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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54 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 ...
2
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1answer
47 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 ...
0
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0answers
28 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) ...
1
vote
0answers
33 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
votes
2answers
71 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, ...
2
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1answer
81 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 ...
2
votes
1answer
29 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 ...
1
vote
1answer
64 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 ...
2
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0answers
24 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 ...
3
votes
2answers
62 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 ...
2
votes
1answer
61 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 ...
1
vote
1answer
38 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
vote
1answer
45 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 ...
2
votes
0answers
49 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: $$[\theta - \rho, \theta + \rho]$$ ...
13
votes
1answer
409 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 ...
2
votes
2answers
47 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? ...
0
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0answers
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 ...
0
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1answer
35 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
237 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} ...
1
vote
1answer
68 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 ...
5
votes
1answer
70 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
votes
1answer
110 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
votes
1answer
41 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) ...
1
vote
0answers
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 ...
6
votes
3answers
96 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 ...
0
votes
0answers
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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0answers
69 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 ...
0
votes
0answers
51 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
votes
1answer
183 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
votes
1answer
79 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: ...
3
votes
1answer
83 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 ...
1
vote
0answers
48 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. ...
1
vote
1answer
64 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 ...
1
vote
1answer
65 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
votes
2answers
153 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
votes
0answers
52 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
votes
2answers
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)} ...
3
votes
2answers
267 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
votes
1answer
200 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
votes
3answers
67 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 ...
1
vote
0answers
183 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) ...
1
vote
0answers
36 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 $k$ ...
3
votes
1answer
121 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
votes
0answers
88 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
votes
2answers
330 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
votes
0answers
89 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
votes
0answers
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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vote
0answers
64 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? ...
13
votes
3answers
730 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 ...
1
vote
0answers
146 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 ...