# Questions tagged [ancillary-statistics]

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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 ...
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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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### 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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### 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 ...
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 ...
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 ...