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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12
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1answer
359 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
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2answers
26 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
votes
1answer
25 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
votes
2answers
127 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
59 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
63 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
78 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
34 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 ...
3
votes
3answers
74 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 ...
0
votes
0answers
38 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
38 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
135 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
68 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
64 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
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. ...
1
vote
1answer
51 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
54 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
138 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
50 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
87 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
196 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
170 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
60 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
126 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
33 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 $k$ ...
0
votes
0answers
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
votes
1answer
116 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
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
votes
2answers
276 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
73 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
39 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 ...
1
vote
0answers
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? ...
13
votes
3answers
579 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
129 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 ...
1
vote
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. ...
3
votes
0answers
234 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 ...
1
vote
1answer
486 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} ...
0
votes
0answers
63 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
votes
1answer
122 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
votes
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 ...
1
vote
0answers
157 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 ...
4
votes
1answer
93 views

Distribution of Extreme Spread for n, sigma

Simple form provided by WHuber: What is the distribution of the diameter of n points in the plane drawn iid from a bivariate Normal distribution? (Diameter is the greatest distance among any pair of ...
1
vote
0answers
106 views

Showing that a statistic is ancillary for a parameter

Working through a HW problem, and a hint is that for a decision rule $$T(X) = \frac{X_{(1)} + X_{(n)}}{2}$$ Then $$T - \bar{X} $$ is ancillary. Intuitively this makes complete sense, but I am ...
1
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0answers
81 views

Order statistics: Probability that a normal random variable is k-th out of n when ordered

My question is almost the same as that of Order statistics: probability random variable is k-th out of n when ordered with the exception that the underlying distribution from which $X_1$ is drawn, ...
3
votes
1answer
134 views

Order Statistics-Expected Value of Random Length

Let $Y_1<Y_2 $ denote the order statistics of a random sample of size 2 from a distribution that is $N\left( \mu,\sigma^2 \right) $, where $\sigma^2$ is known. Compute the expected value of the ...
0
votes
0answers
47 views

Order statistics without independence assumption

I want to derive an expression for the cdf of the min of a set of $k$ random variables $X_i$ (with same cdf $f_i(x)$) which are identically distributed but not necessarily independent. So far, I got ...
0
votes
1answer
229 views

Maximum Likelihood for shifted Geometric Distribution

Really struggling with this please help. Find MLE for p and c \begin{equation} \ {f}(x,p,c) = (1-p)^{x-c}p \end{equation} x=c,c+1,c+2,..... p is between 0 and 1 c is element of the integers I am ...