Questions tagged [factorisation-theorem]

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4
votes
2answers
263 views

Finding UMVUE for a function of a Bernoulli parameter

Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ I'm interested in finding the UMVUE of $(1-\theta)^{1/k}$, when $k$ is a positive integer. . I know $\sum X_{i}$ is a ...
2
votes
1answer
67 views

Problem on sufficient statistics

Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ...
2
votes
1answer
177 views

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 ...
0
votes
0answers
58 views

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 ...
1
vote
1answer
93 views

When a function of sufficient statistic is itself sufficient?

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
5
votes
2answers
586 views

Puzzled by definition of sufficient statistics

I am learning about sufficient statistic from Mood, Graybill, and Boes's Introduction to the Theory of Statistics. I am slightly confused by the book's definition of a sufficient statistic for ...
-1
votes
1answer
41 views

Proving sufficiency by showing ratio of statistic pdf to sample pdf is independent of unknown parameter

Let $X_1,...,X_n$ be iid random variables with densities given by $$ f_{x_i}(x|\theta)=e^{i\theta - x}\mathbb{I}_{(i\theta,\infty)}(x), $$ when $x>i\theta $ and $x=0$ otherwise. Let $T$ be the ...
1
vote
0answers
134 views

Minimal Sufficient Statistic for Gaussians with different means

I have the following problems on my Statistics course (using Casella and Berger's book) problem set: 1) Let $Y_{i} = X_{i}'\theta + U_{i}$ where $\theta \in \mathbb{R}^k$ and $U_{i}$ are iid $N(0,...
1
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1answer
349 views

Showing sufficiency using the Fisher-Neyman factorization theorem

I have derived a likelihood function for $\theta$ as follows: $$L(\theta)=(2\pi\theta)^{-n/2} \exp\left(\frac{ns}{2\theta}\right)$$ Where $\theta$ is an unknown parameter, $n$ is the sample size, ...
0
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0answers
48 views

Is an injective statistic always sufficient?

New to the concept of sufficient statistics. Does it follow, in general, from the factorization theorem that any injective statistic is necessarily a sufficient statistic? I cannot find anything ...
2
votes
1answer
199 views

Can the Fisher factorization theorem be understood as a product of densities?

Let $T$ be some random variable on a probability space $\Omega$. Then we have, for $x\in\Omega$: $$P(x) = P(x|T=T(x))P(T = T(x))$$ This equation is nonsense in an arbitrary probability space but ...
1
vote
1answer
687 views

Manually computing fisher test with Big Numbers in Cells

Hi all I have a 2x2 contingency table like this: ...
3
votes
1answer
725 views

Sufficient statistics for a continuous distribution

Below is a discussion of sufficient statistics for a continuous distribution, taken from the third edition of Lehmann's Testing Statistical Hypotheses. I understand the discussion until the underlined ...
2
votes
1answer
505 views

Does Fisher's factorization theorem provide the pdf of the sufficient statistic?

From Wikipedia Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...