All Questions
Tagged with data-sufficiency or sufficient-statistics
138 questions
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Pivot From a (Sufficient) Statistic
Asking for help in approaching a question from a Statistics textbook:
Let $X_1, X_2, ..., X_n$ independent and identically distributed with
density function $f_ {\theta}(x)$ and $T_n(X_1,X_2,...X_n)$ ...
- 713
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43
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Is every statistic sufficient in this case?
I’m working on the following exercise:
Exercise: Let $g$ be a positive integrable function defined on $(0, \infty)$. Define the probability density function
\begin{align}
f_{\theta, \eta}(x) =
\begin{...
- 363
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56
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Possible fallacy in the proof by Casella and Berger?
I'm confused about the proof in Chapter 6 of the textbook "Statistical Inference" by Casella and Berger. Below is a summary (by me) of the argument the authors gave:
Two experimenters assume ...
- 41
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Why is it called a sufficient statistic and not a sufficient estimator? [closed]
I think the question is clear enough. We say unbiased estimator, efficient estimator, consistent estimator, why not sufficient estimator? All estimators are, by definition statistics, although not all ...
- 433
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Why the idea of sufficiency even work
The Experiment
Let there be 1000 Green doors, 99 Red doors, 1 blue doors. A door is chosen such that the probability that a red door is picked is $p$, a blue door is picked is $\frac{p}{2}$, a green ...
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Factorization theorem for sufficient statistics: interpretation as pdfs?
By the factorization theorem (Fisher-Neyman), we have that a statistic $ T(X) $ is sufficient if and only if there exists a factorization: $ f(x|\theta) = g(T(x)|\theta)h(x) $. Notation follows ...
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What's wrong with my counterexample: minimal sufficient statistic for Cauchy
It is well known that the order statistics are minimal sufficient for the Cauchy distribution. On the other hand we have the theorem that $T(X)$ is a minimal sufficient statistic if the ratio $f(X|\...
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UMVUE for geometric distribution when counting the number of failures before success
I have encountered the following problem:
Let $X_1$, $\ldots$, $X_n$ be an independent random sample from the distribution with p.d.f.
$$f(x;p)=p(1-p)^x, ~~x=0,1,2,\ldots,~~0<p<1.$$
(a) Find a ...
- 157
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73
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Vector of order statistics is sufficient
I was trying to solve the following problem:
Let $X_1, \dots, X_n$ be i.i.d with some continuous distribution $F$. Show that the vector of order statistics, $(X_{(1)}, \dots, X_{(n)})$ is sufficient ...
- 363
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230
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Order statistics are minimally sufficient for double-exponential
In the problem of finding a minimal sufficient statistic for the model: Let $X_1, \ldots, X_n$ be a random sample whose common density is
$$
f_\theta(x) = \frac{exp\{- | x - \theta| \}}{2}, x \in \...
- 175
2
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1
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151
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Probability density function of the sufficient statistic
I am studying Hogg and McKean's "Introduction to Mathematical Statistics" and the following is a theorem whose proof he has asked in his exercise.
Exercise 7.5.8. If $X_1, X_2, \cdots , X_n$ ...
2
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206
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Do minimal sufficient statistics always exist?
In Mathematical Statistics: Basic Ideas And Selected Topics - Vol 1, a sufficient statistic, $T(X)$, is said to be minimally sufficient if for any other sufficient statistic, $S(X)$, there exists a ...
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2
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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 ...
- 6,428
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61
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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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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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