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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1answer
54 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
39 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: ...
2
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1answer
38 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
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0answers
41 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
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1answer
26 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
39 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
103 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
38 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
86 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
111 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
106 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
49 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
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0answers
56 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
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0answers
29 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
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0answers
21 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
99 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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0answers
79 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
167 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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0answers
57 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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0answers
38 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
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0answers
48 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
330 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
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0answers
109 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
76 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. ...
2
votes
0answers
154 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 ...
0
votes
1answer
286 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
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0answers
56 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
102 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
810 views

Expected value of minimum order statistic from a normal sample

UPDATE Jan 10th 2014: the mistake was found - a math typo in one of the sources used. Preparing correction... UPDATE Jan 25th 2014: the mistake is now corrected. Please ignore the calculated values ...
1
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0answers
77 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
87 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
76 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 ...
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0answers
58 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
118 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
42 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
150 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 ...
1
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0answers
120 views

Order statistics of independent NOT identically distributed random variables [closed]

Can I find the p.d.f of the order statistics (min for example) from a set of independent, but not identically distributed random variables? (the analytical p.d.f. of the other variables is at hand)
-1
votes
1answer
107 views

Order Statistics

What is the motivation behind the use of order statistics in parameter estimation. In a very general sense, the first order statistic is considered to be an initial estimate to the location parameter. ...
1
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0answers
73 views

CDF of largest observation in normal distribution [duplicate]

Let $X_1,...,X_n$ be a random sample from a $\mathcal{N}(\mu,1)$ distribution. Only the largest observation $Y = \max(X_1,...,X_n)$ is reported. What is the density of $Y$? How do I get there?
2
votes
0answers
66 views

Obtain order statistics using uniform order statistics

This is a homework questions. Can you guys give me some hints? Let $U_{(1)}<\cdots<U_{(n)}$ be the order statistics of a sample of size $n$ from a Uniform$(0,1)$ population. Show that ...
1
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0answers
91 views

Questions about the order statistics of uniform distributions

I refer to the Simes (1986) paper found here. In this setting, $P_{(1)}$ through $P_{(n)}$ are the order statistics of $n$ independent Uniform$[0,1]$ random variables and, for $0\le \alpha \le n$, ...
1
vote
0answers
59 views

Chi-squared distribution as sum expressed

Let $X_1, \ldots, X_n$ be i.i.d. exponentially distributed random variables with density $$\eqalign{\theta^{-1} e^{-x/\theta}, &x \ge 0 \\ 0, &x \lt 0} $$ and let $Y_i = X_{(i)}$ denote the ...
1
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0answers
403 views

Getting marginal effects after a panel oprobit regression in Stata (using gllamm package)

I am trying to estimate the number of companies entering certain markets using panel data. To do so, I ran an ordered probit regression in Stata using the ...
2
votes
0answers
106 views

Best estimate for a decile or quintile mean from a known distributional family?

Suppose you have a population drawn from a known distributional family f with a vector of unknown parameters θ. You may also assume that the distribution is strictly non-negative and skewed, with a ...
2
votes
1answer
208 views

Finding the Bayes estimator of $\theta$ - having trouble with likelihood calculation

Let $Y_n$ be the nth order statistic of a random sample of size n from a distribution with pdf $f(x|\theta) = 1/\theta$, $0<x<\theta$, zero elsewhere. Take the loss function to be $L[\theta, ...
3
votes
1answer
120 views

Order statistics of equal correlated continuous random variables

Suppose that $X_1, \ldots, X_n$ are mutlivariate normal with equal correlation $\rho$ and each of them are marginally distributed as $N(0,1)$. Let $X_{(1)}, \ldots, X_{(n)}$ be the corresponding order ...
1
vote
0answers
102 views

Density function of the max of smallest and largest observation

Consider $n$ independent uniform random variables $X_i \sim U(-\theta,\theta)$, and let $Y_1 = \min(X_1, \ldots, X_n)$ and $Y_n = \max(X_1, \ldots, X_n)$ . What is distribution of $Z = \max ...
3
votes
0answers
84 views

Distribution of variable

How to find the distribution of $$\sum_{i=1}^n (X_i - X_{1:n}),$$ where $X_i$ are i.i.d. random variables and $X_{1:n} = \min(X_1,X_2,...,X_n)$? I need to find the distribution in a particular case, ...
2
votes
1answer
55 views

Probability proofs using ordered samples

Given an ordered i.i.d sample $X_{(1)}, \dots, X_{(n)}$ from a continuous distribution $F(x)$. How can it be shown that: (1) $\text{Pr}(X_{(k)} \leq x) = \text{P}r(N(x) \geq k)$ where $N(x)$ is the ...
0
votes
1answer
178 views

Rank data from 1 to N (N may not be equal to number of elements in the dataset)

I calculate percentile ranks based on the method at http://www.psychstat.missouristate.edu/introbook/sbk14m.htm I need to assign ranks to a dataset where ranks are ...