{2\theta_2}\sum_{i = 1}^n ...
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6.36 of Theory of Point Estimation, second edition

the question is: $$ X_1, X_2, ..., X_n $$ i.i.d random variables of uniform distribution U(a,b), where a<b. Show that $$ Z_i = \frac{X_{(i)}-X_{(1)}}{X_{(n)} - X_{(1)}}$$ ,i = 2,...n-1, are ...
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$X_i\sim \mathrm{UNIF}(0,\theta)$. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion.

Consider a random sample from a uniform distribution $X_i\sim \mathrm{UNIF}(0,\theta)$, where $\theta$ is unknown. Show that $S=X_{n:n}$ is sufficient for $\theta$ by the factorization criterion. ...
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Likelihood ratio as minimal sufficient statistics in infinite parameter space

Consider a family of density functions $f(x|\theta)$ where the parameter space for $\theta$ is finite, that is, $\theta \in \{\theta_0, \cdots, \theta_p\}$. Assume that $\theta_0$ is such that $f(x|\...
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Prove hypergeometric distribution belongs to a complete family

I was asked to prove the following statement: Let $X$ be a discrete random variable with $$P_\theta(X=x)=\frac{{\theta\choose x}{N-\theta\choose n-x}}{{N \choose n}},\ x = 0, 1, 2, \dots , \min(\theta,...
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When is sufficiency and completeness of a statistic preserved?

I have been given these definitions in my statistical inference class: Let $(X_1,...,X_n)$ be a simple random sampling of $X\rightarrow\{P_\theta:\theta \in \Theta\}$ and $T\equiv T(X_1,...,X_n)$ a ...
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Can the sample size $n$ be in either factor when doing Fisher–Neyman factorization?

I was looking for a sufficient statistic for a random sample of this distribution when a question came to mind. $$ f_X(x)=\frac{\left(\theta-1\right)^{x-1}}{\theta^x}, \ \ \ x=1,2,...\ and \ \ \theta ...
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why should conditioning on a random variable subtract information from it?

The defintion of a sufficient statistic is as follows: A statistic $t = T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $X$, given ...
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prove that maximum and minimum is sufficient statistic for uniform($\theta$ , $\theta$+1)

I'm trying to understand the proof that $T(X_1,..., X_n)=(max\{X_1,..., X_n\}, min\{X_1,..., X_n\})$ , is a sufficient statistic for $\theta$, given that $f_{\theta}=Uniform(\theta, \theta+1)$ from ...
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Does the number of parameters change the form of the sufficient statistic?

I'm given the following instructions; Let $H:(0,\infty)$ be an integrable function and $f(x;\theta)$ the PDF that is defined by $$f(x;\theta)= \left\{ \begin{array}{ll} \alpha(\theta)⋅H(x) & ...
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Show that $\bar{Y} - \min(Y_{1}, \dots, Y_{n})$ is independent of $\min(Y_{1}, \dots, Y_{n})$

Suppose that $Y_1, \dots, Y_n$ are i.i.d observations from the density $f(y, \theta, \beta) = \beta e^{-\beta(y - \theta)}I_{[y \geq\theta]}$ where $\beta \gt 0$, $\theta \in \mathbb{R}$ are unknown ...
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Find a complete sufficient statistic for Uniform$(\theta, \theta + 1)$

Let $X_1, \cdot\cdot \cdot, X_n$ be a random sample from $U(\theta, \theta + 1)$, $\theta \in \mathbb{R}$. Find a complete and minimal sufficient statistic $Y$ for $\theta$. I proved that $T(X) = (X_{(...
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Need a sufficient unbiased estimator of a parameter be unique?

In a recent multiple choice examination I encountered a "select all that apply" type question, which had this statement among others: If a sufficient estimator exists, it is always unique. ...
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Is a sufficient statistic for a parameter $\lambda$ in the exponential family also sufficient for the reciprocal of $\lambda$?

If $\{E(\lambda)\}_{\lambda > 0}$ denotes the exponential family, then $E(\frac{1}{\theta}), \theta > 0$ is an alternative parameterization. I think a sufficient statistic for $\lambda$ is also ...
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Minimal Sufficient Statistics

Minimal sufficient statistics are supposed to have the lowest dimension among all sufficient statistics. Then, is any sufficient statistic having the same dimension (as the mss) a minimal sufficient ...
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How do I find the mass function of this zero-inflated Poisson distribution and show the given statistic is sufficent?

For $\theta \in [0, 1]$, $\lambda > 0$, I have the zero-inflated Poisson distribution $$f_{\theta,\lambda}(x)=\begin{cases} \theta+(1-\theta)e^{-\lambda} & x=0 \\ (1-\theta) \frac{\...
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An example of sufficient statistic and how to understand it.

I am learning sufficient statistics, and the general idea seems to be straightforward but it turns out to be really confusing when I take a closer look (and it is probably because of the conditional ...
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Sufficient statistic for normal distribution by a bijective map

For $X_i$~$\mathcal{N}(\mu,\sigma^2)$, I know that since $T=(\Sigma^{n}_{i=1}x_i,\Sigma^{n}_{i=1}x_i^2)$ is a sufficient statistic for $(\mu,\sigma^2)$, $\bar{T}=(\bar{X}_n,S_n^2)$ ($\bar{X}_n$ is the ...
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Does the pushforward of a sufficient statistic induce unique probability measures?

Consider a collection of probability measures $\{P_\theta | \theta \in \Theta\}$ and a sufficient statistic $T$, that is for all $A \in \Sigma$ (the $\sigma$-algbera): $\mathbb{E}_\theta(1_A|T)$ is $\...
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Probability density function of a sufficient statistic

I'm writing a text on Allan Birnbaum's proof that the likelihood principle is implied by the conjunction of the sufficiency principle and the conditionality principle. Here, we call \begin{equation} E=...
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How is $\{\mathbf{X} = \mathbf{x}\}$ a subset of $\{T(\mathbf{X}) = T(\mathbf{x})\}$ ($T$ is a sufficient statistic)?

(I went through some related StackExchange questions but none addresses the one I have. If you do find one that addresses mine below, I'd be happy to look into that.) In Chapter 6.2 of Casella and ...
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MLE and Sufficient estimator for $N(\mu,1)$ when range of the parameter $\mu$ is restricted to non-negative real numbers.

Let $X_1 , X_2 ,....,X_n$ be i.i.d random variables with $N(\mu,1)$ distribution. Assume that $\mu \in [0,\infty)$. Let $\hat{\mu}$ be the MLE of $\mu$, then which of the following statements are true ...
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Max. likelihood and sufficient statistic of exponential distribution.

Consider the following probability function of a random variable $Y$: $$ f(y \mid \theta)=e^{-(y-\theta)},\quad y\ge\theta $$ and $0$ otherwise. We take a random sample $(Y_1,Y_2,...,Y_k)$ and want to ...
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How much datapoints must be in a subset of a dataset before the subset is representative of that parent dataset?

It makes sense to me that a randomly sampled subset of a dataset should still be theoretically representative of its parent. When you take data and split into training and test sets, you assume that ...
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Equivalent estimators and transformation of joint density

I am currently reading a paper about acceptance sampling (Sampling Plans for Inspection by Variables, G. Lieberman & G. Resnikoff), where I have come across some things I am not fully sure as to ...
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Given $T=(X_{(1)},\sum[X_i-X_{(1)}])$ is minimum sufficient statistics for exp(a,b).

Given $T=(X_{(1)},\sum[X_i-X_{(1)}])$ is minimum sufficient statistics for exp(a,b). Show that $X_{(1)}$ and $\sum[X_i-X_{(1)}]$ are independently distributed as $Exp(a,b/n)$ and $\frac{1}{2}b\chi_{(n-...
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Nonexistence of UMVUE

In Mathematical Statistics written by Jun Shao(2003), exercise 3.22 claims that Exercise 3.22. Let $\left(X_{1}, \ldots, X_{n}\right)$ be a random sample from $P \in \mathcal{P}$ containing all ...
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Sufficient statistic and the maximum likelihood estimator of the probability of having an infectious disease when people are grouped and tested

Suppose N students arriving at a college are all equally likely to have a particular disease with an unknown probability p. The disease status (affected / not affected) of all students are independent....
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Rao Blackwell. Finding unbiased estimator help

I have done many problems with the Rao Blackwell Theory and finding an unbiased estimator, but this question has stumped me a little Question $Uniform(\theta, 3\theta)|\theta > 0$ $g_{\theta}(x) = \...
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Help understanding sufficient statistic proof

I am having a hard time following this proof, maybe the solutions have jumped some steps but was wondering if someone could help me follow it. The question: let $X_{1},...,X_{n}$ be independent and ...
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How can I prove that $T=\sum_{i=1}^n X_i^6$ is sufficient for $\theta$?

I’m revising for an exam in Mathematical Statistics and I have found the following problem in one of the previous exams (I don’t know the source): Let $X_1,…,X_n$ be a random sample that follows the ...
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Given n iid Pareto distributed random variables, find the UMP one sided test of the first moment

Given $X_1,...,X_n$ ($n\geq 2$) are iid and each have density: $f_X(x) = \frac{c^\theta \theta}{x^{1+\theta}}\mathbb{1}(x> c)$ for known $c$ and $\theta > 1$ then we can easily find the first ...
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Conditional expectation of product of Normal variate given their sum

Given $$X_1,\ldots,X_n\stackrel{\text{iid}}{\sim}\mathcal N(0,1)$$ I would like to compute the conditional expectation $$\mathbb E\Big[\prod_{i=1}^n X_i \Big| X_1+\cdots+X_n=x\Big]$$ for statistical ...
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Tricky exercise on sufficient statistics (undergraduate level)

During my semester, my class and I were subjected to an interesting exercise that challenged my entire understanding of sufficient statistics (It turns out that the median of this exercise was 0). ...
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Why does the unbiased statistic in this example be MVUE immediately?

I am reading "Introduction to Mathematical Statistics" to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. Below are page 427 and 428 that ...
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Fisher-Neyman Factorisation Theorem and sufficient statistic misunderstanding

Fisher Neyman Factorisation Theorem states that for a statistical model for $X$ with PDF / PMF $f_{\theta}$, then $T(X)$ is a sufficient statistic for $\theta$ if and only if there exists nonnegative ...
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Variational Inference for Item Response Models Estimation

I should work on a project about Variational Inference for Item Response Models Estimation. This is my university project but I don´t find relevant information around this topic so I would be very ...
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