# Questions tagged [order-statistics]

The order statistics of a sample are the values placed in ascending order. The i-th order statistic of a statistical sample is equal to its i-th smallest value; so the sample minimum is the first order statistic & the sample maximum is the last. Sometimes 'order statistic' is used to mean the whole set of order statistics, i.e. the data values disregarding the sequence in which they occurred. Use also for related quantities such as spacings.

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### In a sum of high-variance lognormals, what fraction comes from the first term?

If $X_i \overset{\textrm{iid}}{\sim} \text{Lognormal}(0, \sigma^2)$ for $i=1,\ldots,n$ and $Y_1 = X_1 / \sum_{j=1}^n X_j$, then I would expect that a particular* limiting distribution of $Y_1$, ...
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### Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n

For random variables $X_1, \cdots, X_n$, we denote the order statistics by \begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ &...
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### Rough answer for the maximum of absolute value of $n$ standard gaussians (Computer Age Statistical Inference Problem 1.3)

I am working through "Computer Age Statistical Inference" as a self-study and am stuck on the follow exercise (1.3): The details of equation 1.6 are unimportant for the exercise, so far as ...
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### Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution

Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$. Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$ How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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### How should I best to use reported stats on the Tippy-top?

Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
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### Can I estimate the mean of a dataset if I have its standard deviation and a portion of the full data that is higher than some threshold?

I have a partial set of measurement data that is limited due to my tool's sensitivity. I know that the data is approximately normally distributed and I have a standard deviation from another data set ...
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### Showing that $X_{(1)}$ is sufficient for shifted exponential distribution

If the pdf of a random sample is $f(x)=e^{-(x-θ)}$ where $x \geq θ$, Show that $T=X_{(1)}$ is a sufficient statistic for $θ$. Can one show that $T$ is a sufficient statistic for $θ$ in the following ...
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### Lower bound on distance between ordered statistic

I have found the following inequality in a manuscript: Let $S_{t}^{i}$ and $S_{t}^{i^{\prime }}$ denote the $i/N$-th and $i^{\prime }/N$-th cross sectional order statistics. We require: \begin{align} \...
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### Distribution of minimum gap of $n$ points in the unit interval

I am trying to find the expected minimum and maximum distance between consecutive points on the unit positive rea interval .I have tried the following so far :Given $n$ uniform random variables on the ...
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### Which statistical test is suitable to compare order statistics of two independent samples?

Say I want to compare two order statistics (say the 2nd largest value or min value) of two samples. Let's not make any distributional assumptions except that the variance is finite? Is something like ...
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### Why does R say 'cannot compute exact p-values with ties' when I can do it with pen and paper?

Suppose I have two sets of three numbers: $x_1, x_2, x_3$ and $y_1, y_2, y_3$ and I want to test the Null hypothesis that they are drawn from the same distribution using the Wilcoxon-Mann-Whitney test....
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### Link between the Beta and Exponential distribution

Let $n \geq 1$ be an integer. Let $X \sim \operatorname{Beta}(i, n - i + 1)$ where $i \in \{1, ..., n\}$. Therefore: $$X = \frac{A_n}{A_n + B_n}$$ where  A_n = \sum_{r = 1}^i Z_r, \qquad B_n = \...
This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...