# Questions tagged [sufficient-statistics]

A sufficient statistic is a lower dimensional function of the data which contains all relevant information about a certain parameter in itself.

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### How to derive a pdf of Complete Sufficient Statistic of exponential family

While studying Mathematical statistics through "Introduction to Mathematical Statistics 7th" (by Hogg and Craig), I've been stuck in the Theorem above. The answer of the exercise 7.5.8 is not given in ...
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### replication of minimal sufficient statistic

Suppose we have a minimal sufficient statistic for observations $X_1, ...,X_n$ that are i.i.d from distribution $f(X|\theta)$, namely $T(X) = (T_1,...,T_k)$ which is a $k$ dimensional statistics. Now ...
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### Dimension of minimal sufficient statistic

Is this true that "dimension of every minimal sufficient statistic is less than any sufficient statistic (minimal or not)"?
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### How to identify one-one correspondance in Sufficient Statistics?

The correct answer to the given question is (1),(3) and (4). I understood how 3 and 4 are correct but I could not understand how (1) is also a correct answer. I know that here $\sum_i X_i$ is a ...
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### Is the sample mean complete sufficient for the Expectation when $X \sim F$ where $F$ is some symmetric distribution?

Let $X_1,X_2... X_n$ be iid $\sim F$ where $F$ is any symmetric continuous distribution and let $\mid E(X)\mid<\infty$. Is $\bar{x}$ complete sufficient for $E(x)=\int{xf(x)dx}$? Assume that all ...
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### Is it true that Fisher information for a statistic and the sample are equal if and only if the statistic is sufficient?

According to https://en.wikipedia.org/wiki/Fisher_information#Sufficient_statistic we have if and only if, but according to https://projecteuclid.org/download/pdfview_1/euclid.imsc/1362751193 we don'...
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### What is the broadest context where $\bar{X}$ is complete sufficient for $\mathbb E(X)$?

$\bar{X}$ is complete sufficient for $\mathbb E(X)$ if $X$ is Normal with known standard deviation $\sigma$. Are there broader contexts? Like for exponential families in general or more general than ...
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### Prove that the Pitman estimator is itself complete sufficient

It's a homework question, but I just have no idea about it ... Let be random variables according to a distribution having joint density ,where is a location parameter. Assume that there exists a ...
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### Sufficient Statistic and Unbiased Estimate in Exponential Family

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he deals with sufficient statistics in exponential distributions ...
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### Proof of Rao Blackwellization

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he introduces the idea of minimizing the variance of an unbiased ...
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### Minimal sufficient statistics: how should we define and interpret it? [duplicate]

Through my studies of statistics inference, I came into the concept of minimal sufficient statistics. However, I find it a little bit cumbersome. Could someone provide me its definition and how should ...
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### Showing the sample mean is a sufficient statistics from an exponential distribution

Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\theta$), where $\theta>0$ is unknown. If we observe $\overline x = 5.2$ for a sample of $20$ light ...
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### Conditional Testing of Two-way Contingency Independence

I am reading about testing independence in two-way contingency tables from Mood Graybill and Boes's Introduction to the Theory of Statistics and is confused about testing independence. We have a two-...
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### Expectation of Sufficient Statistic for Beta Distribution

I am looking at question 1b of the following notes: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1/asst1.pdf In 1a, I have shown that the Beta distribution has a density that can be written in the ...
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### Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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### Binomial distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be random sample from $X\sim\text{Binomial}(2,\theta)$. (a) Find the least variance from the set of all unbiased estimators of $\theta$. (b) Find a sufficient ...
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### Prove that the order statistics are minimal sufficient for a random sample from an unknown density $f$ [duplicate]

This is Exercise 6.29 from Casella and Berger's Statistical Inference, so I'll just post the question in full, and I'll also post the answer included in the solutions manual. I'll make the part that I ...
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### Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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### Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem

Question: Given, $X\sim N(0,\sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $\sigma^2$. My Attempt: This problem is very easy if we use Fisher–Neyman ...
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### obtain (minimal) sufficient statistic for $\gamma$ knowing the canonical statistic $\theta(\gamma)$

A statistical model for a data set y is an exponential family , with canonical parameter vector $\theta= (\theta_1,\theta_2,..\theta_k)$ and canonical statistic $t(y) = (t_1(y),t_2(y),..t_k(y))$ if ...
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### Checking if a minimal sufficient statistic is complete

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[-\theta, 2\theta]$ with $\theta \in \mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete. I found ...
Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
I've been asked to show the gamma distribution can be written in the form $p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$ where $T(x)$ is a sufficient statistic. .... I have ...