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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40 views

How to find the density range function given the max and min random variables of order statistics? [closed]

I would like some advice on this problem, as I'm trying to understand order statistics. What do I need to do to find the density range function? Maybe calculate the pdf and distribution function of ...
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35 views

Finding the pdf of $X_{(1)}$ of the two-parameter exponential distribution

I have to find the pdf of the smallest order statistic $X_{(1)}$ of two-parameter exponential distribution whose pdf is: $f(x; \theta_1, \theta_2) = \frac{1}{\theta_2} \exp\{-\frac{x-\theta_1}{\...
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24 views

pdf of a maximum order statistic with a uniform distribution

I have the following problem, $X_1,X_2,...,X_n$~$U[\theta,2\theta]$ I am tasked with finding the pdf of the mle. First off, I know that $f(x)=\frac{1}{\theta}$ just by definition and after some ...
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Unbiased Estimator of Largest Mean of Two Normal Distributions

Given samples from two normal distributions: $X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X)$ for $i = 1,...,n$ $Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y)$ for $i = 1,...,n$ How ...
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22 views

Ranks from a sample written as a function

Can sample rank be defined as a function of a sample? Suppose $X_1, X_2, \dots X_n \sim F$ where $F$ is defined on space $\mathcal{X}$. Consider the order statistics $X_{(1)}, X_{(2)}, \dots, X_{(n)}...
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54 views

Joint pdf of Order Statistics

Let $X_1, ... X_n \stackrel{i.i.d} \sim Unif(0,2).$ Find $P(Y_1 < \frac12 < Y_n),$ where $Y_1 = \min \{X_1, ..., X_n\} $ and $ \ Y_n = \max\{X_1, ..., X_n\}$ When I attempted this problem, I ...
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Argmax of betas

Let $\theta_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. What is the distribution of $i^\ast = \operatorname*{argmax}_{i \in I} \theta_i$? Let $[X]$ denote the CDF of a random variable $X$. ...
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What are some of the application of order statistics in engineering/data science /statistics? [closed]

I am looking for an interesting application of order statistics in engineering/data science /statistics. I know many applications where order statistics is used as a mathematical tool. However, I ...
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54 views

Formula for difference in order statistics [closed]

Is there a specific formula one can use to compute the differences in order statistics, say $x_i - x_{i-1}$ when the underlying distribution of $x$ is standard normal? Also what is the asymptotic ...
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Variance of $k$th order statistic of normal vector [duplicate]

Let $Z \sim \mathcal{N}(0, I)$. Let $Z_{(k)}$ be the $k$th order statistic of $Z$. Is it true that $\text{Var}(Z_{(k)}) \to 0$ as $n\to \infty$ for $1 \leq k \leq n$? Any estimate on the rate? What ...
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What is $\text{cov}(U,V)$ where $U = \max\{X_1,…,X_n\}$ and $V=\min\{X_1,…,X_n\}$? [duplicate]

The $X_i$'s are iid Uniform on $(-\frac{1}{2},\frac{1}{2})$. Let $U=\max\{X_i\}$ and $V=\min\{X_i\}$, what is the covariance between $U,V$?
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Generalization of rank to two dimensions?

Converting numbers to ranks spreads the values out in one dimension so that they are equally spaced, while maintaining order. Is there an analogue for two dimensions, which spreads points out in a ...
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31 views

Probability that the same r.v. generates the rth order statistic in one noise-added set, and the sth order statistic in another noise-added set

(Note: The title is confusing, as I have no idea if a name / short description exists for the setting below. I'm open to pointers and/or suggestions.) Setting Let $X_1, ..., X_N \overset{i.i.d.}{\...
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Estimating error for a sample minimum

Let's say I'm benchmarking some computer program and, due to non-randomness in the input data, I'm interested in the minimum running time (as opposed to the average over random inputs). In addition to ...
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Difference order distribution of extremal values of two uniform sequences

Fix an integer $m$. Pick two random integer sequences ($a_1$ to $a_n$ and $b_1$ to $b_{n'}$) uniformly independently from $[1,m]$. Denote $a(i)$ to be $i$th smallest value of $a$ and $b(j)$ to be $j$...
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Distribution of order statistics from the same sample? Distribution of difference betwen order stats

Suppose we have $i$-th and $j$-th order statistics from a sample of i.i.d. random variables. How to derive the difference between these order statistics? For example, it's obvious that, if $j>i$, ...
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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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Computing posterior distribution $\theta|x_{1:n}$

Assume that $X_1, ..., X_n, X_{n+1}| \theta \sim \text{iid } Exp(\theta) $ and the posterior is $\theta \sim Gamma(\alpha, \lambda)$. Task is to compute the posterior $\theta|x_{1:n}$. $\pi(\theta|x_{...
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Ranks of a set of maximal invariant statistics

Let $\{\mathbf{x}_l\}_{l=1}^L$ be a set of i.i.d. (continuous) random vectors sharing the same density $p_X$. Let $\{Q_l\}_{l=1}^L$ be a set of (positive and continuous) random variables representing ...
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How to compute the joint density of three ordered random variables? [duplicate]

I am trying to understand ordered statistics and I don't understand how I can compute the joint density of three ordered random variables. Assume ${X_1,...,X_n}$ are i.i.d with a distribution $F(x)$ ...
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1answer
57 views

How to graph distribution of Order statistics?

Is there a software that can graph the pdfs and Cds of an arbitrary number of order statistics or is there some code such software? How to do it? I'm trying to understand the distribution of order ...
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Estimating median from quantized data

I have quantized measurements and I would like to estimate the median of the underlying distribution. Can I do better than taking the median of the quantized measurement?
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Hypothesis test for $\theta$ in a $uniform(0,\theta)$ distribution

Suppose $X_1, \ldots, X_n$ is a random sample drawn from a $uniform(0,\theta)$ distribution. We will test $H_o: \theta = 3$ and $H_a: \theta = 2$. Use the test statistic $X_{(n)}$ and reject $H_o$ if ...
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53 views

Expectation of the k-th order statistic of a standard Gaussian sample

Let $(X_1,\dots,X_n)$ be independent random variables with common distribution $\mathcal{N}(0,1)$. The order statistics satisfy $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$. I am interested in the ...
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Is Dixon's Q statistic ancillary for normal data?

Dixon's Q statistic is the ratio of the "gap" between an outlier and the nearest value, over the range of the data. I would like to know is if this is ancillary to the parameters of the normal ...
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Asymptotic behaviour of order statistic $x_{(n-k+1)}$ when k is $n^{\alpha}$

I am interested in the asymptotic behavior of the top k-th order statistic $x_{(n-k+1)}$ from n i.i.d. standard normal samples, when k is $n^\alpha$ where $\alpha\in (0,1)$. I just wonder if we can ...
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What is the probability that the student who finishes last takes over twice as long as the student who finishes first?

Three students independently attempt to solve a statistics problem. Assume the times taken (in minutes) by each student to solve the problem are identically distributed on $U(0,30)$. What is the ...
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Double Integral involving Beta Functions (about Pareto Distribution)?

I have tried evaluate $(m_i,m_j)$th product moment of $X_{(i)}$ and $X_{(j)}$ order statistics of Pareto Distribution, that is $E[X_{(i)},X_{(j)}]$, where $i\le j$ , $X_1,X_2,...,X_n$ i.i.d. from ...
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168 views

PERMANOVA test and its assumptions

I have collected data based on a 5 point Likert scale (very low, low, neutral, high, very high) on factors considered by individuals before making investment decisions. There are five factors (D.Vs) ...
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23 views

Identifying interaction terms in nonlinear data whose underlying function may be unknown

This is the data that I am using to frame and ask this question (code written in R): ...
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39 views

Distribution of the minimum and maximum order statistics under a partial ordering

Let ${\bf{x}} = (x_{11},x_{12}, x_{13},\ldots,x_{nm})$ and $f({\bf{x}})\propto 1_A({\bf{x}}) \prod_{i,j} {x_{ij}}^{\alpha-1} (1-x_{ij})^{\beta-1}$ for $i = 1,\ldots, n$ and $j = 1, \ldots, m$. That ...
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How to fully estimate a probability density from only a sample of minimum values?

We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. By means of ...
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1answer
59 views

Coin flipping with revision of probability

Suppose two coins $x$ and $y$ have "H" heads probability $p_x$ and $p_y$. $p_x$ and $p_y$ are independently drawn according to a uniform distribution over $[0,1]$. Say that we know $p_x\geq p_y$. So, ...
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Testing against non-dominance of discrete distributions

I have two discrete distributions $A$ and $B$ with independent draws. What tests can I use against $H_0$: $A$ does not first-order stochastically dominate $B$?
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Distribution of ranges of normally distributed variables

I have four independent variables $x_i$, each of them normally distributed with $\mu = 0$ and $\sigma = 1$. What is the distribution of the range of the four variables, i.e., $\max(x_i) - \min(x_i)$? ...
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Probability of selecting maximum in bivariate correlated order statistics?

In a testing, ranking, or selection scenario, we have samples of size n where a measurement is correlated 0<r<1 with some second variable of interest; they are bivariate normally distributed. We'...
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How can a probability densitiy be estimated based on the maximum entropy principle, with constraints in the order statistics?

Let's say we are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. The ...
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How do I find the expected values and covariance matrix of the order statistics of iid random variables sampled from the standard normal distribution?

Recently I was trying to learn more about Normality tests and came to know about Shapiro-Wilk test for Normality. I understood most part of it but one thing I didn't understand is that how do I find ...
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What is the variance of the mean, conditional on being between two order statistics or quantiles?

Suppose I have a simple random sample drawn from a population with a known distribution on some population characteristic like height or income, with probability density function (pdf) $f(x)$. Order ...
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Is my interpretation of the results of my ordered logistic regression right?

I am currently writing my master's thesis. To analyse the results of my survey, I conducted an ordered logistic regression using Stata. My outcome variable is whether someone wants to start a business ...
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what are “adversarial distributions”

I ran into a paper that talked about some k selection algorithms working better or worse with "adversarial distributions", full excerpt here: 1.3. Organization In this paper we present three ...
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94 views

What is the distribution of a bivariate normal component conditional on the max of the other component?

Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
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1answer
81 views

Spacings between continuous uniform random variables

Let $U_1, \cdots, U_n$ be $n$ i.i.d continuous uniform random variables on $(0,1)$ and their order statistics be $U_{(1)}, \cdots, U_{(n)}$. Define $D_i=U_{(i)}−U_{(i−1)}$ for $i=1, \cdots, n$ with $...
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1answer
34 views

Expected value of CDF evaluated at sample min/max, with respect to sample size

I've been using simulation to investigate the behavior of cumulative density function evaluated at extreme values of a sample. Functionally, my study has $n$ instrumented fish migrating downstream ...
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1answer
48 views

Order statistics uniformly spaced?

Suppose you have $n$ i.i.d. random variables $X_i$ that take values in $[0,1]$, and have an absolutely continuous distribution. Let $X_{(1)}\le X_{(2)}\le \dots \le X_{(n)}$ be the random variables ...
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1answer
68 views

Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
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37 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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1answer
36 views

Probability of having an increasing trend in normal variates

Let $x\sim N(\mu,\sigma)$ and $x_i$ is ordered instances of random variate of $x$ for $i=1...n$. What is the probability that the series is in increasing (or decreasing) order? The problem is finding ...
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1answer
74 views

The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
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26 views

Understanding the solution to a problem about a homogeneous Poisson process

This is probably easy, but right now I can't figure it out, so bear with me. The question is: Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...

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