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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".

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 .

137 questions
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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 $\theta$ where $X_{(1)}$ and $X_{(n)}$ stands for the minimum…
student
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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) $f(x|\theta) = e^{-(x-\theta)}, \quad \theta < x < \infty, \quad -\infty < \theta < \infty$. This…
Leonidas
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Examples of sufficient statistics for non-exponential family distributions?

I know that the Pitman-Koopman-Darmois theorem says that only exponential family distributions have sufficient statistics whose dimension stays constant as the sample size increases. I further know that the Fisher-Neyman factorization theorem says…
Aaron
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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 derivative can be written as $$…
nvm
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5
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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 reasons.
Xi'an ні війні
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5
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2 answers

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 $T(\mathbf{X}) = (X_{(1)}, X_{(n)})$ is minimal sufficient but not complete. I…
elbarto
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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 moment which is $E(X)=\mu$ given by the…
s l
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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 suffcient, the idea is checking the completeness by…
Small_shizi
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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 intuition guess that the answer is…
Lesley
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Show that $\bar{Y} - \min(Y_{1}, \dots, Y_{n})$ is independent of $\min(Y_{1}, \dots, Y_{n})$

Suppose that $Y_1, \dots, Y_n$ are i.i.d observations from the density $f(y, \theta, \beta) = \beta e^{-\beta(y - \theta)}I_{[y \geq\theta]}$ where $\beta \gt 0$, $\theta \in \mathbb{R}$ are unknown parameters. Let $(T_1, T_2) = (\min(Y_1, \dots,…
Oscar24680
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1 answer

Max. likelihood and sufficient statistic of exponential distribution.

Consider the following probability function of a random variable $Y$: $$ f(y \mid \theta)=e^{-(y-\theta)},\quad y\ge\theta $$ and $0$ otherwise. We take a random sample $(Y_1,Y_2,...,Y_k)$ and want to find a sufficient statistic and a maximum…
Logi
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Nonexistence of UMVUE

In Mathematical Statistics written by Jun Shao(2003), exercise 3.22 claims that Exercise 3.22. Let $\left(X_{1}, \ldots, X_{n}\right)$ be a random sample from $P \in \mathcal{P}$ containing all symmetric distributions with finite means and with…
Eulerid
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3
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2 answers

Fisher-Neyman Factorisation Theorem and sufficient statistic misunderstanding

The Fisher Neyman Factorisation Theorem states that if we have 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 functions $g_{\theta}$ and $h(x)$ such…
FD_bfa
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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 $t>0$, $\Bbb P(X_1 \leq x_1, \cdots, X_n \leq x_n \lvert T \leq t)$ does not…
Kilkik
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3
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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 $i = 1, .... , n$ be an independent, but not identically…
John
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