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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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Find the probability that a rv is equal to its order statistics $P(X_i = Y_j)$

Let $X_i$, $i=1...n$ be independent non identically distributed random variables and let $Y_i$ be its order statistics. I want to find the probability that $X_i$ is equal to $Y_j$ for each i, j $i....
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Expectation of exponential variable

What approach do I have to use in order to solve the following integral: $$\sum_{i=1}^N\int_0^\infty \log_2(1+g_ip_i)dg_i, $$ where $\mathbf{g}$ has an exponential pdf. And $$g_1\geq g_2\geq , \dots,...
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In a Poisson process,given$ N(t_s)=n$, how to show the difference of the time of the events is conditional independent? [closed]

If $N(t)$ is a Poisson process, and we know that $N(t_s)=n$. And denote the $T_1,T_2,...,T_n$ as the time points of events happened in the process in time ordering.(so $0<T_1<T_2<...<T_n&...
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25 views

Intrinsic Bayes factor

Here on the page 238 in the formula: the arithmetic intrinsic Bayes factor, $$B^A_{10}=\frac{1}{L}\sum_{x(\ell)}B_{10}^{(\ell)}=B_{10}(x)\frac{1}{L}\sum_{x(\ell)}B_{01}\left(x_{(\ell)}\...
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A question about stochastic order.

If two random variables $X,Y \in \mathbb{R}^n$ with nondecreasing function $f:\mathbb{R}^n \to \mathbb{R}$ such that \begin{equation} \mathbf{E}f(X) \leq \mathbf{E}f(Y) \end{equation} Then we call ...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution [duplicate]

Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f. f(y) = e^(-y), 0 < y < ∞, zero elsewhere. Show that Z1 = Y(2) and Z2 = ...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f.

Help me to solve this problem please.. Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 ...
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21 views

Finding a fair weighting for “most popular responses”

apologies if this has been asked before - I am not sure how pose my question in a searchable way! I want to collect votes on people's "favorite album" for a band, and then I want to show the band's ...
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42 views

Fundamental theorem of calculus on double integrals

The problem: Let $ X_1,X_2,\cdots ,X_n $ be a random sample from the uniform distribution with pdf $f(x;\theta _1,\theta _2)=\frac{1}{2\theta _2} , \theta _1-\theta _2<x<\theta _1+\theta , $ ...
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1answer
72 views

Joint Distribution of Uniformly Distributed Independent Random Variables

The problem is as follows: Suppose we have have independent random variable $Y_1, Y_2, ... ,Y_n$, and they are uniformly distributed over the closed interval [0,1]. If $V$ and $W$ are the smallest ...
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2answers
33 views

Transformations of Order Statistics

Consider the ordered sample $X_{(1)} < X_{(2)} < X_{(3)}$ from a distribution with PDF $f_X(x) = 2x, 0<x<1$. Show that $Y_1 = \frac{X_{(1)}}{X_{(2)}}$, $Y_2 = \frac{X_{(2)}}{X_{(3)}}$ and $...
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Order Statistics Bivariate transformation

Hey can anyone please help me with this. Find the probability range that a random sample of size $4$ from $U(0,1)$ is less than $\frac{1}{2}$. I.e find $P\left(y_4-y_1<\frac{1}{2}\right)$ where ...
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58 views

Order Statistics Sum

Hey can anyone please help me with this. Find the probability that range of a random sample of size 4 from $U(0,1)$ is less than $\frac{1}{2}$. i.e to find $ \mathbb{P}(Y_{4}-Y_{1})< 1/2$ ...
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Estimation of parameters from independent but non-identical random variables

I am looking for some reference (books/papers/slides, etc) for estimating parameters from independent but non-identical random variables by using order statistics. The model is the following. There ...
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1answer
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Limiting distribution of first order statistics ${X}^{n}$

Question from "Introduction to probability and mathematical statistics" by Bain and Engelhardt Salutations! I am attempting the problem above.Specifically I am having a problem with part c). I do not ...
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1answer
151 views

UMVUE of $\frac{\theta}{1+\theta}$ and $\frac{e^{\theta}}{\theta}$ from $U(-\theta,\theta)$ distribution

Let $X_1,X_2,\dots, X_n$ be rvs with pdf: $$f(x\mid \theta)=\frac{1}{2\theta}I(-\theta<x<\theta)$$ Find UMVUE of $(i)\dfrac{\theta}{1+\theta}$ and $(ii)\dfrac{e^{\theta}}{\theta}$. Note ...
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2answers
57 views

Uniform distribution of points in hyperball: deterministic arrangement?

I am currently self-studying Elements of Statistical Learning (2nd ed), by Friedman, Hastie & Tibshirani. I have a question with regards to Equation 2.24, which states the median distance of the ...
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154 views

Take $n$ i.i.d. Gaussians and remove largest $m$ and smallest $m$ points. What is the variance of the mean of the remaining points?

Let $n,m\in\mathbb{Z}$ with $0 \le 2m < n$. Let $X_1, \cdots, X_n$ be i.i.d. standard Gaussians and let $X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)}$ denote their order statistics (i.e., $\{X_1, X_2,...
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Weak convergence(i.e convergence in distribution) of first order statistics # problem 1.1(ch 6) of “Intermediate course in Probability” by Allan Gut

For each $n = 1, 2, ....$, suppose that $X_n$ is a continuous random variable with density $$\hspace{10mm}\mathrm{f}(x) = \begin{cases} \frac{1}{2}(1+x)e^{-x}, & \text{if $x \ge 0$ } \\[2ex] 0, &...
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1answer
27 views

Asymptotic rate of the largest order statistic.

Let $X_1, \cdots, X_n$ be i.i.d. random variables with distribution $P$. Let $g$ be a measurable function with $P g = 0$ and $P g^2 = 1$. Show that $\max_{1 \leq i \leq n}|g(X_i)| = o_p (\sqrt{n})$. ...
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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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24 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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66 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
42 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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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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36 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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29 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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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
51 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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1answer
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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
35 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
35 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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39 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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60 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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8 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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63 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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30 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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23 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
61 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
125 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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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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30 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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35 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
72 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
47 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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1answer
49 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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25 views

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 ...