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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52 views

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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27 views

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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14 views

Expectation of stochastically ordered random variables

I'm reading the paper Invariant directional ordering, and I'm confused by two propositions. The basic definition is If it holds, then My first question is, in proposition $(1.6)$, shouldn't it be $...
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32 views

Expectation of the ith highest draw from a uniform distribution

I can across the following lemma in my class lecture notes: The ith highest draw from a uniform distribution on [0,x] has expectation $\frac{n + 1 - i}{n + 1}$. I'm now attempting to prove this to ...
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54 views

Probabilty that none of n i.i.d uniform samples from R^2 are larger in both coordinates than first.

I am randomly sampling n elements from a square within R^2. They are independently uniformly distributed. What is the probability that for any arbitrarily but fixed previously selected sample index, ...
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48 views

Why is $\operatorname{Var}(X_{(1)}) = \operatorname{Var}(X_{(n)})$ for i.i.d $X_1, \ldots, X_n \sim U(0,1)$?

This is an order statistics question. I'm using the notation found on https://en.wikipedia.org/wiki/Order_statistic. We have $n$ IID random variables $X_1, \cdots, X_n$ that are uniformly distributed ...
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Showing that $U=\min(X,Y)$ and $V = X - Y$ are independent [duplicate]

Let $X,Y$ be independent random variables with the same geometric distribution $\{q^kp\}$, and $U$ be the smaller of $X$ and $Y$, and put $V = X - Y$. Show that $U$ and $V$ are independent. I found ...
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36 views

Testing a discrete random variable for uniformity through some order statistics

I would like to develop a simple test for the uniform distribution of a discrete random variable, but I did not manage to find on Wikipedia or here the relevant informations, and I am pretty sure that ...
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43 views

what is the expected sum of max and min of n positive random variables?

Given a set $R = (R_1, R_2, ..R_n)$ with $n$ positive random variables. The sum is a fixed constant $C$. What's the $E(min(R)+ max(R))$ ? $R_1, ..R_{n-1}$ is uniform distributed from $(0, \frac{C}{...
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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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$k$th Moment of the $j$th Order Statistic

Expectation of Order Statistic Let $X$ be a random variable with probability density function (pdf) given by $f\left( x\mid\alpha, \beta\right) = \frac{\alpha\beta\exp \left[ \alpha\left( 1-1/x^{\...
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Independence of functions of order statistics when the random variables are uniformly distributed

Let $X_1$,$X_2$,…,$X_n$ be $n$ i.i.d. random variables with $f(x)$ as the pdf and $F(x)$ as the cdf in interval $[0,1]$. Let $F$ be uniformly distributed. Let $X_{i:n}$ be the $i^{th}$ order statistic ...
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37 views

Expectation of product of two order statistics

Let $X_1, X_2,\ldots, X_n$ be $n$ i.i.d. random variables with $f(x)$ as the pdf and $F(x)$ as the cdf in interval $[a,b]$. Let $X_{i:n}$ be the $i^\text{th}$ order statistic such that $X_{1:n}\leq X_{...
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40 views

For r.v. Z = max(X1, …, Xn), what is $f_Z(z)$, given X1, X2, …, Xn are independent.

I'm trying to understand the problem in the picture above. What happened to the "dz" between this step: $$f(Z)~ dz~ \Bigg( \int \limits_{-\infty}^{z} f(x)~dx \Bigg)^{n-1}$$ and this step: $$f_Z(z) ...
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31 views

Asymptotic distribution of exponential order statistics

Let $X_1,...,X_n$ be i.i.d. random variables and $X_{(1)}<...<X_{(n)}$ be the order statistics. Assume $X_i\sim \text{exp}(1)$. Find a sequence of constants $a_n$ such that $X_{(n)}-a_n$ ...
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25 views

Asymptotic distribution

Let $X_1,...,X_n$ be i.i.d. and $X_{(1)}<...<X_{(n)}$ be the order statistics. Assume $X_i\sim Unif(0,1)$. For any fixed k, find the asymptotic distribution of $nX_{(K)}$ as $n\rightarrow \...
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21 views

Convergence of uniform distributed order statistic

I am working on this problem: Suppose $X_1, X_2...$ i.i.d~ $U(0,1)$, prove that $n^{-1}\log(1-X^{n-1}_{(n)})$ converges to $0$ in probability. I can derive the distribution function of $X^{n-1}_{(n)}$,...
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70 views

Find the asymptotic distribution of $nX_{(k)}$

Assume $X_i\sim \mathrm{Uniform}(0,1)$ and $X_{(1)}<\dots<X_{(n)}$ the order statistics. For any fixed k, find the asymptotic distribution of $nX_{(k)}$ as $n \rightarrow\infty$. What we ...
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Sampling exercise basic

I have to resolve the follow problem about basic sampling concepts. I've tried a solution, I want to know if I've solved the problem well, please In planning an office network study, the following ...
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30 views

Probability distribution of n'th-order statistic when sampling without replacement.

I have been trying to understand the derivation from the UMVUE for the German Tank Problem. We have $n$ values sampled without replacement from a population $\{1, 2, \cdots, N\}$ of unknown size $N$, ...
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42 views

Finding the peak of this unimodal sequence

Consider the function: $h\left(m,n,p\right):=\frac{n-1}{m}\left(\begin{array}{c} n-2\\ n-m-1 \end{array}\right)\cdot\int_{0}^{p}x^{n-m-1}\left(1-x\right)^{m-1}dx$ for $m=1,\ldots,n$, $n\geq2$ and $p\...
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20 views

Expected number of years that have record high or low rainfall

Exercise problem from Introduction to Probability by Joe Blitzstein, Let X1,X2, . . . be the annual rainfalls in Boston (measured in inches) in the years 2101, 2102, . . . , respectively. Assume that ...
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Expectation of the largest order statistic of non-standard normal distribution

Is there an asymptotic or approximate expression for the expectation (maybe even the variance, but my main question is about the mean) of the n-th order statistic, i.e. the maximum, of n independent, ...
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35 views

Stochastic OR-function [closed]

I need a function where the input is a list of probability values from 0 to 1, and the output is the chance that one of these probabilities comes out true. eg. 0.5 + 0.5 = 0.75 0.9 + 0.9 + 0.9 = 0....
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34 views

A random variable independent of its order statistic

Suppose I have $n$ random variables $X_i \overset{iid}{\sim} f(x_i) $. Is it possible that the $r^{th}$ order statistic $X_{(r)}$ is independent of arbitrary random variable $X_i$? I really want this ...
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40 views

Reasons for variations in sufficient statistic where order statistics $X_{(1)},X_{(2)},…,X_{(n)}$ are involved

I need to understand an elementary part of sufficient statistics. $X_1,X_2,\ldots,X_n \space$ are a random sample. Let \begin{align} & (i) & & X_1,X_2,\ldots,X_n \sim U(0, \theta), \ ...
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31 views

Unbiased estimator of mean of exponential distribution

$X_1,X_2, .. ,X_n$ is a random sample of an exponential distribution with mean $\theta$. Show $nX_{(1)}$ is an unbiased estimator of $\theta$ When I calculated $X_{(1)}$, there's a $n$ in the ...
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32 views

Does expectation of max - expectation of min always equal expectation of max - min?

Suppose $X_1,X_2,\dots,X_N$ are iid. Does $E[X_{max}]-E[X_{min}]$ equal $E[X_{max}-X_{min}]$ for any distribution? For uniform, is does. Is there a counterexample?
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64 views

Let $X_1, X_2, X_3$ be i.i.d exponential random variables with mean $1$. What is $\operatorname{Pr}(X_1 < X_2 < X_3)$?

Ive been working on this question and just want to know if i'm on the right track of if im completely off. The join pdf for the order statistic is $f_{x_{(1)}x_{(2)}x_{(3)}}(y_1,y_2,y_3)$ = $3!e^{-(...
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38 views

Probability that the nth highest of a set of k numbers selected from m numbers has rank greater than the observed rank within that superset

I'm trying to calculate the importance of a subset of genes within a ranked set of genes for a process known as gene-set enrichment analysis (GSEA). GSEA enrichment scores are usually calculated by ...
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3answers
67 views

Expectation of sample range for an exponential distribution

$X_1, \ldots , X_n$, $n \ge 4$ are independent random variables with exponential distribution: $f\left(x\right) = \mathrm{e}^{-x}, \ x\ge 0$. We define $$R= \max \left( X_1, \ldots , X_n\right) - \min ...
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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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31 views

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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64 views

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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39 views

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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Order Statistics and Convergence in Probability

Let $X_1, ..., X_n$ be iid continuous random variables with cdf $F$ and define $Y_n(x) \equiv \sum_{i=1}^n 1(X_i \leq x)$ Define the inverse cdf as $F^{-1}(y) = \text{inf}\{x \in \mathbb{R} : F(x) \...
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35 views

$\mathbb{P}(X_{(r)}\leq u\leq X_{(r+1)})$ order statistics

Let $X_1,\dots,X_n$ be iid random continuous variables with cdf $F$ and $X_{(1)}\leq \dots\leq X_{(n)}$ their order statistcs. I know that $\mathbb{P}(X_{(r)}\leq u)=F_{X_{(r)}}(x) = \sum_{j=r}^n {n\...
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49 views

Convergence in probability of $r_n$th order statistic to associated quantile.

Suppose $X_1,\dots,X_n$ is a random sample from a distribution $F$. Let $0<p<1$. Suppose that $q$ is such that $F(q-)<p<F(q)$. Show that $$P(X_{[r_n]}=q) \rightarrow1$$ if $$(r_n-np)n^{-...
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31 views

Distribution of the maximum of a noisily sorted, incomplete list

Let $Y$, $X$, and $\epsilon$ be random variables with $Y = X + \epsilon$. $X$ and $\epsilon$ are independent and $\epsilon$ is mean zero. I make $n$ independent draws of $X$ and $\epsilon$: $X_1, \...
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30 views

limiting distribution of first order statistics

Let $YI$ denote the first order statistic of a random sample of size $n$ from a distribution that has the p.d.f. $$f(x) = \begin{cases}e^{-(x - \theta)}&\theta < x < \infty\\ 0&\text{...
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1answer
22 views

Is sample minimum an unbiased estimator for population mean?

Given $\mu$ as the population mean and $X_{(1)}$ as the lowest value of a sample extracted from this population, I want to know if $X_{(1)}$ is an unbiased estimator for $\mu$, i.e., if $E(X_{(1)}) = \...
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28 views

Expected value of 2nd-bigger number out of n independent random variables in uniform distribution [r,1], r>0

I am having a problem on which I have 4 players bidding in second price auction with reserved price (r). I need to find the expected value of 2nd bigger number when there are 3 players or 4 bidding ...
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60 views

Maximize $\sum_{i=1}^n\ln\left(\frac{2x_i}{\theta}\mathbf{1}_{[0,\theta)}(x_i)+\frac{2(1-x_i)}{1-\theta}\mathbf{1}_{[\theta,1]}(x_i)\right)$

I'm trying to solve the following problem: Consider a sample of $n$ i.i.d observations drawn from a distribution characterized by the density function $$f_{\theta}(x)= \begin{cases}{\frac{2 x}{\...
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32 views

Partial Sum of Random Variables - Order Statistics

Let $U_1, U_2,\ldots,U_n$ be iid uniform random variables on $[0,1]$. $U_{1,n}\leq U_{2,n}\leq\cdots\leq U_{n,n}$ be the order statistics. Show that, as $\frac{k_n}{n}\to p$ and $0\leq p\leq 1$ $$\...
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49 views

General order statistics - quantile and convergence

Let $U_1, U_2,...,U_n$ be iid uniform random variables on $[0,1]$. $U_{1,n}\leq U_{2,n}\leq...\leq U_{n,n}$ be the order statistics. $U_{0,n}=0$ and $U_{n+1,n}=1$ The spacings are $Q_{i,n}=U_{i+1,...
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112 views

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 ...
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3answers
75 views

How to calculate the expected value of the differences between nearest ordered values?

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 ...
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59 views

Expectation of the first order statistic conditional on the value of the second one.

Let $𝑋_{(1)},\ldots,𝑋_{(𝑛)}$ be the order statistics of a set of $𝑛$ independent uniform [0,1] random variables. Find the conditional expectation of $𝑋_{(1)}$ given that $𝑋_{(2)}=x_2~$, i.e. $~\...
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57 views

Probability that one Gaussian RV exceeds all others

Imagine we have $k$ Gaussian RVs $$ X_i \sim N(\mu_i, \sigma_i^2) \text{ for } i=1, \ldots, k $$ and we sample from each of them independently to produce a vector, $\vec{x} = (x_1, \ldots, x_k)$. For ...

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