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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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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8 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
81 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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219 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 ...
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68 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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33 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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54 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 ...
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26 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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119 views

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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113 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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19 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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48 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 ...
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64 views

Find pdf for minimum of 4 random variables

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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123 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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28 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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50 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, ...
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102 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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21 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
123 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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108 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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48 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 $$ ...
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216 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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57 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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38 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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70 views

Why is my (own) shapiro test inaccurate?

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 ...
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74 views

From moments product matrix to covariance matrix of normal order statistics

I'm trying to compute the exact covariance matrix of normal order statistics. Well known formulas (listed in Zakkula Govindarajulu, 1962) allow us to compute moments of order statistics, as well as ...
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222 views

Order statistic for beta distribution

Let $x_1,\dots,x_n$ be i.i.d. draws from $Beta\left(\frac{k}2,\frac{k-p-1}{2}\right)$. How are the minimum and maximum order statistics distributed, respectively? I would greatly appreciate a ...
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11 views

Conditional probablity of k-th order statistic of a column given the k-th order statistic of the sum of the columns?

Suppose A is a random matrix. Each row is a series of i.i.d random variables. I like to know if we can calculate the conditional probability (for a given $k$) $$P\big(A^i_{(k)} \mid (\sum ...
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211 views

Order Statistics problem: why doesn't law of total expectation (Adam's law) work?

This is the problem The opening prices per share, $Y_1$ and $Y_2$, of two similar stocks are independent random variables, each with a density function given by $$f (y) = ...
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132 views

Independence of Sample mean and Sample range of Normal Distribution

Let $X_1,\dots,X_n$ be i.i.d. random variables with $X_1 \sim N(\mu,\sigma^2)$. Let $\bar X =\sum_{i=1}^n X_i/n$ and $R = X_{(n)}-X_{(1)}$, where $X_{(i)}$ is the $i$ the order statistic. Show that ...
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56 views

How to rank monthly data, using both trends and averages

I have a very large data set containing the daily searches for some Wikipedia entries. I am using the number of searches as proxy of popularity and want to rank the entries. Lets say I have entities ...
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52 views

Conditional Expectation of Order Statistics

Given $X_1,...,X_n \sim f(x)$ How do I find $E(X_{(1)} | X_{(2)})$? Would I have to find the conditional pdf and integrate wrt x? I get the conditional distribution to be $f_{X|Y}(x|y) ...
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43 views

Find meaningful next comparison for total ranking on the fly

I want to obtain a total ranking from pairwise binary comparisons. For this, I can use algorithms like Balanced Rank Estimation or Bradley-Terry Model. However, I wonder if you need fewer comparisons, ...
3
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2answers
151 views

Maximum of Independent Gamma random variables?

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows gamma distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...
3
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1answer
374 views

Proof that n-order statistics are sufficient for a sample of size n

This is problem 1.5.8 in Mathematical Statistics by Bickel and Doksum. It seems straightforward, but I am not sure if my proof is lacking in some way. It doesn't seem quite correct. Question Let ...
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1answer
36 views

Probability of obtaining a greater-than or equal set of observations from a Poisson RV

I have a suspicion this might be fairly trivial, but for some reason I cannot obtain a satisfiable answer today. Assume a Poisson random variable $X$ with known parameter $\lambda$ (though I suspect ...
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1answer
67 views

PDF of sum of ordered weighted exponential RVs

Let $X_{(1)}, X_{(2)}, ..., X_{(N)}$ be the order statistics of an iid exponential RVs with parameter $\lambda$, where $X_{(1)} \geq X_{(2)} \geq ...\geq X_{(N)}$. Any hints on how to find the PDF of ...
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30 views

Hypothesis testing for vector of order statistics

I have a process that generates n values and returns the k largest. I would like to test if the results my process generates are ...
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80 views

Probability that the range includes the mean in a sample of $n=4$ from a normal distribution?

If we select one random sample with 4 elements from a normal distribution, and we denote the minimum value among the sample with $a$, and denote the maximum value among the sample with $b$, what is ...
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1answer
71 views

probability distribution of the maximum

Let T be a random variable giving the time to failure of led lights that follow exponential distribution with a mean value of 15 000 hours. We put three new lights at the same time. Find the ...
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1answer
64 views

Finding expected order statistics from a normal with known parameters [duplicate]

I'm interested in finding the expected value for the kth ordered observation of a normally distributed variable with known standard deviation, mean and n. Could someone let me know the formula for ...
1
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1answer
55 views

Unbiased Estimators

So I've been banging my head against the wall trying to figure out where to go with these problems, and I'm looking for a little direction. Suppose that $Y_1, Y_2, Y_3$ is a random sample where the ...
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78 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
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712 views

Same Mean, Different Variance

Suppose you have eight runners run a race; the distribution of their individual run times is Normal and each has mean $11$ seconds, say. The standard deviation of runner one is the smallest, two the ...
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99 views

Distribution of proportions relative to sum of random variables

Let $X_1,...,X_n$ be iid lognormally distributed variables and $X_{sum} = X_1+...+X_n$. What is the distribution of $\frac{X_k}{X_{sum}}$ for each $k$ in $1..n$? What are their density functions? ...
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68 views

First order statistic of folded normal

Are there any good approximations or tail bounds for the first-order statistic of the folded normal, or the closely related chi-square distribution with $k$ degrees of freedom? It seems that the ...
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448 views

Determine the limiting distribution of Uniform Order Statistic

I have a random sample of size $n$ from a uniform distribution $$U(0, \theta)$$ And I've proven that the pdf of $Y_n$, the n-th order statistic of the sample is: $$ f_{Y_n}(y) = \frac{n}{\theta^n} ...
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1answer
82 views

Maximum of uniformly distributed random variables using iterated expectations

I'm working through the problems in Wasserman's 'All of Statistics'. The chapter on expectations and conditional expectations ends with a (seemingly) easy problem: Let $Y$ be the maximum of $n$ iid ...
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84 views

Is $X_{(1)} + X_{(n)}$ a good estimator for $\theta$?

Problem 8.7 From Van der Vaart's Asymptotic Statistics: Given a sample of size $n$ from the uniform distribution on $[0,\theta]$, the maximum $X_{(n)}$ of the observations is biased downwards. ...