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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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 ...
3
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57 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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54 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
24 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 ...
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1answer
34 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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35 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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1answer
375 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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2answers
31 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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15 views

Getting a “friendly” tailbound from a closed-form description of the probability density (the case of the n-th order statistic)

Suppose I have a probability distribution of an $n$-th order statistic $X_n$ with mean $\mu$ and density $f_n(x)$, where $n$ scales to infinity. If one wants a concrete example, the one I care about ...
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1answer
31 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 ...
4
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2answers
153 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
63 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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1answer
66 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. ...
4
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1answer
92 views

estimating the upper bound on a uniform distribution from max order statistic

I have a question. Suppose that $X_1,\ldots,X_n$ are iid $U(0,\lambda)$ and let $X(n)$ denote the nth order statistic. Suppose $\lambda$ is unknown and should be estimated from the sample. Take ...
4
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1answer
37 views

Sampling Order Statistics for Numerical Integration

This may be a stupid question. I want to do Monte Carlo integration over a region $$ {\int}_{D_{1} \geq D_{2} \geq ... \geq D_{m} \geq 0} g(d_1,\ldots,d_m) f(d_1) f(d_2) \cdots f(d_m) ...
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6 views

Collision in randomly generated String [duplicate]

I have a function which gives me randomly a combination of letters from A-Z and numbers from 0-9. I want to generate 20000 keys out of this. My question is now that I want to determine the probability ...
3
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3answers
76 views

$x_{1}…x_{n}$ are independent continuous random variables with common distribution function $F(x)$,compute $E(F(x_{(n)})-F(x_{(1)}))$

$x_{1}...x_{n}$ are independent continuous random variables with common distribution function $F(x)$,consider the order statistics $(x_{(1)},...,x_{(n)})$, compute $E(F(x_{(n)})-F(x_{(1)}))$ I have ...
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16 views

$f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}$, find P(X<Y<Z) and P(X=Y<Z)

$\begin{equation}f_{X,Y,Z}(x,y,z)=\begin{cases}e^{-(x+y+z)} &\mbox{x,y,z>0}\\0 &\mbox{otherwise}\end{cases}\end{equation}$ $Find \ P(X<Y<Z)\ and\ P(X=Y<Z)$ What I have done is ...
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44 views

How to quantify the similarity between two samples using quartiles only?

I have several sets of samples I would like to compare. Each set is comprised of two samples, for which I only have the quartiles, min/max values and sample size for each sample. I would like to ...
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0answers
40 views

Hypothesis testing and order statistics

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
2
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1answer
146 views

First order statistics (min) of n non-identical but independent normal variates [duplicate]

While I have seen papers and posts about mean and variance of n i.i.d normal random variables, I am looking for the first order statistics of $n$ (specifically $11$) normal, non-identical (different ...
2
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1answer
76 views

Show that a statistic is ancillary

Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Show that $$ \frac{X_{(1)}}{X_{(n)}}$$ Is ancillary for theta I coulxnt find a way of doing it that looks convenient. Any idea? P.s: ...
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1answer
68 views

Asymptotic distribution of uniform order statistics

It can be shown that for an iid sample from a Uniform(0, 1) distribution, \begin{equation} n(1-U_{(n)}) \rightarrow exp(1) \\ n(U_{(1)}) \rightarrow exp(1) \end{equation} To see this just try finding ...
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47 views

probability that a variable is ONE OF the top k out of n when ordered

Suppose ($h_1,h_2,...,h_n$) is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a non-random variable which can vary across $i$ and $X_i$ is a random variable with Pareto Type I distribution. ...
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1answer
54 views

[Revised]Proving the expected \bold{density} of being the Nth order statistics is decreasing in sample size

(Sorry that I've previously formulated the question in a wrong way, which confused everyone including myself. This is a better version of the question. Thanks!) Here's another order statistics ...
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1answer
55 views

Joint density of first r order statistics

Let $X_i \sim^{iid} F$ for $i=1,...,n$, where $F$ is a continuous distribution. I want to find the pdf for $X_{(1)},X_{(2)},..., X_{(r)}$, with $r\leq n$. We know that $f_{X_{(1)},X_{(2)},..., ...
3
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2answers
148 views

Proving some properties of expected first order statistics with respect to sample size

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)= ...
2
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0answers
51 views

Estimating a joint distribution from observed max and min samples

Suppose that you have jointly distributed $N$ (~100) random variables, $\{X_1,\ldots,X_N\}$, and this distribution is unknown to you. However you do know that their sum is zero by construction. Having ...
3
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90 views

Is $F(E[Y_n]) \approx E[F(Y_n)]$ a reasonable approximation?

Studying the asymptotic distribution of order statistics I came across this approximation: $$F \left( E \left[ Y_n^{\left(n \right)} \right] \right) \approx E \left[ F \left( (Y_n^{\left( n \right)} ...
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219 views

Limiting distribution of the first order statistic of a general distribution

Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)>0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting ...
4
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1answer
184 views

Compute pdf of a k-th order statistic

How to compute the density function of the k-th order statistic of a sample of $X_1, X_2, ..., X_n$ random variables distributed independently but not identically (i.e., $X_i \sim F_i$ with $F_i\neq ...
4
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3answers
65 views

$\phi$-divergence?

I am frustrated of looking for a simple explanation of this term $\phi$-divergence, but I cannot find any. Therefore I would be really grateful if somebody could introduce a reference or write a ...
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160 views

Marginal distribution of a function of order statistics

From the joint distribution of any two order statistics, say $Y_j$ and $Y_k$, $j<k$ I would like to derive the distribution of $Z=F(Y_k)-F(Y_j)$. The initial pdf is: $$f_{Y_j,Y_k} (y_j,y_k) ...
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34 views

Feature relationship based class separability

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
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25 views

What is the diff between singly censored and progressive censored data in survival analysis?

I have a question regarding survival analysis . To my understanding, the singly censored data are those if there is one point in time, i.e, say, if the patient died (bulb is still working?) after ...
3
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1answer
118 views

A question in order statistics of continuous type distribution

Let $X_1,X_2,\dots$ be a sequence of random variables from a continuous type distribution and $m$ and $n$ be two integers such that $m<n$, and $2\le n-m$. How can I show the probability that the ...
0
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83 views

MLRP of random variables and order statistics

Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy the monotone likelihood ratio ...
4
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2answers
301 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

The following question is part (1/4) of a 2.30h written exam for the course "Probability and Statistics" in a school of engineering. So, although tricky and difficult (because the Professor is really ...
3
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75 views

How to measure the reliability of a consensus ranking (problem from Kemeny-Snell book)

Suppose that $k$ experts are each asked to rank a set of $n$ objects in order or preference. Let allow ties in the rankings. John Kemeny and Laurie Snell in their 1962 year book "Mathematical models ...
2
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41 views

What is a robust way to find the max of $n$ independent, non-identical random variates?

Suppose I observe $n$ random variates along with their variance (but not mean) and I'd like to select the one with the largest mean as frequently as possible. The procedure must be memoryless--you ...
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62 views

Estimating distribution given top k order statistics and unknown n

This problem occurred to me a couple of days ago, in the context of a game with a leaderboard. I wondered, given only the leaderboard, could I estimate parameters for the distribution of scores? ...
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631 views

Can I reconstruct a normal distribution from sample size, and min and max values? I can use mid-point to proxy the mean

I know this might be a little ropey, statistically, but this is my problem. I have a lot of range data, that is to say the minimum, maximum and sample size of a variable. For some of these data I ...
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133 views

maximum gap between order statistics of normally distributed random variables

Hello Cross Validated community, I am currently working on a not-that-easy problem involving order statistics. As I am unsure as to how I could solve it, I thought it might already possess a ...
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2answers
82 views

Ordered gamma variables led to an ugly integral

Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f. $$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$ and let $Y_1,...,Y_n$ be the order statistics for these variables. a) Find the conditional p. ...
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246 views

Better understanding of GARCH and ARCH models

I want to make a function that does GARCH and ARCH in python for calculating variance. But I only have a general understanding of the model. Are there any good papers that can be recommend to give me ...
2
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1answer
516 views

Likelihood Ratio of two-sample Uniform Distribution

Consider two uniform distributions: $$f \left( x, \theta_i \right) =\begin{cases} \frac{1}{2\theta_i} \quad -\theta_i<x<\theta_i, -\infty<\theta_i<\infty \\ 0 \quad \text{elsewhere} ...
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66 views

I do not understand the terminology of $Q_n$

I am trying to understand what looks to be some simple terminology issue, involving $Q_n$ from my book here: Specifically, I do not understand what they mean when they say $m = {n \choose 2}$, and ...
4
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1answer
128 views

Joint pdf of functions of order statistics

Let $ Y_1 < Y_2 <\ldots <Y_{10}$ be the order statistics of a random sample from a continuous type distribution with cdf $F(x)$. How would I begin to show that the joint distribution of ...
2
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1answer
2k views

Expected value of minimum order statistic from a normal sample

UPDATE Jan 25th 2014: the mistake is now corrected. Please ignore the calculated values of the Expected Value in the image uploaded - they are wrong- I don't delete the image because it has generated ...
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191 views

Approximation of the variance of the first order statistic (min) of normal random variates

I'm looking for a closed form approximation of the variance of the minimum order statistic for normal random variates. Can anyone point me to a reference, or an approximation? I've seen the post ...