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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 ...

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order statistic, distribution of a partial sum

Please let me know if I can clarify my question in any way. I want to figure out the distribution of a partial sum of k largest observations in a n sample from a non-central Chi-square distribution. I ...
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16 views

The CDF of the maximum of some function of the maximum two order statistics

Let the random variables $X_1,\,X_2,\,\ldots,\,X_K$ be i.i.d. exponential random variables with parameter 1. Also, let the random variables $Y_1,\,Y_2,\,\ldots,\,Y_K$ be defined similarly. Now let $...
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1answer
29 views

Add component to equation so a value turns into a negative?

This is a variation of another question I posted. The difference is that here we have an equation that mostly works. We just need help finding how to push a Case. Honestly, we're just trying out ...
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60 views

On the convergence in probability of $n^{-d}\sum\limits_{k=1}^n (X_{(k)} - Y_{(k)})^2$ to $0$, for every $d>0$

For $n \in \mathbb{N}$, let $(X_1, \dots, X_n)$ and $(Y_1, \dots, Y_n)$ be iid. samples from the same distribution. I write $X_{k:n}$ the $k$-th order statistic out of a sample of size $n$. I am ...
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1answer
40 views

The CDF of the summation of independent and dependent random variables

I want to evaluate the following probability $$\text{Pr}\left\{\frac{Y_1}{X_1}+\frac{Y_2}{X_2}\leq z\right\}$$ where the support of all random variables is $[0,\,\infty)$, but $Y_1\leq Y_2$, i.e., ...
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14 views

Order statistics of a truncated distribution

Say I have some exponential distribution with rate parameter 1. The expected value of the order statistics for this has a nice closed form see here. Now say I want to truncate this distribution to the ...
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33 views

Sufficiency with order statistics

What if we need to look for a sufficient statistic. We do the maths and we end up with a specific formule (with help of the factorization criterion) and we have the random variables X,i bounded; 0 <...
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25 views

The CDF of a summation of random variables with order statistics

Suppose I have the random variables $Z_k=X_k/Y_k$ with a PDF $f_{Z_k}(z_k)$ for $k=1,\,2\,\ldots, K$, where $\{X_k, Y_k\}$ are i.i.d. random variables. I can find $$\text{Pr}\left[\sum_{k=1}^3Z_k\leq ...
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1answer
24 views

Analysis of a stochastic process for shuffling an ordered list

Recently I needed a way to shuffle an ordered list with a controllable degree of randomness. To state it more formally, given an ordered list $\mathcal{X}=(x_1,x_2,\dots,x_n)$ and a number $r\in[0,1]$,...
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4answers
49 views

Is it true that $\max(-x_{(1)},x_{(n)})=\max_{1\le i\le n}|x_i|$?

Let $x_1,x_2,\cdots,x_n$ be a set of $n$ observations where $x_i\in(-a,a)\,,i=1,2,\cdots,n$ for some $a>0$. Suppose $x_{(1)}<x_{(2)}<\cdots<x_{(n)}$ are the ordered observations. Is it ...
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What is the interpretation of (y bar square/variance) in 'Nominal is Best' method of Taguchi Design?

If y bar is the mean and s is the standard deviation then what is the interpretation of their ratio? Why they are squared?
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probability of 1st order statistics is smaller than $x$ and 2nd order statistics is larger than $x$

Suppose that there $X_1,X_2,\cdots,X_n$ are drawn independently drawn according to a CDF $F$ and pdf $f$. Let $X_{(k;n)}$ be the $k$-th order statistics. so that we have $X_{(k;n)}\leq X_{(k+1;n)}$. ...
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1answer
34 views

Mean of top two realizations when one of them is known (order statistics)

suppose that $x_1,x_2,x_3$ are independently drawn according to a CDF $F$. I understand the mean of highest or second highest value can be found using the order statistics. My question is if we ...
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1answer
30 views

Expected value of minimum of $n$ samples from a $\chi^2$ distribution

Suppose I have $n$ sets of empirical data, each with Gaussian noise with unit variance $\sigma^2=1$, and each containing $\nu$ points. I fit some model to each dataset, and find that the sums of the ...
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23 views

The CDF of multiple order statistics

Let $X_{(1)}\geq X_{(2)}\geq\cdots X_{(K)}$ be the order statistics of the random variables $X_1,\,X_2,\,\ldots,X_K$, which are independent and identically distributed exponential random variables ...
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52 views

Relationship between order statistics and equality with uniform distribution

I have read somewhere (unfortunately I cannot locate the exact text anymore) that there exists an equality in distribution between the order statistics of any continuous distribution and the uniform ...
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6 views

Probability to uniformly draw smaller than minimum/order statistic of uniform r.v.

I have a solution for my problem but am unable to verify it by simulations. Is there anything wrong with my reasoning? Assume the following setting: We have $N$ i.i.d. uniformly distributed ...
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24 views

Lower/Upper quartile problems

I have lots of problems with the idea of the lower quartile. Firstly, consider this example: Find the lower quartile of 1,2,3,4,5,6,7,8,9,10 On using the formula (n+1)/4 we achieve (10+1)/4=2.75 ...
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29 views

Equal probability of rank implies identically distributed?

Let $X$ and $Y$ be independent, continuous random variables. It is easy to see that if $X$ and $Y$ are identically distributed then $P(X < Y) = 1/2$. Is the converse true? That is, for $X,Y$ ...
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22 views

Standardising maximum of Uniform distribution

Let $M_n = \max(U_1,\ldots,U_n)$ , the maximum of a sample of size n from $U(0,1)$ distribution. We want to see what happens with the distribution of $M_n$ (properly standardised or normalised) as $n ...
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1answer
48 views

Maximum Likelihood Estimator of : $f(x) = \theta x^{-2}, \; \; 0< \theta \leq x < \infty$

Exercise : Find a maximum likelihood estimator of $\theta$ for : $f(x) = \theta x^{-2}, \; \; 0< \theta \leq x < \infty$. Attempt : $$L(x;\theta) = \prod_{i=1}^n \theta x^{-2} \mathbb{I}_{...
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1answer
79 views

Convergence of Truncated Expectation of Order Statistics $E[Y_{k:N}|Y_{k:N}>v]\rightarrow v$

Setting Let $(X_i)_{i\leq N}$ be a set of i.i.d. random variables, with $X_i$ mapping to some interval $[a,b]$. Let $Y_{k:N}$ be the $k$th order statistic of this set and $v\in[a,b]$. Denote by $f_X,...
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23 views

Is there an explicit expression for the density of a data point minus the sample median?

Let $X_1, X_2,\ldots, X_n$ be i.i.d. random variables with a nice density function (for example the normal(0,1) density). Denote $M_n = \text{median}(X_1,X_2,...X_n)$ and assume that $n$ is odd for ...
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24 views

Finding complete sufficient statistic

let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic. I know that $T=(X_{(1)}, X_{(n)} )$ is a sufficient statistic for $\theta$.Also i know T is ...
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32 views

A random vector of continuous random variables and its Order Statistics

I am reading a book Introduction to Probability by Joe Blitzstein, Jessica Hwang. I was going though a section on Order Statistics, which I have mentioned below. Let $X_1, X_2, \cdots, X_n$ be i.i.d ...
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1answer
67 views

Distribution of a sample IQR

Let $X_i ∼ U(0, 1), i = 1, . . . , 20, iid$. IQR = $F^{−1}(.75)−F^{−1}(.25)$ = $X_{(15)}−X_{(5)}$ in this example as n = 20. a. Find the distribution of the random variable W = IQR. b. ...
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1answer
45 views

Show that $Y_1=\frac{X_{(1)}}{X_{(2)}},Y_2=\frac{X_{(2)}}{X_{(3)}},\dots, Y_{n-1}=\frac{X_{(n-1)}}{X_{(n)}}$, and $Y_{(n)}=X_{(n)}$ are independent [duplicate]

I am into order statistics lately, and I have a problem here. Let $X_1,X_2,..,X_n$ be a random sample from $f(x)=1 , 0<x<1$. Show that $Y_1=\frac{X_{(1)}}{X_{(2)}},Y_2=\frac{X_{(2)}}{X_{(3)}},\...
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47 views

Showing that $\frac{X_{(i)}}{X_{(n)}},i=1,2,…,n-1$ and $X_{(n)}$ are independent for a population with df $F(y)=y^{\theta}$

Let $X_1,X_2,...,X_n$ be i.i.d with df $F(y)=y^{\theta}, 0<y<1, \theta>0$. Show that $\frac{X_{(i)}}{X_{(n)}}$, for $i=1,2,...,n-1$ and $X_{(n)}$ are independent. I found the population ...
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32 views

What is the probability that the inequality $Y_{(m')}<X_{(m+1)}<Y_{(m'+1)}$ will hold?

$X_1,X_2,...X_n$ and $Y_1,Y_2,..Y_n$ are independent random samples taken from the same continuous distribution with distribution function $F$.What is the probability that the inequality $Y_{(m')}<...
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Probabilities in Matching to the closest point

Suppose that, in a unit interval, there are $n$ red dots and $n$ blue dots. Red dots are drawn independently from a CDF $F$ and blue dots are drawn independently from a CDF $G$. Suppose that we ...
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72 views

For a random sample from the distribution $f(x)=e^{-(x-\theta)} , x>\theta$ , show that $2n[X_{(1)}-\theta]\sim\chi^2_{2}$

Show that for a random sample of size $n$ from the distribution $f(x)=e^{-(x-\theta)} , x>\theta$ , $2n[X_{(1)}-\theta] \sim \chi^2_{2}$ distribution and $2\sum_{i=2}^{n}[X_{(i)}-X_{(1)}]$ also has ...
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30 views

Expected shortest distance in a rideshare problem

In relation to this question, suppose that there are $n$ passengers and $n$ taxicabs in a town of $[0,1]$. Passengers are independently located according to a CDF $F$ and pdf $f$, and taxicabs are ...
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155 views

Obtaining a positive definite covariance matrix of order statistics

Suppose $X_1,\dots,X_n$ are independent samples from some distribution with known absolutely continuous CDF $F:\mathbb{R}\rightarrow[0,1]$. Let $X_{(1)},\dots,X_{(n)}$ denote the order statistics, i.e....
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2answers
45 views

The joint pdf of three or more order statistics is given by $f_{x_{(1)},x_{(2)},…x_{(n)}}(x_1,x_2,..x_n)=n! f(x_1)f(x_2)..f(x_n) $

The joint pdf of three or more order statistics is given by $f_{x_{(1)},x_{(2)},...x_{(n)}}(x_1,x_2,..x_n)=n! f(x_1)f(x_2)..f(x_n) , \ -\infty<x_1<x_2<...<x_n<\infty $ How can I derive ...
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1answer
51 views

Distribution of the closest sample to a certain point

Suppose there are $n$ samples drawn from a CDF $F$ and pdf $f$ with support $[0,1]$. The distribution of a sample that is closest to zero can be found using the theory of order statistics. If I ...
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24 views

Order statistics of order statsitics

I have these random variable $g_1,\,g_2,\cdots,\,g_K, h_2,\,h_2,\,\cdots,\,h_K$, where all random variables are i.i.d. Suppose $\{g_k\}$ are ordered as follows $$g_{(1)}\leq g_{(2)}\leq\cdots\leq g_{(...
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31 views

Asymptotic result of $\mathbb{E}\left[\max_{i=1,\cdots,K} |h_i|^2 \right]$?

I am trying to understand the following asymptotic results in one of the article which gives no proof (and no reference either). I just can't see how trivial this is. Given i.i.d. $h_i \sim \mathcal{...
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1answer
21 views

$0<x<y<z<1, P(z-x<1/3)=$?

I'm trying to solve a question about 'order statistics'. $X, Y, Z$ follow uniform distributions between 0 to 1. $(X , Y, Z \sim U(0,1))$ An unequality is given that $0< X < Y < Z < 1$. ...
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1answer
36 views

Joint pmf of $(X_{(1)},X_{(n)})$ when $(X_i)_{1\le i\le n}$ is a random sample from discrete uniform population

Let $(X_1,X_2,\cdots,X_n)$ be a random sample of size $n$ drawn from a population having pmf $P(X=j)=\frac{1}{N}\mathbf1_{j\{1,2,\cdots,N\}}$. What is the joint pmf of $X_{(1)}$ and $X_{(n)}$ ? Let $...
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101 views

Complete Statistics for Uniform Distribution

Let $X_1, X_2, ..., X_n$ be independent and identically distributed uniform $U(0, \theta)$ distribution where $0 < \theta < \infty $. Show that $T(X)=max X_i$ is a complete statistics. My ...
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1answer
27 views

A not complete Statistics

Let $X_1, \dots,X_n$ be i.i.d. from the uniform distribution on the interval $(0, \theta)$. How can I show that for $\theta>1$ the $X_{(n)}$ statistic is not complete? I know and that if $\theta&...
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1answer
17 views

Deriving order statistics with respect to sample size

I am considering a sample of n iid random variables, distributed according to a law F. I am interested in deriving the CDF of the i-th order statistics, $F_i(t)=\sum_{k=i}^n \binom{n}{k}F(t)^k(1-F(t))^...
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Exercise 2.20 from “Mathematical Statistics - Jun Shao”

Let ${ \left\{ X_{ i } \right\} }_{ i=1 }^{ n } \sim E(a,\theta)$ where $a \in {\rm I\!R}$, and $\theta > 0$. Show that the smallest order statistic, $X_{(1)}$, has the exponential ...
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Proving expected value of X as sum of 1-CDF [duplicate]

I'm fairly new to stats. I was wondering how to do this practice question that I seem to be stuck on.
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1answer
20 views

Number of sampling and expected value of 3rd-best draw

Suppose $n>m$. Person 1 samples $n$ times (i.i.d.) Person 2 samples $m$ times (i.i.d.) $x_{(n-2)}$ is the 3rd-highest number obtained from $n$ sampling from distribution $F$ $y_{(m-2)}$ is the ...
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17 views

Expected value of a random variable after transformation and taking maximum with another random variable

Two coordinates in the domain of a bivariate surface. $(S,O)_1$ and $(S,O)_2$ $S\sim U(0,1)$ and $O \sim U(0,1)$ $S$ and $O$ are independent and identical Those are two students that have ...
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0answers
41 views

expectation of maximum of iid random variables from normal distribution. [duplicate]

1) How to find expectation of max of random variables , i.e : $\mathbb{E}[max(x_1,x_2,\dots,x_n)]$ where $x$ are IID random variables from $\mathcal{N}(\mu,\sigma^2)$. I know that CDF is $F(x)^n$ ...
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2answers
85 views

Intuitive solution of an order combinatorics problem

I am seeking a quick and intuitive one-or-two-step solution to the following combinatorics/probability problem. Suppose $m$ men and $w$ women compete in a tournament. All rankings are equally likely. ...
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1answer
33 views

Formulating an algorithm for generating the smallest $k$ among $n$ i.i.d. uniform variables

Let $X_1,\ldots,X_n\sim U[0,1]$ be independent uniform variables, and let $X_{(1)}, X_{(2)},\ldots,X_{(n)}$ be their order statistics (i.e., $X_{(i)}$ is the $i$'th smallest among $X_1,\ldots,X_n\sim ...
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1answer
47 views

Proving an identity of partial binomial sum from a statistical context without using combinatorics

The identity is: $$\sum_{i=m}^n {n \choose i}p^i(1-p)^{n-i}=m{n \choose m}\int_0^p {t^{m-1}(1-t)^{n-m}}\,\mathrm dt\quad(0{\le}m{\le}n)$$ How I met it: $$$$The CDF of $m$-th order statistic is $$P(X_{(...