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".

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

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 ...
user avatar
1 vote
0 answers
41 views

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 ...
user avatar
  • 358
4 votes
0 answers
27 views

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 ...
user avatar
  • 41
5 votes
1 answer
51 views

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 ...
user avatar
0 votes
1 answer
59 views

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). ...
user avatar
  • 91
0 votes
0 answers
16 views

Beta binomial regression model: sufficient statistic of the regression coefficients

I have the following beta-binomial with logit link function model: $$f(y_i|\pi_i,\phi)\sim\operatorname{binomial}(p_i), \text{with }p_i \sim \operatorname{beta}(\frac{\pi_i }{\phi},\frac{(1-\pi_i)}{\...
user avatar
2 votes
0 answers
32 views

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 ...
user avatar
  • 21
0 votes
0 answers
26 views

Find fisher information matrix for minimum estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
user avatar
  • 655
0 votes
2 answers
45 views

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 ...
user avatar
  • 1,147
1 vote
0 answers
45 views

Find fisher information matrix for minimizes function (Mathematical Statistics)

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
user avatar
  • 655
0 votes
1 answer
23 views

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 ...
user avatar
1 vote
0 answers
47 views

Complete statistics and Sufficient statistics

I was aolving the following problem: Find the statistics of the following density that is complete: $$\text{Let } y_k \text{ be i.i.d random variables with the density: }\\f(y_k;x)=(x-1)y_k^{-x}\...
user avatar
  • 47
0 votes
0 answers
22 views

How to understand sufficient statistics of 'set of density'

Exercise 1.5.12 in Mathematical Statistics(Bickel&Doksum) says Let $\mathrm{P} = \{P_{\theta} : \theta \in \Theta\} $ where $P_\theta$ is discrete concentrated on $X =\{x_1,x_2,\cdots \}$.Let $p(...
user avatar
  • 1
3 votes
1 answer
59 views

Showing that max of uniform laws on $[0,\theta]$ is sufficient statistic with definition

Let $X_1, \cdots, X_n$ be i.i.d. $Unif(0,\theta)$ and $T = \max\{X_1,X_2,···,X_n\}$. Show that T is a sufficient statistic using the definition. So I need to show that for $t>0$, $\Bbb P(X_1 \leq ...
user avatar
  • 1,052
0 votes
0 answers
19 views

Minimal sufficient for $X \sim Pois(\beta_o+\beta_1x_i;x)$

Find the minimal sufficient statistics for $y_i, i = 1, ..., n$ are independent Poisson with mean $\mu_i$ with $$\log(\mu_i) = \beta_o+\beta_1 x_i$$ where $x_i$'s are known predictors and $(\beta_0,\...
user avatar
0 votes
0 answers
34 views

Sufficient statistics of $p_{\theta}(x) = (2\theta)^{-1}e^{-|x|/\theta}$

Suppose $x_1 \cdots x_n$ are iid sample from the double-exponential distribution with density: $$p_{\theta}(x) = (2\theta)^{-1}e^{-|x|/\theta}$$ It is obvious from the factorization theorem that $t(x) ...
user avatar
0 votes
0 answers
31 views

Showing order statistic is sufficient

Suppose the model $P_θ$ is the class of all continuous distributions; this is called a ‘nonparametric family’, where the unknown parameter $θ$ is the whole distribution function. Let $x_1,...,x_n$ be ...
user avatar
1 vote
0 answers
16 views

Variance of a Multivariate Gaussian Sufficient Statistics

A Multivariate Gaussian is part of the Exponential Family $$p(x,y|\eta)=h(x,y)\exp\left\lbrace \eta^TT(x,y)-A(\eta)\right\rbrace $$ Where the Sufficient Statistics are $$T(x,y)=\begin{bmatrix}x\\y\\xx^...
user avatar
  • 140
7 votes
1 answer
219 views

Minimal Sufficient Statistic for $f(x) = e^{-(x-\theta)}, \; \theta < x < \infty, \; x \in \mathbb{R}$

My question comes from Exercise 6.9(b) of Statistical Inference by Casella and Berger: 6.9: Find a minimal sufficient statistic for $\theta$ (b) $f(x|\theta) = e^{-(x-\theta)}, \quad \theta < x &...
user avatar
  • 826
0 votes
0 answers
45 views

An incorrect application of the Rao-Blackwell theorem

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") Note:- here $Y_1$ is a sufficient ...
user avatar
  • 802
0 votes
1 answer
30 views

why sufficient statistics is a vector in this case?

why sufficient statistics is a vector in this case? Let $\mathbf{X}=\left(X_{1}, X_{2}, \ldots, X_{n}\right)$ be a random sample from the two-parameter exponential distribution with pdf $$ f(x)=\...
user avatar
4 votes
1 answer
130 views

An exercise in "Mathematical Statistics Jun Shao" about the completeness of a 'modified' exponetial family

It is not the first time meeting this problem in StackExchange and I have read the answer to it(the original solution is copied at the bottom, also available in Show a statistic is complete but not ...
user avatar
0 votes
1 answer
32 views

How can I formally reason that the data itself is a sufficient statistic? [closed]

Intuitively, it is very clear that the entire data itself is a sufficient statistic for a parameter of interest. Formally, if random variable $X$ represents data, $S(X)$ is a sufficient statistic ...
user avatar
  • 7,877
2 votes
1 answer
136 views

Sufficient statistic for normal distribution (not iid)

I am a bit confused with this exercise, since I never worked with samples of this type. I would appreciate if you can help me. The exercise is as follows: Let $\{Xi\} \sim N(iθ, 1)$ for $i = 1, .... ,...
user avatar
  • 21
1 vote
0 answers
57 views

Complete and sufficient statistic

Let $X_1\dots X_n$ be iid observations with pdf $f(x|\theta)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$ for $-\infty<x<\infty$ and $-\infty<\theta<\infty$. I need to find a complete sufficient ...
user avatar
2 votes
0 answers
44 views

Minimal Sufficient Statistics and MLE for Parameters in a Piecewise Random Variable [duplicate]

Problem Setting: $X_i$ is i.i.d. from a piecewise distribution which is $$ f_{\theta_1, \theta_2}(x) = \frac{1}{\theta_1+\theta_2}e^{-\frac{x}{\theta_1}}I_{[x>0]} + \frac{1}{\theta_1+\theta_2}e^{\...
user avatar
0 votes
0 answers
56 views

Sufficient but not Complete statistics for binomial experiment

Consider a binomial experiment with probability of success $p$ in which $m$ trials are conducted resulting in $R$ successes. A further set of trials is then conducted until $s$ further successes have ...
user avatar
0 votes
1 answer
58 views

Equally Distributed Data Set Measurement

I will be creating my own dataset with scores ranging from 50.00 to 100.00. How will I say that the dataset I chose is equally distributed and unbiased ? Is there a formula to know this?
user avatar
0 votes
1 answer
137 views

Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete.

I just proof that $T=(X_{(1)},X_{(n)})$ is not complete but now I want to show a more general case. To be more specific I want to show that if a function of a sufficient statistic is ancillary, then ...
user avatar
0 votes
1 answer
95 views

finding UMVUE for $\theta_x$/$\theta_y$

Let Xi ~ Exp($\theta_x$), Yj ~ Exp($\theta_y$), i = 1; ... ; n1, j = 1;...;n2. Find UMVUE of $\theta_x$/$\theta_y$. Since $\bar{X}$ and $\bar{Y}$ are compelete sufficient statistic, by using Lehmann-...
user avatar
  • 53
0 votes
1 answer
89 views

minimal sufficient statistic

Let X ~ Ber(n1; p), Y ~ Ber(n2; p^2), where X and Y are independent. Find a minimal sufficient statistic T and, using a nontrivial function, show that it is not complete. I get confused by having two ...
user avatar
  • 53
2 votes
0 answers
40 views

Finding minimal sufficient statistics for this family coming from the given probability mass function

Let $X_i\big|_{i = 1...n}$ be random sample from the PMF: $P(X_i = 0) = \frac{1-\theta}2;\;P(X_i = 1) = \frac12 ; P(X_i = 2) = \frac\theta2$ where $\theta\in(0,1)$. Find the minimal sufficient ...
user avatar
  • 2,265
3 votes
1 answer
115 views

Is $\max\{-X_{(1)},X_{(n)}\}$ a one dimensional or two dimensional statistic?

Is statistic $\max\{-X_{(1)},X_{(n)}\}$ one dimension or two dimension? I was trying to find the minimal sufficient statistic for $U(-\theta,\theta)$ from $n$ $i.i.d$ random variables $X_i$. The ...
user avatar
  • 521
1 vote
0 answers
54 views

Self-Study, Minimal Sufficient Statistic, MLE, Beta, Correct Argument

The following example is taken from Hogg Introduction to Mathematical Statistics 7e and the exercise is to show that the MLE is a minimal sufficient statistic. I am not 100% sure about my argument. ...
user avatar
0 votes
1 answer
57 views

Self-Study Sufficient Statistics, Pdf with Indicator Function

The example is from the Book Hogg Introduction to Mathematical Statistics Page 384, Chapter 7.2 Sufficient Statistics. Please let me know if my argument for the solution is correct, since I used a ...
user avatar
0 votes
1 answer
57 views

Data Processing Inequality for sufficient statistic case

Consider a Markov chain $ X \rightarrow Y \rightarrow Z $ and assume $Z$ is a sufficient statistic for $X$ (i.e $I(X;Y)=I(X;Z)$), do we have a case for $X, Y$ and $Z$ where $H(Y) > H(Z)$? Here is ...
user avatar
  • 621
1 vote
1 answer
283 views

Is sample variance a complete statistic for variance of a normal distribution if the mean is known?

Suppose $X \sim N(\mu,\sigma^2)$. I know that $T(X)=(\bar X, S^2)$ is a complete sufficient statistic for $\mu, \sigma^2$ if $\mu, \sigma^2$ are unknown. But if $\mu$ is known, is $S^2$ still a ...
user avatar
1 vote
1 answer
529 views

Question of the minimal sufficient statistics of beta-distribution

The beta distribution with parameters $\alpha>0$ and $\beta>0$ has density$$f(x|\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1$$ ...
user avatar
  • 905
1 vote
1 answer
164 views

Sufficient statistic by factorization theorem

Suppose we have a random sample $X_1,\dots,X_n$ of $X$, where $X$ has the following pdf: $$f_{\mu,\sigma}(x)=\left(\pi\cdot\sqrt{(x-\mu)(\mu+\sigma-x)}\right)^{-1}$$ where $x\in(\mu,\mu+\sigma),\mu\...
user avatar
0 votes
1 answer
91 views

Showing a minimal sufficient statistic

If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$ with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample. Then how can we show that the ...
user avatar
-1 votes
1 answer
120 views

Finding Sufficient Statistics

Let X1, . . . , Xn be a random sample from the following pmf. P(X = 0) = θ, P(X = 1) = 2θ, P(X = 2) = 1 − 3θ, 0 < θ < 1/3 Find a non-trivial sufficient statistic. I start like this: L(θ)=L(θ)=∏...
user avatar
0 votes
1 answer
351 views

When does a sufficient statistic not exist by the Factorization Theorem?

The Neyman Factorization Theorem states the following: Let $f(x_1, ..., x_n; \theta)$ denote the joint pmf or pdf of $X_1, ..., X_n$. Then $T = t(x_1, ..., x_n)$ is a sufficient statistic for $\theta$ ...
user avatar
  • 1,603
1 vote
0 answers
73 views

Minimally Sufficient Statistics Partition Intuition

I am trying to understand the intuitive idea of a minimally sufficient statistic. It is my understanding that a statistic $T$ is minimally sufficient for $\theta$ for a family of populations $X\sim P_\...
user avatar
  • 1,174
1 vote
1 answer
97 views

Sufficient statistic for $(\theta,j)$ when $X_i\sim f_{\theta,j}$

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables with pmf $f_{\theta,j}(\cdot)$ where $\theta \in (0,1)$ and $j=1,2$. $f_{\theta,j}:$ pmf of Poisson $(\theta)$ when $j=1$ and $f_{\theta,j}:$ pmf of ...
user avatar
2 votes
0 answers
45 views

Minimally Sufficient Statistics

I'm trying to find the minimally sufficient statistic where $\{X_i\}_{i=1}^{n}$ are iid from the following family of populations: $$P=\{U(0,\theta): \theta>0\}$$ I looked at the ratio of the ...
user avatar
  • 1,174
4 votes
1 answer
993 views

Full Rank Exponential Families

I am trying to better understand the importance of full rank exponential families of distributions i.e. a family of populations dominated by a $\sigma$-finite measure such that the radon-nykodym ...
user avatar
  • 1,174
1 vote
1 answer
1k views

Complete Sufficient Statistic for double parameter exponential

I am trying to show that $(X_{(1)}, \sum_{i=1}^{n}(X_i-X_{(1)})$ are joint complete sufficient for $(a,b)$ where $\{X_i\}_{i}^{n}\sim exp(a,b)$. I know the joint pdf is $$\prod_{i=1}^{n}\frac{1}{b}...
user avatar
  • 1,174
1 vote
1 answer
110 views

Can someone clear my understanding of sufficient statistics?

The definition of sufficient statistics says that the conditional distribution of a sufficient statistic, say $S$, must be independent of the unknown parameter,say $\theta$. Consider the $Ber(\theta)...
user avatar
0 votes
1 answer
43 views

Is the range of a sample of size n a "sufficient statistic"?

I know that every order statistic themselves are sufficient statistic. The range is the max minus the minimum. Is the range also a sufficient statistic because it is a function of two sufficient ...
user avatar
1 vote
2 answers
1k views

Sufficient statistic for Double Exponential

Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ...
user avatar