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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31 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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21 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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14 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
76 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 ...
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0answers
68 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 ...
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2answers
119 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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34 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 ...
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33 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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0answers
37 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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3answers
257 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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0answers
75 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
71 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
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0answers
75 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 ...
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1answer
160 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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0answers
55 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
93 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
432 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 ...
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42 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 ...
3
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1answer
65 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 ...
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0answers
63 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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48 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
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1answer
100 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 ...
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35 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 ...
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1answer
125 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 ...
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70 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)
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1answer
100 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. ...
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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
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0answers
65 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 ...
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15 views

Arrangement of the ranges .

To derive the distribution of $i^{th}$ order statistics $$f(x_{(i)})=n!f(x_{(i)})\int_{-\infty}^{\infty}\ldots \int_{-\infty}^{\infty}f(x_{(1)})\ldots f(x_{(i-1)})f(x_{(i+1)})\ldots ...
0
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18 views

If $n$ order stats are iid from Uniform(0,1), why does dividing by the highest order stat give $n-1$ order stats iid from Uniform(0,1)? [duplicate]

As the title states: If $P_{(1)}, ... ,P_{(n)}$ are order statistics of $n$ independent uniform $(0,1)$ random variables, why are $P_{(1)}/P_{(n)} ..... P_{(n-1)}/P_{(n)}$ also order statistics of ...
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87 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 $0\le \alpha \le n$, ...
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0answers
55 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 ...
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0answers
290 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
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0answers
85 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
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1answer
185 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, ...
2
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1answer
95 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 ...
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90 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
77 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
54 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 ...
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1answer
148 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 ...
1
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0answers
128 views

Order statistic of dependent chi-squares

I was wondering if someone could please help me with the following. I am trying to find the pdf of a set of dependent $\chi^2$ random variables. Suppose $x_1,x_2,...,x_n$ are independent normals $x_i ...
3
votes
1answer
117 views

Probability of an order statistic

If I randomize time points $$T=\{266.14646, 107.28526, 108.89631, 593.03129,\\ 118.10284, 425.60470, 39.02817, 291.90210\}$$ into two groups of equal size, then what is the probability that $T_i$ (the ...
3
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1answer
201 views

Spacings between discrete uniform random variables

Let $U_1, \ldots, U_n$ be $n$ i.i.d discrete uniform random variables on (0,1) and their order statistics be $U_{(1)}, \ldots, U_{(n)}$. Define $D_i=U_{(i)}-U_{(i-1)}$ for $i=1, \ldots, n$ with ...
3
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2answers
142 views

Probability of a random variable to be the largest among others

Let us have $N$ random variables generated by uniform distribution. That is, $$u_i \sim \mathcal{U}(0,1),\quad i=1,\ldots,N$$. What is the probability of $u_N$ being the largest? I.e., how can I ...
3
votes
1answer
143 views

Computation of distribution parameters for the maximum of two random variables

Let $X$ be a beta distributed variable with parameters $a$, $b$. Let $Y$ be a beta distributed variable with parameters $c$, $d$. Let $Z = \max(X, Y)$. Does anybody know of a fast way to compute the ...
4
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1answer
150 views

Covariance of INID order statistics

In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...
3
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1answer
246 views

Order statistics of absolute value of bivariate normal distribution

Suppose $X_1$ and $X_2$ are bivariate normal and let $|X|_{(1)}$ and $|X|_{(2)}$ be the ordered version of their absolute value. I am interesting in finding the following probabilities or some bounds ...
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1answer
586 views

How to show order statistic is sufficient

I have some trouble showing sufficiency for largest order statistic ${x}_{n}$. This is from Casella's text, problem 1.6.3. Let ${p}_{\theta}$ be a density function. ${p}_{\theta}{x}=c({\theta})f(x)$ ...
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1answer
117 views

Estimating ratios of quantile means for sorted data

In studying the relationship between income and consumption, it is common to sort the observations by income and observe that high-income households have lower levels of consumption per dollar of ...
3
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1answer
218 views

Relevancy of order statistics to the roll-and-keep dice mechanic?

Throw $x$ dice, each of which has $z$ sides, keep the $y$ highest values rolled, and find their sum $s$. In roleplaying games, this is called a "roll and keep dice mechanic" and notated "$x$d$z$k$y$" ...