# 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. Order statistics are widely used in non-parametric inference.

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### What does “Defining” a maximum or minimum mean in an order statistics problem?

I would like to understand what does this problem mean by "defining" a random variable. I know that $W$ represents the maximum of the order statistics $X_1$, $X_2$, $X_3$, and that $Z$ represents ...
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### Expected time for multiple independent coupon collectors to finish

I have a problem where there are $C$ independent coupon collectors and there are 5 coupons in total. All $C$ collectors have already collected the first three coupons and have to finish collecting the ...
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### How to use Delta Method to find the asymptotic variance of $M_X$

Set up: Let $(U_i)_{i=1,...,n}$ be iid Uniform $[0, 1]$ where $n$ is odd. Let $M_U$ be the sample median, I have already found the mean of this median is $\frac{1}{2}$ and variance $\frac{1}{4n+8}$. ...
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### Is it possible to get the sample distribution $F_X(x)$ out of a second order statistics $F_{X_{(2)}}(x)$?

If I observe the distribution of the second order statistics $F_{X_{(2)}}(x)$, are there ways to back out the sample distribution $F_X(x)$, such that $F_x(x)$ satisfies the properties of a CDF. I ...
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### Non-Parametric statistics Finding distribution

I didn't understand this one question in my book , I looked at the k out of n system but didnt get the method in order to solve Find distribution for n out of k system reliability
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### Question about order statistic in Introduction to Mathematical Statistics 8th edition

Here is the derivation of marginal pdf of an order statistic in page 255 of Introduction to Mathematical Statistics 8th edition I don't understand why it is \begin{align} \int_{y}^{b}[F(w)]^{\beta-1}...
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### Order statistics and indices

Let $X_1,\dots, X_n$ be iid continuous random variables with cdf $F$ with support $[a,b]$. Let $x,y\in [a,b]$ such that $x<y$. Let $k\in\{0,\dots,n\}$. Fix some $i\in\{1,\dots,n\}$. I would like ...
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### Joint PDF of all $n$ order statistics from distinct populations?

Let $X_1, \ldots, X_n$ be $n$ i.i.d. variables with probability density $f(x)$. Let $X_{(1)}, \ldots, X_{(n)}$ be the ordered statistics. Then the joint probability density of all $n$ order statistics ...
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### Gumbel distribution and exponential distribution

The Gumbel distribution term in Wikipedia says: Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution approaches ...
Imagine I generate $N$ real numbers with a uniform distribution between $0$ and $1$. I sort them in ascending order. And I calculate the differences between each consecutive pair. For example, for ...