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Questions tagged [data-sufficiency]

questions regarding sufficiency of information.

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Using statistics in petroleum engineering [closed]

How can engineers use mathematical statistics to determine relationship between number of wells and oil, gas and water production?
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28 views

Party Invitations Problem [closed]

There is a party on some Sunday next year, it is open to all. a)Everyday at some point, a person comes and drops a slip with contact details in a dropbox. b)A person on a given day could comeback ...
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19 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
45 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
44 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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23 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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72 views

Minimum sufficient statistic for logistic regression model

For the question in the link below, I am seeking the minimal sufficient statistic for $\theta$={$\beta_1$,$\beta_2$} in the linear regression model given. I have taken the ratio of likelihoods $...
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1answer
43 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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1answer
47 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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54 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
131 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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30 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
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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
164 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
49 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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0answers
130 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
94 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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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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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
601 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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70 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
76 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
107 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
93 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
12 views

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
244 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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0answers
101 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
772 views

Issues using indicator function to find maximum likelihood estimator

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
131 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
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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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0answers
28 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
192 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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143 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
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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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0answers
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 ...
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1answer
44 views

Representation of real or complex numbers as vector of coefficients of polynomials.

Assuming we have a polynomial $$p(x) = \sum_{k=0}^N c_k x^k, \hspace{1cm} c_k \in \mathbb{N}$$ As far as I know we can create find polynomials having roots being any algebraic real number. Are there ...
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1answer
214 views

Show T is sufficient for exponential statistic using conditional distribution approach.

Given iid $X_1,\dots X_n \sim \exp(\beta)$, use the conditional distribution approach to show that $T=\sum_{i=1}^n X_i$ is sufficient for $\beta$. My attempt: $\mathbb{P}\left(\bigcap_{i=1}^n (X_i = ...
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1answer
135 views

Finding Sufficient statistic

Let $X_1,\dots X_n$ be an i.i.d sample from the distribution with frequency function $$P(X=x) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}$$ for $x = -1,0,1$. Use the factorization ...
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1answer
6k views

Minimal sufficient statistics for uniform distribution on $(-\theta, \theta)$

Let $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$. It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\theta$ ...
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2answers
33 views

Significant figures between SI and non-SI units

This question on Chem.SE addresses the question I feel more suitable for math.SE: A person is running 10 km, and wikipedia states a US land mile corresponds to 1609.344 m. Would it be more sensible ...
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1answer
266 views

sufficient statistics to estimate the unknown parameters

I am a beginner in statistical inference and am learning sufficient statistics. As far as I know the distributions conditional on the sufficient statistics doesn't depend on the unknown parameters. I ...
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2answers
152 views

Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$

Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$ I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one ...
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0answers
80 views

Let $\theta >0$ be unknown and suppose that $(X,Y)$ is uniform over the triangular region with vertices at $(0,0)$,$(\theta, 0)$, and $(0,\theta)$.

Let $\theta >0$ be unknown and suppose that $(X,Y)$ is uniform over the triangular region with vertices at $(0,0)$,$(\theta, 0)$, and $(0,\theta)$. Let $(X_i,Y_i)$ be iid as $(X,Y)$. Find a one ...
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1answer
303 views

Show that $T$ is not a sufficient statistic

Let $X_1\: , X_2$ a random sample for $N(\theta ,1).\:$ Show that $\:T=X_1 + 2X_2$ is not a sufficient statistic. I've tried to prove it by contradiction: I assumed that $T$ is sufficient. That ...
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1answer
227 views

File Compression Statistics

Statically, how is the compression of a file possible? As in, a file n bytes long can have $2^{n*8}$ (the 'times 8' part is because there are 8 bits in a byte). So, if you decrease the size then ...
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1answer
773 views

What are the coordinates of point P? SweSAT Data sufficiency

The points $P = (x_1, y_1)$ and $Q(3, 4)$ are on the line $y=kx+m$. What are the coordinates of point $P$ ? Given $m = -2$ $P$ is on the line $x=2$ We need both information together to solve the ...
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1answer
23 views

SweSAT Data Sufficiency

A product has undergone two price increases with the same percentage, ie by x percent each time. How big was the increase in percent each time? ( 1 ) The total price increase was 450 dollars (2) ...
4
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1answer
91 views

sufficient statistics of a sequence of normal random variable

If $X_1, X_2\ldots,X_n$ are independent variables with $X_i \sim \mathcal N(i\theta,1)$, $\theta$ is an unknown parameter. What is a one dimensional sufficient statistic $T$ of this sample? I have a ...
2
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1answer
4k views

Find the sign of $a,b,c$ in $ax^2+bx+c$ given the graph and a coordinate on it.

So my first approach was that, we see that there are $2$ roots. And one is negative and one is positive. $a$ would be evidently positive. The positive one's modulus is bigger than the negative one's. ...