Questions tagged [data-sufficiency]

questions regarding sufficiency of information.

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52 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\...
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21 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 ...
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60 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)=∏...
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19 views

Select correct sampling rate for a process, data science

For example, lets say I collected a days worth of a several variables every 0.1 seconds, and I want to collect them every day from now on. If I were to keep the 0.1 second sampling rate my database ...
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1answer
88 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 $\theta$ ...
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43 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_\...
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1answer
57 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 ...
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29 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 ...
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1answer
119 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 ...
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1answer
140 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}...
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1answer
30 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)...
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11 views

Intuitive meaning behind the formal definition of sufficient statistic

According to the definition of sufficiency, A statistic is sufficient for a parameter if the conditional distribution of X given a value of statistic does not depend upon the parameter. What I am ...
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20 views

Algorithm for scoring applications threat level

So I am trying to figure out the best way to take a list of CVSS vulnerability scores(0 - 10 range) for all present vulnerabilities based on lets say "Application X" scans and turn that into a single ...
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50 views

How to solve this integral? Is the data given sufficient?

The integral to be calculated is: $$I=\int_{a}^{b}\frac{f(\frac{a}{x})-f(\frac{b}{x})}{x}dx$$ Really this is the only info given. There is nothing about the nature of $f(x)$ or $a,b$. Since I couldn'...
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199 views

Minimal sufficient statistic for Cauchy distribution; a confusion

I have to find a minimal sufficient statistic for cauchy distribution with parameter $\theta$. I have found this question Minimal sufficient statistics for Cauchy distribution in the site but I have a ...
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1answer
26 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 ...
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62 views

Minimal Sufficient Statistic for $U(0, \theta)$

The definition of a Minimal Sufficient Statistic (MSS) denoted $S(X)$ is $$ \frac{L(\theta;x)}{L(\theta;x)} \text{ independent of $\theta$} \iff S(X) = S(Y), $$ assuming the densities exist and $L$ ...
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2answers
449 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 ...
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28 views

can minimal sufficient be independent of observation?

(1)part(a): can minimal sufficient be independent of observation? $\\ $ let $X$ is one observation from $ U(\theta, \theta+1)$, $\theta \in \{0,\pm1 ,\pm 2 ,\cdots \}$ $T=\lfloor X \rfloor$ is ...
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1answer
441 views

Degree of the minimal sufficient statistic for $\theta$ in $U(\theta-1,\theta+1)$ distribution

Suppose $X_1,X_2,...,X_n$ is a random sample from the Uniform distribution over the interval $(\theta-1,\theta+1)$. By the factorization theorem, it is clear that the order statistics $Y_1=X_\left(1\...
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1answer
64 views

Prove that $\sum^n_{i=1} X_i$ and $\prod^n_{i=1} X_i$ are sufficient statistics for the gamma distribution

This question is set in the statistical context, but my difficulty is more ‘pure math’ in nature, so I have posted it here instead of at the statistics forum. I am to prove that $V := \sum^n_{i=1} ...
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31 views

Hogg Intro to Mathematical Statistics Sufficient Statistic Notation Question

When introducing the notation of a sufficient statistic, Hogg uses a peculiar sort of notation at the bottom of page 381 of the 7th edition textbook: given the statistic $$Y_1 = u_1(X_1,X_2,...,X_n)$$ ...
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77 views

Minimal Sufficient Statistics and Sufficient Statistic

The procedure I have to take is to first identify that both U(X) are unbiased for theta, and that U2(X) = E[U1(X)|T2(X)]. My question is How would that relate to U2 having a smaller variance than U1,...
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73 views

Conditional gamma distribution

Let $X_1,X_2,X_3,X_4$ be iid and $X_1\sim \text{Gamma}(\alpha,\beta)$. Let us fix $$T_1(X_1,X_2,X_3,X_4)=\frac1n \sum_{i=1}^4X_i=t_1,$$ $$T_2(X_1,X_2,X_3,X_4)=\frac{\left( \prod_{i=1}^4X_i\right)^{1/...
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108 views

Minimum Sufficient Statistic for Uniform(θ,θ+1) distribution?

I saw that (min(X), max(X)) are minimal sufficient statistic for this. But I was wondering, why can't we just have min(X) OR max(X)? If distribution is Uniform(θ1,θ2), then it makes sense to have ...
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1answer
350 views

Example of a maximum likelihood estimator that is not a sufficient statistic

I am currently researching on providing some bounds on estimation using some information theoretic tools (I won't expend on that here for now, I may make a post about it later) and turns out that ...
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54 views

(Self-Study) UMVUE of the mean of a normal distribution [duplicate]

Let $X_1,...,X_n$ be a random sample from normal(θ,1). Is there an UMVUE of $θ^2$ here? $X^2-1$ is an unbiased estimator of $θ^2$. First thing that came to my mind is to use Lehmann-Scheffe Theorem. ...
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1answer
212 views

Prove $(\sum_{i=1}^{n}X_{i},\sum_{i=1}^{n}X_{i}^{2})$ is not a complete statistic for $N(\mu,\mu^2)$ distribution

Let $X_{1},\ldots,X_{n}\stackrel{\text{ i.i.d }}{\sim}N(\mu,\mu^{2})$. $T=\left(\sum_{i=1}^{n}X_{i},\sum_{i=1}^{n}X_{i}^{2}\right)$ is a sufficient statistic for $\mu$. Also $T$ is minimal sufficient....
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1answer
368 views

Let X1,…, Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for theta.

Help me to solve this problem about sufficient statistics please.. Let X1,..., Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for ...
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3answers
51 views

Where do we use continuity?

If $f$ is continuous on $\mathbb{R}$, $f'(0)=1$ and $f(x+y)=f(x)f(y)$ for all $x \in\mathbb{R}$, show that $f'(x)=f(x)$ for all $x\in\mathbb{R}$. Solution: It is clear that $f(0)=1$. For each $x$ we ...
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241 views

Minimal sufficient statistic for $\theta$ where $f(x;\theta)$ = $2(1+\theta-x) I_{\theta \le x \le\theta+1}$

I am not able to find the minimal sufficient statistic for the following density function: $$f(x_i;\theta) = 2(1+\theta-x_i)I_{\theta \le x_i \le \theta+1}$$ The function does not belong to the ...
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0answers
117 views

Minimal sufficient statistic for $\theta$ where $f(x;\theta)=\frac{\beta^3}{2}e^{-\beta(x-\theta)}(x-\theta)^2\mathbf1_{x\ge\theta}$

I have this density function for which I am not able to find a minimal sufficient statistic, as required. It does not belong to the exponential families distribution as the support depend also on the ...
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46 views

Equivalent defintions of minimal sufficient statistics

Wikipedia claims that the statistic $S(X)$ is minimal sufficient if and only if $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ $\iff$ $S(x) = S(y)$. It is also claimed that this is a ...
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201 views

data processing inequality-mutual information

suppose that we have a family of probability mass functions ${f_\theta }\left( x \right)$ indexed by $\theta$, and let $x$ be a sample from this distribution. Then from the information theory, we have ...
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2answers
2k views

Minimal sufficient statistic for normal distribution with known variance

Let $X_1, ..., X_n$ be a random sample from the $N(\theta,1)$ distribution. Find a minimal sufficient statistic for $\theta$. Now, I can find a sufficient statistic using the factorisation theorem ($\...
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71 views

Correlation coefficient between early procedure and hospitalization.

I'm preparing a poster to present in a scientific meeting. I've done a retrospective research in our hospital database regarding patients undergoing a particular procedure (which is used for people ...
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1answer
158 views

Sufficiency and Completeness of Gamma Random Variable for Normal Distribution

Let $X\sim N(0,\theta)$ for $\theta>0$. Show that $X^2$ is complete and sufficient for $\theta$. I assume this is referring to $\theta$ as the variance of $X$. I'm unsure of how to show ...
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1answer
151 views

sufficient statistic for uniform

Given that $\theta$ is an integer and that $X_1$ and $X_2$ are independent random variables which are Uniformly distributed on the integers $1, 2, \ldots, \theta$, prove that $X_1 + X_2$ is not ...
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1answer
221 views

Knn Classifier - Sample size influence

I'm working with a K-nearest neighbours classifier, using cross validation to determine k. What I'm stuck on is this: How does total sample size N influence the optimal value of k? My thinking was ...
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1answer
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Is my approach correct to check the sufficiency of the static

Let we have $n$ random variables from Poisson distribution with parameters $\lambda$. It is required to check the sufficiency of the following estimators a). $(X_1,\sum_{i=2}^{n}X_i)$ b). $(X_1,\bar{...
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1answer
453 views

confusion on ancillary of gamma distribution

Here is the question. I am concerned about part (ii). I found out, $T$ is complete sufficient statistic for $\beta$. Now I need to show that $X_{(i)}$ is ancillary. But, for of all, I can not find a ...
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132 views

Conditional on k successess for n independent Bernoulli trials

Question A sequence of n independent experiments is performed. Each experiment is a success with probability p and a failure with probability q = 1 − p. Show that conditional on the number of ...
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1answer
1k views

Issues using indicator function to find maximum likelihood estimator [duplicate]

I am having trouble understanding how to use the indicator function to help find the likelihood. Let $Y_1, Y_2, ... , Y_n$ be a random sample from a population with density function $$ f (y | \...
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1answer
149 views

Is the median a sufficient statistic for a uniform distribution on $(-θ, θ)$?

I have a uniform distribution on $(-θ, θ)$ and I have to find a sufficient statistic. I know that the order statistic [$x_{(1)}$, $x_{(n)}$] are jointly minimal sufficient but I was wondering whether ...
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2answers
2k views

Showing that a statistic is minimal sufficient but not complete uniform distribution

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[\theta-\frac{1}{2}, \theta+\frac{1}{2}]$ with $\theta \in \mathbb{R}$ unknown. Show that the statistic $T(\mathbf{X}) = (X_{(1)}, X_{(...
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32 views

Why do we assume all sample $Y\gt 0$ in uniform distribution of $[0, \theta]$?

So I am calculating sufficient statistic for uniform distribution on $[0,\theta]$ and $\theta \gt 0$. Sample $Y=(Y_1,Y_2,...,Y_n)$ has size $n$. I have $L(\theta, Y)=\prod^n_{i=1}\frac{1}{\theta-0} \...
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1answer
223 views

Completeness of order statistics for normal setup

Suppose we have iid $X_1,\ldots, X_n\sim N(\mu,\sigma^2)$ where $\mu$ is unknown. Let $X_{(1)}, X_{(2)},\ldots,X_{(n)}$be the order statistics. Is the order statistics $\textbf{complete sufficient}$ ...
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183 views

UMVUE of Bernoulli distribution [duplicate]

I know how to show that Y is complete and sufficient for part A using the exponential family form, but how do I get the UMVUE for part B? I know we probably use Y and go for an unbiased function of Y,...
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1answer
2k views

How to find UMVUE

I understand that for part A, we can show that Y is sufficient using the exponential family form. I also understand that for B, we must now use this statistic to find an unbiased UMVUE for theta. How ...
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15 views

Drilling up categoric to interval data?

This is a statical question. I have a data set that provides the number of people readmitted to associated hospitals within 30 days of discharge (a common efficacy measure). At immediate level of ...