Questions tagged [ancillary-statistics]

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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 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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30 views

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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1answer
71 views

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 ...
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1answer
83 views

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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0answers
70 views

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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1answer
1k 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 ...
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168 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 ...
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512 views

Is Complete Statistic Uncorrelated with Ancillary Statistic

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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1k views

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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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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258 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 ...
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2k views

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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466 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 ...
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586 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 ...