# Questions tagged [data-sufficiency]

questions regarding sufficiency of information.

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### 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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### 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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,...