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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. 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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$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(...
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How to see this order statistic result and find my error

Let W be a random variable with pdf $f(w)=\theta B^{-\theta}w^{\theta-1}$ for $0 \lt w \lt B$ and 0 otherwise. Assuming Independence, Show that , $W_{n:n} \to B$ as $n \to \infty$ where $W_{n:n}$...
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Comparing ordering of sample and conditional distribution based on that sample

I don't have a specific question, but just a request for resources and/or guidance. Suppose we take $n$ draws from a distribution $F$. Suppose a particular sample is given by $\vec{\theta} = (\...
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Is there a closed-form solution for ratios of order statistics?

Is there a closed-form solution for the expected value and variance of the ratios between specified order statistics drawn from a large sample from a known parametric distributional family? Actually, ...
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16 views

Ordered statistic calculation with a different kind of cdf [closed]

Let $X_1, X_2,...,X_n$ be I.I.D random variables, and let $Y_{(1)}\leq Y_{(2)} \leq ... \leq Y_{(n)}$ be the ordering of the random variables.The distribution function is given as- \begin{align*} F(x) ...
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Proving completeness of highest-order statistic using Leibnitz' Rule

Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$ I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
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68 views

Basic calculations with Order Statistics

I've come across the following problem, and I am tempted to delve into order statistics to solve this. I would greatly appreciate any help! Suppose you draw 6 independent samples from a continuous ...
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Reference line for QQPlots: robust regression vs quartiles method

Using the car package, I drew two qqPlots of 144 p values from correlated tests on one dataset. The top plot uses a reference line drawn by "quartiles" and the bottom plot uses a line drawn by "robust"...
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Relation between a Gamma prior and posterior in terms of paramters

I am doing a maths exercise and I have found out that the prior of my parameter is is a inv.gamma (alpha, beta), the likelihood is an exponential distribution. Finally I have discovered that my ...
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27 views

Tail probability of minimum to maximum ratio among $n$ i.i.d. half-normal random variables

I have $n$ i.i.d.$\sim\mathcal{N}(0,1)$ random variables $X_1,\cdots,\ X_n$. For my research, I am interested in finding bounds (upper and lower) of the tail probabilities of the ratio $\frac{|X|_{(1)}...
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Does having the same order statistics imply the same distribution?

I'm stuck at the following problem: Say $X_i \overset{\text{iid}}{\sim} \operatorname{Exp}(\lambda)$ for $i = 1, \ldots, n$. Denote $X_{(1)}, \ldots, X_{(n)}$ the order statistic from the $n$ ...
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101 views

Correlation between an observation and its rank in a random sample

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d random variables with an absolutely continuous distribution. We say the observation $X_i$ has rank $R_i$ if $$X_i=X_{(R_i)}\quad,\,i=1,2,\ldots,n,$$ where $X_{(...
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138 views

Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The ...
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stochastic ordering of counting processes

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
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Order statistics for log series distribution

I am trying to obtain the probability mass function for various order statistics of a log series distribution for a given n. To do so, I tried modifying the code given in this question: Simulating ...
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Order statistics are minimal sufficient for the set of all continuous distributions

Previous question exists here, but no answer has been posted. I don't believe this is a duplicate of existing questions concerning minimal sufficiency of order statistics that have an answer. Problem:...
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Estimating Expected Order Statistics

I have a fairly basic question that I'm looking for a reference for. First, a couple definitions. Let's say $X_1,\ldots,X_n$ are IID samples from a distribution $F$ over $[0,1]$. For any $k\in\{1,\...
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Minimum of i.i.d. Random Variables [duplicate]

What importance does the minimum of an identical and independently distributed random variables play in probability distribution?
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55 views

sampling distribution of order statistics

Let, $X_{1},X_{2},\ldots,X_{n}$ be an i.i.d. sample from $R(1-\theta,1+\theta)$. Show that, $(X_{(1)},\bar{X},X_{(n)})$ is sufficient for $\theta$. Ans: $f(x_{1},x_{2},\ldots,x_{n})=\frac{1}{(2\theta)...
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Cumulative distribution of X_{1} = min{X_{1},X_{2}}

So I have $X_{1},X_{2} \sim^{iid} Unif(\theta,1)$, hence the density function is $f_{X_{i}}(x) = \frac{1}{1-\theta}\mathbb{1}_{(0,1)}(x)$ What is the cumulative distribution of $X_{1} = min\{{X_{1},...
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Limiting distribution of first order statistic x

Good evening everyone, I am attempting a problem on limiting distributions. The problem is as seen in the picture above. Could you please help me with part c) of this problem. The answer as per the ...
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1answer
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$X_i \sim \text{Uniform}(0, \theta)$ iid; $Y = \max{(X_1,..,X_n)}$. Why is $\theta$ necessarily larger than $y$?

I'm going through Statistical Inference by Casella & Berger, and on page 419, in the intro section of interval estimation there is the following example (note: most of the text was left out as it'...
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45 views

maximum likelihood estimator

Suppose, $X_1,X_2,\ldots,X_n$ be a random sample from $U(\theta-2,\theta+2)$. Define, $X_{(n)}=\rm{max}\{X_1,X_2,\ldots,X_n\}$ and $X_{(1)}=\rm{min}\{X_1,X_2,\ldots,X_n\}$. Then which of the ...
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Is there a canonical probability distribution on an ordering/permutation

What is a good way of defining a non-uniform probability distribution on a permutation of k objects? For example, suppose the parameter was an ideal ordering, and the probability of an ordering was ...
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62 views

Distribution of maximum over minimum of Weibull(alpha, 1)

Let the PDF of Weibull $(\alpha, 1)$ be $$ f_X (x) = \alpha x^{\alpha-1} \exp\left( - x^{\alpha}\right) $$ I know from probability integral transfrom that $$ X_{(1)}^{\alpha} \overset{d}{=} Z_{(1)...
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CDF of top $k$ value among $n$ RV's when the value of one of them is known

Suppose that $X_i$ is a random variable whose CDF if $F$ and pdf is $f$, $i=1,\cdots, n$. Assuming independency, the $m$-th highest value can be found using order statistics. What would be the CDF ...
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PDF for the ith ORDERED uniformly random sample compared to an evenly spaced sample

Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1. Let $R_1 < R_2 < ... < R_N$ denote a set of ...
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232 views

a fast uniform order statistic generator [closed]

Can someone provide me with the mathematical expression for this code/function as a fast way to generate $n$ sorted $U[0,1]$ random numbers: ...
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21 views

Distributing elements over multiple buckets of a varying size

If one has b buckets that each have a size of b_i and one has to place n elements in the buckets. How can I mathematically described the amount of different permutations the elements can be placed in ...
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1answer
124 views

Density estimation using (different) order statistics

I need to estimate a univariate distribution $F$ as flexibly as possible. However, I do not observe draws from $F$ directly. Each observation $x_i$ is the minimum of $a_i$ draws from $F$, where $a_i$ ...
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1answer
126 views

Mean of maximum of exponential random variables (independent but not identical)

I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the ...
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Order Statistics invovling Continuous Distribution [closed]

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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Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $(X_1,\cdots,X_n)\stackrel{\text{i.i.d}}{\sim}\text{Pareto}(k,a)$

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a $\text{Pareto}(k,a)$ population with density $f(x)=\frac{ak^{a}}{x^{a+1}}\mathbf1_{x>k}$ where $a,k>0$. What is the distribution of $\...
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1answer
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Expectation and variance of range (x(n)-x(1)) uniform [closed]

I am working on calculating the expectation and then variance of the range from a Uniform(-theta, theta) distribution, but have gotten stuck. Basically the first page I show how I get the pdf and ...
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1answer
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Asymptotics for the expectation of the standard normal order statistics

What is an asymptotics of the expectation of the order statistics of the standard normal distribution $$e(r:n) \approx \Phi^{-1}\Big(\frac{r-\alpha}{n-2\alpha+1}\Big)$$ as $n\rightarrow\infty$? By ...
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What is the motivation behind this sampling algorithm

I was recently reading on methods for sampling from linear distributions of directions. This distribution on a direction $X$ is defined by the density $$ p(x)=\frac{1}{2}(1-3\kappa)+3\kappa\frac{1+x}{...
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What is the probability of a random variable larger than or equal to any other independent random variables? [duplicate]

Suppose random variables $X_1$, $X_2$,..., and $X_n$ are independent and normally distributed (with varied means and variances), what is $P(X_i\ge \max_{j\in [1,n], j\neq i} \{X_j\})$?
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Order statistic, probability

Let $X_{1},X_{2},..., X_{10}$ be i.i.d. with continued, strictly increasing distribution function $F$. Let $X_{1:10},X_{2:10},...,X_{10:10}$ be the order statistic. Let $m_{X}$ be the median of random ...
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166 views

Non-parametric version of German Tanks problem?

The German Tank Problem assumes that we observe k i.i.d. random variables $X_1,...,X_k$ uniformly distributed over $[0,N]$, where $N$ is unknown. The goal in this problem is to estimate $N$ given the ...
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Conditional Distribution of Order Statistics [closed]

Let $X_1,...,X_n$ be independent and identically distributed random variables with commom cdf F. Let $X_{(i)}$ be the ith order statistics of $X_i^,s$.Then prove that for $j>i, \{X_{(j)}|X_{(1)}=...
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Closed-form multivariate Gaussian probability

I am looking for a closed-form version of this formula: \begin{align*}\text{P} &= \int_{-\infty} ^{\infty} \left[1 - \left(\int_{-\infty} ^{t}\right.\right.\ldots\\ &\left.\left.\ldots\int_{-...
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Covariance of order statistics convergence?

Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of $$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
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Derivation of an estimate for CDF given ordered sample

Given a sample $x_1,\ldots,x_n$ one may order them $x_{(1)},\ldots,x_{(n)}$ and an estimate for the CDF $P(X\leq x_{(i)})$ is $\frac{i}{n}$ which is valid given that the CDF of a sample is uniform. ...
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1answer
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What is the pdf of the sampling distribution of the sample maximum of iid standard normal?

I found here that The cdf for the max is the cdf for the normal raised to the power of the sample size. [say, $n$] Since the cdf for the normal is the error function, $$F_X(x)=\frac{1}{\sigma\...
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0answers
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How to find approximation of variance of $i^{th}$ order statistic [duplicate]

Given PDF and CDF of a distribution, how does one find an approximation of $(\operatorname{Var}(X_i))$ using a normal approximation of $(X_i)$? According to "Mathematica Laboratories for ...
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725 views

Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0<x<1$ and $0$, otherwise. Using Chebyshev's inequality, find the lower bound of $P\left(|X_1 + X_2-1| \le\...
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1answer
135 views

Finding the mean of the max order statistic drawn from standard normal

Let $X=\max\{X_1, X_2, \cdots, X_N\}$, where each $X_i \sim N(0,1)$ and are independent. What is the approximate value of $X$ for large $N$. The term "approximate" isn't defined very clearly. I'm ...
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383 views

expectation of exponential order statistics

Suppose $x_1, x_2, ......, x_n$ be i.i.d. random variable of exponential distribution $exp(1)$, i.e., $f(x)=e^{-x}, x>0$. Given the order statistics $x_{(1)} \le x_{(2)} \le......\le x_{(n)}$, it ...
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Percentiles of $Z=\max(X,Y)$, where $X$ and $Y$ are correlated bivariate normal random variables

Suppose $X \sim N(\mu_x,\sigma_x^2)$, $Y \sim N(\mu_y,\sigma_y^2)$, and $\operatorname{Corr}(X,Y)=\rho$. I am interested in calculating percentiles of $Z = \max(X,Y)$. We can assume bivariate ...
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Densitiy of order statistic when in a certain interval

$X$ is distributed with $F$, i.i.d. and with densities. I am trying to discern an expected value for a certain order statistic $X_{k}$ under the condition that $X_{k}$ is closest to some value $\...