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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30 views

Joint PMF of two order statistics with discrete parent distributions

Let $X_1, X_2$ be i.i.d from a discrete distribution with finite support with cumulative distribution $F(x)$ and probability mass function $f(x)$. Let $X_{1:2}$ and $X_{2:2}$ represent the order ...
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Largest order statistics of non-identical distributions when extra information is available

Suppose we have two independent draws, one from a distribution $F_1$ and the other from a distribution $F_2$. The two distributions have the same support, say $[0, 1]$. The distribution of the largest ...
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Estimator for a particular statistic involving Order Statistic

Let$ X_{1}, X_{2}, \cdots, X_{n} $ be a random sample from a continuous life distribution $ F $ be with survival function $ \bar{F},$ density $ f $ and finite mean $ \mu. $ While doing some ...
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Prove that $\mathbb E [T_n(x) ~ T_n(y)] = nF_X(x) + n(n - 1)F_X(x)F_X(y)$, for $x<y.$

Let $X_1, \cdots,X_n $ be iid random variables with distribution $F. T_n(x)$ denotes the number of elements $\le x; x \in \mathbb R$. Prove that $\mathbb E [T_n(x) ~ T_n(y)] = nF_X(x) + n(n - 1)F_X(x)...
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Proving convergence in distribution between order statistics and quantiles

The random variable of continuous type $X$ has CDF $F(x)$ $X_1, X_2, \cdots, X_n$ is a random sample of size $n$ from the distribution of $X$ Function $h(y)$ is defined as $h(y) = F^{-1}{(1-e^{-y})}I_{...
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Approximate distribution of a complicated function of a random variable

If $X$ is a random variable cdf $F(x)$ such that $F$ is invertible then we have the standard method of finding the pdf of any function of $X$, say, $\sin(X) $ or $ X^3+1 $.However,in many situations ...
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1answer
60 views

Distribution of Maximum of Geometric random variable

Let $X_1, X_2, ... X_n$ be geometric random variables with density $$P(X=x)=pq^{x-1} , x=1,2,3,...$$ What will be the distribution of $Y=\max(X_1, X_2, ..., X_n)?$ Will the distribution of $Y$ be ...
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Estimate population mean from "best of N" samples

If I have a data set for which I know all measurements represent the largest of N observations, is there a good method for estimating the mean of all observations? So for example if N=10 and I have 3 ...
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Distribution of a Minimum of iid random variables to a power

Assume that I have $Z_{1},Z_{2},\dots,Z_{N}$ independent and identically distributed random draws following distribution $F(z)$ which are positive-valued. Define random variables $Y = \min_{i \in N} \...
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Quantitative/statistical comparison of two orderings/permutations

There is a couple, say, Theresa and Robert. They assess their preferences on 5 books by ranking them from the most attractive to the least one. Theresa: [0,4,3,2,1]...
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Distribution of percentile rank of largest value in sample

Let's imagine that I sample 100 values from some probability distribution $Distribution$ over the real numbers. Out of these samples, I pick the maximum value $m$. It seems intuitive (and apparently ...
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1answer
25 views

Solved: Relative Efficiency of Average versus Maximum Order Statistic on a Uniform Distribution

$\newcommand{\szdp}[1]{\!\left(#1\right)}\newcommand{\eff}{\operatorname{eff}}$ Problem Statement: Let $Y_1, Y_2, \dots, Y_n$ denote a random sample from the uniform distribution on the interval $(\...
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1answer
268 views

computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$

Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
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What are E(max(X1, X2)) and Var(max(X1, X2)) when the Xs are normal random variables? [duplicate]

Let X = (X1, X2) be normally distributed random variables with mean m = (m1, m2) and covariance matrix S. Y = max(X1, X2) = X1 + max(0, X2 - X1) = X1 + D (X2 - X1), where D = 1 if X2 > X1 and 0 ...
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exponential parameter estimtion from the smallest k-th order statistics

Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered ...
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What is the expectation of a random variable satisfying some conditions?

How to find the expectation E[X.I(Y<x,X<x)], where X and Y are independent random variables with respective cumulative distribution functions F(.) and G(.) respectively. x is a positive value. ...
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257 views

Convergence in distribution to a degenerate distribution

This question came up based on a disagreement I had with a TA. This was the specific example: Let $X_{1},...,X_{n}$ be an iid random sample from a population with pdf $f(x)=3(1-x)^2, 0<x<1$. The ...
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1answer
114 views

Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
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Order statistics and sample size

How do I estimate the odds of the highest value elements of samples from two populations, A and B, exceeding a threshold value, where the same size of A is larger than that of B? Even if A and B have ...
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21 views

Probability that a random variable is the k-th smallest

Suppose we have random variables $X_1, \dots, X_n$ drawn independently from different distributions. How do we calculate the probability of some $X_i$ is the $k$th smallest of those random variables? ...
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32 views

Distribution of the minimum of the components of a multivariate normal random variable [duplicate]

Let $\mathbf{X} = (X_1, \dots, X_p)^\mathsf{T}$ be a $p$-dimensional random variable following a multivariate normal distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{...
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2answers
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Find relation between Categorical dependent variable and continuous independent variable

I have one sample group of data where I found the difference between two categorical data (this is my dependent variable) and continuous - numerical data (this is my independent variable). I want to ...
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123 views

variance estimation using order statistics

I have four largest samples drawn from a distribution of N i.i.d Gaussian R.V. with standard deviation (Sigma) where sigma is unknown. N is known to be between 50-200. Mean is given to be 0. How do ...
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2answers
84 views

Marginal distributions given the distribution of range

I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution. The distance of a point from the diagonal line y = x is $\frac{\sqrt{(x_0-y_0)^...
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Entire set of order statistics

Let's say that we have $X_{1}$,...,$X_{n}$ iid discrete random variables (let's say that there are 15 possible values in the support of the discrete distribution) and n is a large number (let's say 50)...
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What is the intuition behind the Independent Spacings Theorem? [duplicate]

The IST states: Let X1, X2, ,,, Xn be iid Exponential(μ). Then the random variables (where Xsubscript(0) = 0) Ysubscript(j) = (n - (j-1)) (Xsubscript(j) - Xsubscript(j-1)) for j = 1, 2, 3, 4, ,,, n - ...
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Order Statistics Problem: Wackerly/Mendenhall/Scheaffer, 5th Ed., Problem 6.58

Problem Statement: Suppose that the number of occurrences of a certain event in time interval $(0,t)$ has a Poisson distribution. If we know that $n$ such events have occurred in $(0,t),$ then the ...
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Complete sufficient statistics for bivariate observations

We have observations $(X_i,Y_i), 1\le i\le n$ from a family of distributions $\mathcal F$, consisting of all absolutely continuous bivariate distributions. I wish to show that $(X_{(i)},Y_{\text{...
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1answer
122 views

Find the pdf of the Sample Median M when n is even. (Order Statistics)

I have a problem about order statistics. I am trying to find the probability density function of the median $M=\frac{X_{\left(\frac{n}{2} \right)} + X_{\left(\frac{n}{2} +1\right)}}{2}$ where $n$ is ...
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1answer
73 views

The variance of variance-covariance matrix

Looking at answer on the standard error of the variances and this answer on the standard error of the covariances, and knowing that both are part of the variance-covariance matrix, $\Sigma$, I am ...
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1answer
44 views

Order Statistics with joint density

I have an guess in a larger stochastic problem. I assume following: Let $x,y$ be two variables, with $y<x$ and let $f(\cdot)$, $F(\cdot)$ be the a continous PDF and CDF with support $[0,z]$. I ...
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1answer
39 views

Finding conditional distribution when matching ordering

Suppose we draw two values $x_1,x_2$ according to a CDF $F$. Independently, we draw another two values $y_1,y_2$ according to another CDF $G$. Both $F$ and $G$ has support $[0,1]$. Among those four ...
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2answers
108 views

Distribution of ratio of minimum and maximum of uniform

I have a problem in which $X_1, X_2, .., X_n$ follows $U(0, \theta)$. We have $X_{(n)}$ as the maximum and $X_{(1)}$ as the minimum. I am required to compute the correlation between $X_{(n)}$ and the ...
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1answer
22 views

Estimating Sample CDF from 1st Order Statistics

I have a process where I can only measure the 1st order statistics, but would like to know something about the underlying sample CDF. I understand that I can calculate the CDF of my 1st order ...
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1answer
26 views

Joint density of order statistics from dependent normal random variables

Suppose that $𝑋_1,…,𝑋_𝑛$ are mutlivariate normal with correlation $\rho_{ij}$ and each of them are marginally distributed as $𝑁(0,1)$. Let $𝑋_{(1)},…,𝑋_{(𝑛)}$ be the corresponding order ...
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1answer
93 views

Expected value of a Gumbel variable conditional on Gumbel being the maximum of N iid Gumbel

I found the following results in Hanemann (1984) which I cannot find a proof for. I checked through simulation that it is right, but I would like to see an analytical proof... Hanemann, W. M. (1984). ...
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Why doesn't Stdev take absolute value of x- xbar? [duplicate]

Newbie here. Curious why standard deviation subtracts x from xbar and then ^2's them instead of skipping hte squaring/square-rooting and instead takes the ABS value of each x-xbar Thanks mods
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1answer
64 views

Order statistics and Poisson process

Let $N(t)$ be $PP(λ)$.Given that $N(t)=n$, compute the probability of a) Last event before $t$ occurs before $3t/4$. b) First event after $t$ occurs after $t+h$, $0<h$. c) $...
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1answer
48 views

Expected value of sum of normal random variables given one of them is the largest

Suppose $X_1\sim N(\mu,\sigma^2)$ and $X_2,X_3,...,X_n\sim N(0,\sigma^2)$, is the following identity correct? $$E(X_1+X_2+...+X_n|X_1>\max(X_2,X_3,...,X_n))=E(X_1+X_2+...+X_n)$$ It seems that it is ...
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1answer
278 views

Expected value of max of two discrete random variables

I'm reading this paper An Efficient PTAS for Stochastic Load Balancing with Poisson Jobs. Which is solving a makespan minimizing job-shop problem for Poisson job sizes. Basically, schedule the minimum ...
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1answer
52 views

MLE based on bivariate data

Let $ x \sim Exp({\lambda}_{1}) , Y \sim Exp({\lambda}_{2})$ and are independent . We observe Z and W with Z = min(X, Y) and $W = \begin{cases} 1 &, if Z=X \\ 0 &, if Z = Y. \end{cases} $ Now ...
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0answers
36 views

Kth Order statistic of a multivariate distribution

The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic. How does this concept ...
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2answers
73 views

Order statistics of independent but non-identical uniform distribution

Suppose we have independently distributed $X_i \sim \text{Uniform}(0,a_i)$ where the $a_i>0$ are fixed numbers. I want to obtain the probability that $X_j=X_{n-i+1,n}$ where $X_{n-i+1,n}$ is the $n-...
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0answers
30 views

Bootstrapping the sample range - When the bootstrap fails [duplicate]

I'm trying to convince myself that bootstrapping fails when estimating extreme order statistics (and thus functions thereof). This is a classic shortcoming of the bootstrap laid out in some detail by ...
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8 views

Expected value of conditional choice probability [duplicate]

I would like to compute the conditional choice probability which is defined as follows \begin{equation} p_k=Prob(f(m_k)+u_k>f(m_i)+u_i, \forall i\neq k|m) \end{equation} where $u_i, i=1,...,N$ are ...
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48 views

Expected value of the conditional choice probability

I would like to compute the conditional choice probability which is defined as follows \begin{equation} p_k=Prob(f(m_k)+u_k>f(m_i)+u_i, \forall i\neq k|m) \end{equation} where $u_i, i=1,...,N$ are ...
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0answers
30 views

minimum value of square of normal distribution

I am given that $X_1, X_2,....,X_n)$ are $N(0,\sigma^2)$, and A=minimum value of $(X_1^2, X_2^2,...X_n^2)$, and need to find the smallest value of n such that $P(A<0.004)>0.90$ when $\sigma^2=1$....
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1answer
30 views

What does the $k$the order statistic mean?

I'm writing an implementation of the $Q_{n}$ estimator as proposed by Rousseeuw and Croux, as an alternative to the Median Absolute Deviation. The estimator statistic is defined as $$Q_{n} = d \{\left|...
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1answer
56 views

Order Statistics Question [closed]

A random sample of size 10 is drawn from a Uniform distribution on [0,1]. Calculate the probability that the third order statistic is greater than 0.50.
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247 views

Expected squared distance between order statistics?

Suppose $p(\cdot)$ is a smooth probability distribution over $\mathbb R$. Suppose we draw two collections of $k$ i.i.d. samples from $p(\cdot)$, yielding random variables $(X_1,\ldots,X_k)$ and $(Y_1,...

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