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 ...

learn more… | top users | synonyms

0
votes
0answers
4 views

What type of statistical models would be useful for evaluating an athlete's performance?

Suppose a performance athlete wants to use statistics to design an optimal exercise regimen. The goal of the statistical model should be to essentially maximize the performance of the athlete by ...
2
votes
1answer
56 views

Order Statistics, Expected Value of range, $E(X_{(n)}-X_{(1)})$

$X_1, X_2,...,X_n$ is a random sample from $U(0,\theta)$. Find $E(X_{(n)}-X_{(1)})$. I attempted this question by first finding the CDF of $X_{(n)}-X_{(1)}$ using the formula: $$F_{U}(u)= ...
0
votes
0answers
14 views

Independence of ordered statistics part 2

In relation to my previous question, I have a second problem. How to show that $U_1+U_2$ is independent of $Y_3$? I can find out the pdf of $U_1+U_2$, as well as of $Y_3$, however please help me ...
5
votes
2answers
119 views

Independence and Order Statistics

I have a problem at hand, which I am not being able to proceed. Can someone help me begin? $Y_1<Y_2<Y_3$ :An order statistic of size 3 from distribution having pdf $$ f(x)=2x\ \ \ ...
0
votes
0answers
9 views

The probability distribution of rth order statistic after transformation

If I have a sample from a Rayleigh distribution, then I transform this sample to a sample from Gamma distribution by using the fact that the summation of the square of a Rayleigh variable is a Gamma ...
0
votes
1answer
28 views

When is further reduction than the order statistics not possible?

Let X1, ...Xn be a random sample from a population with location pdf f(x-θ). Show that the order statistics, T(X1, ...Xn) = (X(1), ...X(n)), are a sufficient statistic for θ and no further reduction ...
0
votes
1answer
39 views

How to show $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator

If $X_i \sim$ uniform$(0,\theta)$, how can I show that the estimator $(n+1) \max(X_1, X_2, \ldots , X_n)/n$ is a method of moments estimator for $\theta$? For example, we know the first sample moment ...
3
votes
1answer
41 views

Trouble understanding result in Simes (1986) $pr\{ P_{(j)} > \frac{j\alpha}{n} ; j = 1,…,n \} = 1-\alpha$

I'm referencing this paper http://www-stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/MultipleComparision/Simes86pdf.pdf On page 752, a theorem is presented which states that If ...
1
vote
0answers
23 views

Inverse of ordering function

Lets assume we have p by n matrix.We can generate an output matrix, w (p x p) such as w_ij represent how many times i_th rows number is bigger than j_th (can be at most n obviously). My question is ...
0
votes
1answer
28 views

Is there a statistically best place to be in a line to choose lowest number from a pool?

Here's the situation: There is a line of 500 people. Each will be given a random number from 1 to 500. I want to find out if there is a best place to be a line to get the best odds at getting a low ...
0
votes
0answers
21 views

Reward of c best tasks

I'm trying to create an analytical model for performance of a greedy scheduler in a particular domain. I've created a preliminary model, but I think it's very confusing to others and was wondering ...
1
vote
1answer
35 views

What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
0
votes
0answers
11 views

How to decide about the number of looks (window size for ensemble averaging) in SAR images?

This question has frustrated me for a while. In order to find an answer I sent an email to prof. Yamaguchi, the author of the paper Four-Component Scattering Power Decomposition With Rotation of ...
5
votes
1answer
85 views

Distribution of differences in beta-distribution

I want to get an analytic solution to the difference of the highest and second highest of a beta distribution. More simply, I have some datapoints on which I assume a beta-distribution. Analytically ...
7
votes
3answers
232 views

What is the distribution for the time before K successes happen in N trials?

What is the distribution for the time before K successes happen in N trials? Suppose there is a telephone center, and N people, each of whom will either call the telephone center in time T with ...
3
votes
0answers
70 views

Joint CDF of random variables vis-a-vis that of their order statistics

Suppose $\{X_i\}_{i\in 1\ldots n}$ are $n$ independent, non-identically distributed RV's. Let $X_i \sim f_i(x) \mathbf{1}_{[0,1]}$, where $f_i$ is the $i$-th parent supported on $[0,1]$. I am ...
3
votes
1answer
36 views

Calculate the mean of the minimum of any k samples

Given a fixed list of N numbers from an unknown distribution, and a k (k <= N), say we ...
5
votes
0answers
59 views

What is the ratio distribution of a spacing and the sample mean?

Let $X_1,\dots,X_n$ be a sample of iid exponential random variables with mean $\beta$, and let $X_{(1)},\dots,X_{(n)}$ be the order statistics from this sample. Let $\bar X = \frac{1}{n}\sum_{i=1}^n ...
3
votes
1answer
58 views

Sort $X$, then scale the first differences by $\bar X$: what, if anything, is this used for?

Sort data vector $X$, take first differences of the sorted data, and divide by $\bar X$. I came across this transformation in someone's notes, without any citation. It would be applied to ...
1
vote
0answers
30 views

how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
7
votes
2answers
128 views

Transforming Order Statistics

Assume random variables $X_1, ... , X_n$ and $Y_1, ..., Y_n$ are independent and $U(0,a)$-distributed. Show that $Z_n= n\log\frac{\max(Y_{(n)},X_{(n)})}{\min(Y_{(n)},X_{(n)})}$ has an $\text{Exp}(1)$ ...
8
votes
1answer
128 views

Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this ...
0
votes
0answers
23 views

Distribution of Spacings

If I'm understanding my notes correctly, the distribution for any finite collection of spacings is approximately Exponential with mean 1/n(f(F^-1(k/n)). Can anyone help me understand the proof of ...
2
votes
1answer
52 views

Nonparametric Identification from Order Statistics

Suppose a vector of random variables $(X_1,...,X_n,Y_1,...,Y_m)$ is such that $X\sim F(\cdot)$ and $Y\sim G(\cdot)$. So $X$ are distributed independnetly and identically as $F(\cdot)$ and $Y$ as ...
1
vote
0answers
72 views

Find pdf for minimum of 4 random variables

Let $X_1, X_2, X_3$,and $X_4$ be four mutually independent random variables, each with p.d.f. $$f(x) = 3(1-x)^2\quad \mathbb{I}_{(0,1)}(x)$$If $Y$ is the minimum of these four variables, find ...
5
votes
1answer
133 views

Sums of Random Variables from Order Statistics of Dice Rolls

Let's say you have a set of order statistics $ X_{(1)}, \dots, X_{(N)} $ drawn from a discrete uniform distribution $ \text{unif}(1,S) $. If you choose $ X_{(n_1)}, \dots, X_{(n_k)} $ from this set, ...
0
votes
1answer
33 views

Ordered response with nominal predictors

I have a data set with ordered response variables (ten levels) and nominal independent variables (sex, year of birth, educational background etc. of the participants in a questionnaire). How do I ...
1
vote
0answers
109 views

The distribution of the maximum of N independent but not identically distributed Gumbel random variables

I am interesting in determining if there is a closed form expression of the CDF and PDF of the maximum on $N$ Gumbel distributions that are independent but not identically distributed. In particular, ...
3
votes
1answer
113 views

Inter-arrival time of subsampled Poisson point process

Suppose that I draw $n$ points from a Poisson point process of rate $\lambda$, i.e. with inter-arrival times distributed i.i.d $\sim \text{Exp}(\lambda)$. Now suppose that I choose $m < n$ of ...
0
votes
0answers
31 views

Ranking of Rates of Return as dependent variable

I am working on a study for my university. I am trying to test a model that some companies uses to rank countries as a top-down model for investing. The objective of the model is to rank, explained ...
3
votes
1answer
150 views

Find the unique MVUE

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388. Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta<x<2\theta,$ ...
2
votes
1answer
114 views

joint probability distribution of $k \le n$ order statistics

For $X_i \sim$ iid random variables: For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest ...
4
votes
1answer
51 views

Partition data into two sets such that the difference of their variance is minimal

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ ...
3
votes
1answer
346 views

Distribution of sum of order statistics

The question is from a problem I am trying to solve in Robert Hogg's introduction to Mathematical Statistics 6th version problem 7.2.9 in page 380. The problem is: We consider a random sample ...
0
votes
0answers
58 views

Sort mean values with deviation

I'm trying to determine what physical attributes of a syllable are important in determining stress. So I have some recordings from different people saying a word, each person multiple times. And for ...
1
vote
1answer
42 views

Order Statistics Conditional Distribution of Affiliated System

We have a system with $M (M\ge 2)$ random variables. The M variables are related as follows. For each i, 1 to M, $X_i = I_i+Z$, where $I_i$, Z are independent uniform random variables. What is the ...
1
vote
0answers
81 views

Why is my (own) shapiro test inaccurate?

Linked to my previous question : From moments product matrix to covariance matrix of normal order statistics , i coded an EXACT shapiro-wilk test for normality. Using the related literature; i coded ...
2
votes
1answer
80 views

From moments product matrix to covariance matrix of normal order statistics

I'm trying to compute the exact covariance matrix of normal order statistics. Well known formulas (listed in Zakkula Govindarajulu, 1962) allow us to compute moments of order statistics, as well as ...
7
votes
2answers
297 views

Order statistic for beta distribution

Let $x_1,\dots,x_n$ be i.i.d. draws from $Beta\left(\frac{k}2,\frac{k-p-1}{2}\right)$. How are the minimum and maximum order statistics distributed, respectively? I would greatly appreciate a ...
0
votes
0answers
13 views

Conditional probablity of k-th order statistic of a column given the k-th order statistic of the sum of the columns?

Suppose A is a random matrix. Each row is a series of i.i.d random variables. I like to know if we can calculate the conditional probability (for a given $k$) $$P\big(A^i_{(k)} \mid (\sum ...
-1
votes
1answer
224 views

Order Statistics problem: why doesn't law of total expectation (Adam's law) work?

This is the problem The opening prices per share, $Y_1$ and $Y_2$, of two similar stocks are independent random variables, each with a density function given by $$f (y) = ...
5
votes
2answers
167 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 ...
1
vote
1answer
58 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
votes
0answers
53 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
50 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
233 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, ...
3
votes
1answer
551 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 ...
3
votes
1answer
41 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
69 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
votes
0answers
31 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 ...