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 ...

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Expectation Value of Standard Deviation from “k” Closest Samples

The question that I'm am trying to answer is how to determine the expectation value of the standard deviation of 'k' closest samples to value 'A'. The standard deviation and mean of the underlying ...
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Real life examples of applications of induced order statistics

I am working in the field of order statistics and I need some help in using the concept of induced order statistics in a real application. I will be very grateful if anyone can suggest some real life ...
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Problem involving Scheffe's theorem and asymptotic distribution

If $\{ X_n \}$ are independently and identically distributed $U(0,1)$ random variables and $V_n = n(1 - X_{(n)})$ (where $X_{(n)}$ denotes the $n$th or largest order statistic), then how do I derive ...
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Expected value of maximum ratio of n iid normal variables

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$. How would one be able to upper bound the expected maximum of the ratio ...
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Show estimate converges to percentile through order statistics

Let $X_1, X_2, \ldots, X_{3n}$ be a sequence of iid random variables sampled from an alpha stable distribution, with parameters $\alpha = 1.5, \; \beta = 0, \; c = 1.0, \; \mu = 1.0$. Now consider ...
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51 views

Expected value of difference of two order statistics

$X_1,X_2,..,X_n$ is a random sample from the random variable whose pdf is, \begin{align*} f(x)=\lambda e^{-\lambda(x-\mu)},\mu<x<\infty \end{align*} How can we find $E(X_{(2)}-X_{(1)})$, if $n=2?...
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47 views

Estimating R90 (radius of a circle expected to include 90% of impacts)

I want to determine how big a target I can hit with a bow at a certain distance with 90% probability. I place some paper targets at that distance and shoot 20 arrows at them. I have a ruler and a ...
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52 views

Covariance vs variance of order statistics

I wonder if the following statement is true? Let $x_i$ be i.i.d. drawn from distribution $F$ with non-negative support and $f(0)>0$, i.e. the density at zero is positive. Let $X_{i:n}$ be the $i$-...
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15 views

Determine the parameters of a normal distribution knowing the parameters of another related normal distribution

$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$. I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and ...
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17 views

Compute completion time using order statistics

I am trying to figure out how order statics work, I need to estimate the completion time of a computation. The computation consists of n stages and runs at N threads (parallel computing). Assuming ...
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49 views

Correlation of order statistics from Uniform parent

Can someone please help me out with this sum? It says: If $(X_1, X_2, \dots, X_n)$ are a random sample from Uniform(0,1) distribution, find the correlation between the order statistics $X$ ...
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78 views

Approximate Order Statistics for lognormal variables

Are there any known formulas that approximate the expected value of the maximum of $N$ i.i.d. lognormal random variables? I am looking for something similar to: Approximate order statistics for ...
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117 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I'm relatively new to stack exchange and I have already posted this on math....
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1answer
46 views

100-Dollar Game Responses as Predictor

Outcome = Feature Sentiment Participants will rate on 1-5 point scale how much like a particular feature. Potential Predictors = Developments A - E Participants will be asked to imagine they have ...
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1answer
32 views

Pdf of Order statistics

Can anyone please explain the logical reason behind how the joint PDF of a random sample of order statistics of size n is n! times the joint PDF of the random sample? I have derived it mathematically, ...
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67 views

Order Statistics, Expected Value of range, $E(X_{(n)}-X_{(1)})$

$X_1, X_2,...,X_n$ is a random sample from $U(0,\theta)$. Find $E(X_{(n)}-X_{(1)})$. I attempted this question by first finding the CDF of $X_{(n)}-X_{(1)}$ using the formula: $$F_{U}(u)= n\int_0^\...
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Independence of ordered statistics part 2

In relation to my previous question, I have a second problem. How to show that $U_1+U_2$ is independent of $Y_3$? I can find out the pdf of $U_1+U_2$, as well as of $Y_3$, however please help me ...
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143 views

Independence and Order Statistics

I have a problem at hand, which I am not being able to proceed. Can someone help me begin? $Y_1<Y_2<Y_3$ :An order statistic of size 3 from distribution having pdf $$ f(x)=2x\ \ \ 0<x<1$$...
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The probability distribution of rth order statistic after transformation

If I have a sample from a Rayleigh distribution, then I transform this sample to a sample from Gamma distribution by using the fact that the summation of the square of a Rayleigh variable is a Gamma ...
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31 views

When is further reduction than the order statistics not possible?

Let X1, ...Xn be a random sample from a population with location pdf f(x-θ). Show that the order statistics, T(X1, ...Xn) = (X(1), ...X(n)), are a sufficient statistic for θ and no further reduction ...
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47 views

How to show $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator

If $X_i \sim$ uniform$(0,\theta)$, how can I show that the estimator $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator for $\theta$? For example, we know the first sample moment ...
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45 views

Trouble understanding result in Simes (1986) $pr\{ P_{(j)} > \frac{j\alpha}{n} ; j = 1,…,n \} = 1-\alpha$

I'm referencing this paper http://www-stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/MultipleComparision/Simes86pdf.pdf On page 752, a theorem is presented which states that If $P_{(1)},....
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Inverse of ordering function

Lets assume we have p by n matrix.We can generate an output matrix, w (p x p) such as w_ij represent how many times i_th rows number is bigger than j_th (can be at most n obviously). My question is ...
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29 views

Is there a statistically best place to be in a line to choose lowest number from a pool?

Here's the situation: There is a line of 500 people. Each will be given a random number from 1 to 500. I want to find out if there is a best place to be a line to get the best odds at getting a low ...
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21 views

Reward of c best tasks

I'm trying to create an analytical model for performance of a greedy scheduler in a particular domain. I've created a preliminary model, but I think it's very confusing to others and was wondering ...
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1answer
39 views

What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
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17 views

How to decide about the number of looks (window size for ensemble averaging) in SAR images?

This question has frustrated me for a while. In order to find an answer I sent an email to prof. Yamaguchi, the author of the paper Four-Component Scattering Power Decomposition With Rotation of ...
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1answer
90 views

Distribution of differences in beta-distribution

I want to get an analytic solution to the difference of the highest and second highest of a beta distribution. More simply, I have some datapoints on which I assume a beta-distribution. Analytically ...
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234 views

What is the distribution for the time before K successes happen in N trials?

What is the distribution for the time before K successes happen in N trials? Suppose there is a telephone center, and N people, each of whom will either call the telephone center in time T with ...
3
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71 views

Joint CDF of random variables vis-a-vis that of their order statistics

Suppose $\{X_i\}_{i\in 1\ldots n}$ are $n$ independent, non-identically distributed RV's. Let $X_i \sim f_i(x) \mathbf{1}_{[0,1]}$, where $f_i$ is the $i$-th parent supported on $[0,1]$. I am ...
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38 views

Calculate the mean of the minimum of any k samples

Given a fixed list of N numbers from an unknown distribution, and a k (k <= N), say we ...
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67 views

What is the ratio distribution of a spacing and the sample mean?

Let $X_1,\dots,X_n$ be a sample of iid exponential random variables with mean $\beta$, and let $X_{(1)},\dots,X_{(n)}$ be the order statistics from this sample. Let $\bar X = \frac{1}{n}\sum_{i=1}^n ...
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58 views

Sort $X$, then scale the first differences by $\bar X$: what, if anything, is this used for?

Sort data vector $X$, take first differences of the sorted data, and divide by $\bar X$. I came across this transformation in someone's notes, without any citation. It would be applied to non-...
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how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
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Transforming Order Statistics

Assume random variables $X_1, ... , X_n$ and $Y_1, ..., Y_n$ are independent and $U(0,a)$-distributed. Show that $Z_n= n\log\frac{\max(Y_{(n)},X_{(n)})}{\min(Y_{(n)},X_{(n)})}$ has an $\text{Exp}(1)$ ...
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Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this ...
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Distribution of Spacings

If I'm understanding my notes correctly, the distribution for any finite collection of spacings is approximately Exponential with mean 1/n(f(F^-1(k/n)). Can anyone help me understand the proof of ...
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1answer
53 views

Nonparametric Identification from Order Statistics

Suppose a vector of random variables $(X_1,...,X_n,Y_1,...,Y_m)$ is such that $X\sim F(\cdot)$ and $Y\sim G(\cdot)$. So $X$ are distributed independnetly and identically as $F(\cdot)$ and $Y$ as $G(\...
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Find pdf for minimum of 4 random variables [duplicate]

Let $X_1, X_2, X_3$,and $X_4$ be four mutually independent random variables, each with p.d.f. $$f(x) = 3(1-x)^2\quad \mathbb{I}_{(0,1)}(x)$$If $Y$ is the minimum of these four variables, find ...
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142 views

Sums of Random Variables from Order Statistics of Dice Rolls

Let's say you have a set of order statistics $ X_{(1)}, \dots, X_{(N)} $ drawn from a discrete uniform distribution $ \text{unif}(1,S) $. If you choose $ X_{(n_1)}, \dots, X_{(n_k)} $ from this set, ...
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35 views

Ordered response with nominal predictors

I have a data set with ordered response variables (ten levels) and nominal independent variables (sex, year of birth, educational background etc. of the participants in a questionnaire). How do I ...
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138 views

The distribution of the maximum of N independent but not identically distributed Gumbel random variables

I am interesting in determining if there is a closed form expression of the CDF and PDF of the maximum on $N$ Gumbel distributions that are independent but not identically distributed. In particular, ...
3
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1answer
117 views

Inter-arrival time of subsampled Poisson point process

Suppose that I draw $n$ points from a Poisson point process of rate $\lambda$, i.e. with inter-arrival times distributed i.i.d $\sim \text{Exp}(\lambda)$. Now suppose that I choose $m < n$ of ...
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Ranking of Rates of Return as dependent variable

I am working on a study for my university. I am trying to test a model that some companies uses to rank countries as a top-down model for investing. The objective of the model is to rank, explained ...
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1answer
163 views

Find the unique MVUE

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388. Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta<x<2\theta,$ ...
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116 views

joint probability distribution of $k \le n$ order statistics

For $X_i \sim$ iid random variables: For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest ...
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56 views

Partition data into two sets such that the difference of their variance is minimal

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ \left|\frac{1}{k}\sum_{i=1}^k(x_i-\hat{\mu_k})^2-\frac{1}{n-k}\sum_{j=k+1}...
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436 views

Distribution of sum of order statistics

The question is from a problem I am trying to solve in Robert Hogg's introduction to Mathematical Statistics 6th version problem 7.2.9 in page 380. The problem is: We consider a random sample $...
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44 views

Order Statistics Conditional Distribution of Affiliated System

We have a system with $M (M\ge 2)$ random variables. The M variables are related as follows. For each i, 1 to M, $X_i = I_i+Z$, where $I_i$, Z are independent uniform random variables. What is the ...
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Why is my (own) shapiro test inaccurate? [closed]

Linked to my previous question : From moments product matrix to covariance matrix of normal order statistics , i coded an EXACT shapiro-wilk test for normality. Using the related literature; i coded ...