All Questions
Tagged with ancillary-statistics mathematical-statistics
20 questions
2
votes
2
answers
125
views
Why variance of least square estimator in simple regression is conditional on predictor?
I'm a new to statisics and now reading the book Applied Linear Regression by Sanford Weisberg. I may be asking a non-sensical question, but why does variance of least square estimator $\hat{\beta_0}$ ...
- 23
1
vote
2
answers
55
views
Does the factorization theorem prove that the simplest factorization of the PDF is the most informative?
Let me state the factorization theorem as: the existence of a PDF factorization where $X$ depends on the parameter only through $T(X)$, proves $T(X)$ is sufficient, defined as conveying the maximum ...
- 642
7
votes
1
answer
450
views
What is a good journal for submitting my article on a conjecture in theoretical statistics, re: ancillary complement for correlation?
I'm working on a draft of a statistics article, and I'd like to plan for the journal where I'll ultimately submit. My problem is, the article topic is somewhat abstract—it's a conjecture in ...
1
vote
0
answers
55
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Does this distribution belong to the exponential family? [duplicate]
I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
0
votes
1
answer
147
views
Ancillary function of a random vector, which is independent of change of origin and scale
Let
$(X_1,\ldots,X_n)$ be a random vector, whose distribution involves unknown: location parameter $\mu$ and a scale parameter $\sigma>0$. It follows, that any measurable function $f(X_1,\ldots,X_n)...
- 382
3
votes
1
answer
754
views
How can I show that $(\bar{X}, S^2)$ is independent of $(X_{(n)}-\bar{X})/S$?
Let $X_1,\ldots,X_n$ be a random sample from $N(\mu,\sigma^2)$ with both parameters unknown. How can I show that $(\bar{X}, S^2)$ is independent of $(X_{(n)}-\bar{X})/S$?
Work:
I am quite confident ...
- 2,073
1
vote
1
answer
46
views
What are the broadest class of distributions for which the range statistic is ancillary to the expectation of the random variable?
Let $X_1,X_2,X_3$ be iid random variables such that $E(X_1)=\mu$
Define $X_{(3)}$ and $X_{(1)}$ as the maximum and minimum order statistics respectively.
I know that if $X$ is normal, $R=X_{(3)}-X_{...
- 73
3
votes
1
answer
3k
views
Is Uniform distribution [a,b] always symmetric?
I want to know whether any uniform distributed random variable is symmetric on any interval [a,b].
My thinking is it is symmetric on any interval [a,b].
i tried to think about a counter-example. But I ...
- 348
3
votes
1
answer
300
views
What is an approximate ancillary statistic?
In the article Assessing the Accuracy of the Maximum Likelihood Estimator: Observed Versus Expected Fisher Information the authors use the expression "approximate ancillary statistic". This expression ...
3
votes
0
answers
688
views
Is Complete Statistic Uncorrelated with Ancillary Statistic [closed]
By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be ...
- 307
8
votes
1
answer
959
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Does Basu's Theorem require minimal sufficiency?
Casella & Berger state Basu's Theorem (Th 6.2.24) as follows:
If $T(X)$ is a complete and minimally sufficient statistic, then
$T(X)$ is independent of every ancillary statistic.
However, in ...
- 3,800
24
votes
1
answer
1k
views
Is there a general expression for ancillary statistics in exponential families?
An i.i.d sample $X_1,\dots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistic if $S(X)$ depends on the sample only through $\frac{X_1}{X_n},\cdots,\frac{X_{...
- 2,490
6
votes
1
answer
382
views
Ancillary statistics:Beta distribution is free of $\beta$?
I am reading Robert V. Hogg Introduction to Mathematical Statistics 6th Version page 409, second paragraph.
$X_1, X_2$ is a random sample from a Gamma $\text{G}(\alpha,\beta)$
distribution with ...
- 4,886
6
votes
1
answer
3k
views
Basu's Theorem Proof
I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below:
Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
- 541
0
votes
0
answers
631
views
How to show ancillary statistic of normal random sample?
Let $X_i \sim N(\mu,\sigma^2)$ and $X_i$ are independent. Then how to show that:
$$
T = \left(\frac{X_1-\bar{X}}{S},\frac{X_2-\bar{X}}{S},\ldots,\frac{X_n-\bar{X}}{S}\right)
$$
$T$ is an ancillary ...
- 113
6
votes
1
answer
527
views
Ancillary statistic: $X_i \sim N(\theta, \theta^2)$
Let $X_1, X_2, ... , X_n$ i.i.d random variables with probability density function $N(\theta, \theta^2)$. Show that
$$T(X) = \frac{X_{(1)}-X_{(n)}}{X_{(2)}-X_{(n)}}$$
is ancillary to $ \theta$.
My ...
- 1,218
3
votes
1
answer
2k
views
Show that a statistic is ancillary
Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Show that
$$ \frac{X_{(1)}}{X_{(n)}}$$
Is ancillary for theta
I coulxnt find a way of doing it that looks convenient. Any idea?
P.s: $X_{(i)}...
- 1,218
2
votes
1
answer
1k
views
Showing that a statistic is ancillary for a parameter
Working through a HW problem, and a hint is that for a decision rule
$$T(X) = \frac{X_{(1)} + X_{(n)}}{2}$$
Then
$$T - \bar{X} $$
is ancillary.
Intuitively this makes complete sense, but I am ...
- 131
3
votes
1
answer
87
views
Ancillary statistic not containing information about sample distribution?
From a note by Jun Shao
If $V(X)$ is a nontrivial ancillary statistic, then $σ(V(X)) ⊂ σ(X)$ is a nontrivial σ-field that does not contain any information about $P$.
I was wondering in what sense &...
- 19.8k
2
votes
1
answer
196
views
Meaning of "a statistic $U$ is ancillary to another statistic $T$"?
From Wikipedia
Given a statistic $T$ that is not sufficient, an ancillary complement is a statistic $U$ that is ancillary to $T$ and such that $(T, U)$ is sufficient. Intuitively, an ancillary ...
- 19.8k