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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Can anyone suggest some real 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 real application. I will be very grateful if anyone can suggest some real life ...
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28 views

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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93 views

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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61 views

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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50 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 ...
3
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1answer
51 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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16 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 ...
3
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1answer
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 $X$ ...
3
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73 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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2answers
114 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
45 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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1answer
65 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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17 views

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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140 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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1answer
44 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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25 views

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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1answer
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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16 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 ...
5
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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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232 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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0answers
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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37 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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31 views

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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1answer
133 views

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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25 views

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
52 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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73 views

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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139 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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34 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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135 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
116 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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36 views

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 ...
3
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1answer
161 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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1answer
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 ...
4
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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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1answer
417 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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59 views

Sort mean values with deviation

I'm trying to determine what physical attributes of a syllable are important in determining stress. So I have some recordings from different people saying a word, each person multiple times. And for ...
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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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88 views

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 ...