Questions tagged [ancillary-statistics]
The ancillary-statistics tag has no usage guidance.
34 questions
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Why can't one define a complete statistic from a minimal sufficient one by accounting for all first-order ancillaries?
According to this: "If a minimal sufficient statistic $T$ is not complete, then there is a non-trivial first order ancillary statistic $V(T)$, and there does not exist any complete statistic.&...
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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 $\hat{\beta_0}$ ...
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Can someone explain why ancillary statistic is needed in "orthodox statistics"?
I am reading Jaynes' Probability Theory: the Logic of Science. In Chapter 8 Jaynes discusses the issue of ancillary statistic in "orthodox" setting, where he says that ancillary statistic ...
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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 ...
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Why does the sufficient statistic for the bivariate normal not imply a sufficient statistic for the correlation under bivariate normality?
This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven ...
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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 ...
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Show that Sample Mean and Sample Range are independently distributed for a random sample from Normal Distribution
Let $X_{1},\ldots {, X_{n}}$ be iid random variables with $X_{1} ∼ N(µ,\sigma ^{2}).$Let $\bar{X}= \sum_{i=1}^{n} \frac{X_{i}}{n}$, $R=max_{1\le i \le n} \{X_{i}\}$-$min_{1\le i \le n}\{X_{i}\}$.Show ...
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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 ...
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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)...
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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 ...
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What does the assumption of the Fisher test that "The row and column totals should be fixed" mean?
As this source states, one of the assumptions to perform Fisher's exact test of independence is that the row and column totals should be fixed. However, I find the explanation coming with it pretty ...
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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_{...
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Is Dixon's Q statistic ancillary for normal data?
Dixon's Q statistic is the ratio of the "gap" between an outlier and the nearest value, over the range of the data.
I would like to know is if this is ancillary to the parameters of the normal ...
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Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics
Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
user242554
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Proof that the mean is a complete sufficient statistic and the sample variance is an ancillary statistic
I have $X_1, X_2, ..., X_n $ that are random samples from the single variate $N(\mu,\sigma^2) $. I want to prove that the mean $\bar{X}$ and the sample variance $s_x ^2 = \frac{1}{(n- 1)} \sum_{i=1}^...
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Showing these statistics are ancillary
Let $Z_i = X_{(n)} - X_{(i)}$ for $i=1,2,\dots,n$ where $X \sim N(\mu, 1)$, and $X_{(i)}$ is the ith order statistic of the sample.
I want to show $Z=(Z_1,\dots,Z_{n-1})$ are ancillary for $\mu$.
My ...
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Rss and sample variance indipendence in simple linear regression
Suppose that $ (X_1 ,Y_1...X_n,Y_n) $ is an i.i.d. random sample from a simple homoschedastic linear model $Y=\alpha +\beta X+e $ , with $e|X \sim N(0,\sigma_e^2)$.
I want to understand if $ \frac{...
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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 ...
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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 ...
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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 ...
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Is frequentist conditional inference still being used in practice?
I've recently reviewed some old papers by Nancy Reid, Barndorff-Nielsen, Richard Cox and, yes, a little Ronald Fisher on the concept of "conditional inference" in the frequentist paradigm, which ...
user75138
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Can someone explain the concept of ancillary statistics in layman's terms?
I'm having a hard time trying to relate or understand it in the simplest way (without solving).
"Without solving" in a sense that I don't have to solve for the marginal distribution of T2, if for ...
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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 ...
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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_{...
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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 ...
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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 ...
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The range is an ancillary statistic for location families
I am having trouble understanding why the range is considered an ancillary statistic for location family pdfs ... I will try to set up the proof for this and then point out where I am having trouble ...
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What is the difference between conditioning on regressors vs. treating them as fixed?
Sometimes we assume that regressors are fixed, i.e. they are non-stochastic. I think that means all our predictors, parameter estimates etc. are unconditional then, right? Might I even go so far that ...
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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 ...
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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 ...
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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)}...
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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 ...
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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 &...
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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 ...
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