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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Help developing intuition behind sufficient statistics (Casella & Berger)
Migrated to Cross Validated
I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272):
A sufficient statistic captures all of the ...
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Is my understanding about sufficient statistics correct?
Definition: a statistic $T(X)$ is sufficient for a parameter $\theta$ if the conditional distribution of the sample data $X=(X_1,...,X_n)$ given $T(x)$ does not depend on $\theta$.
My question is: I ...
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Complete Sufficient Statistis with pdf $f(x;\theta) =\theta(1+x)^{-(1+\theta)}$
[Bain Problem 31 p.356] Let $X_1,X_2,\cdots,X_n$ be a random sample of size $n$ from a distribution with pdf
$$f(x;\theta) = \left\{\begin{array}{rr} \theta(1+x)^{-(1+\theta)}, &x>0 \\ 0, &...
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Find minimal sufficient statistic of this random sample with cursed support
Suppose $X_1,X_2,...,X_n$ is a i.i.d random sample with probability mass function $p(x_i,\theta)$ where $x_i \in \{\theta,\theta+1,\theta+2,...\}$ and $\theta \in \mathbb{R}$. I claim that minimal ...
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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?
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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 ...
