All Questions
Tagged with extreme-value order-statistics
47 questions
5
votes
1
answer
216
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In a sum of high-variance lognormals, what fraction comes from the first term?
If $X_i \overset{\textrm{iid}}{\sim} \text{Lognormal}(0, \sigma^2)$ for $i=1,\ldots,n$ and $Y_1 = X_1 / \sum_{j=1}^n X_j$, then I would expect that a particular* limiting distribution of $Y_1$, ...
- 195
1
vote
0
answers
39
views
How should I best to use reported stats on the Tippy-top?
Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
- 3,247
0
votes
0
answers
170
views
How to find the MGF of the max of a set of i.i.d. exponential random variables
As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
0
votes
1
answer
88
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Max of the running average of the kth through nth elements for a given probability distribution
This question is based slightly on https://www.reddit.com/r/AskStatistics/comments/16bqit0/calculating_probability_when_phacking_is_allowed/
Given a variable $X$, let $A_j$ be the average of $X_1$ ...
- 111
0
votes
0
answers
42
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Is modeling the extreme value of a distribution a basic probability result?
I was reading briefly about the field of EVT - extreme value theory, and the associated distributions that arise from modeling the maximum of a finite sample. It's not quite clear to me the nature of ...
- 64.8k
2
votes
1
answer
178
views
CDF of max of $n$ cauchy variates
Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf
$$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$
Define:
$$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
- 447
2
votes
1
answer
259
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Distribution/estimation of maximum change of a stationary time series
Any help on this would be much appreciated.
Let $x_{t} = b x_{t-1} + u_{t}$, where $u_{t} \sim N(0,1)$ and $\lvert{b}\rvert < 1$.
What can we say about the distribution of $y_{t} = \max(x_{t+2},x_{...
- 21
1
vote
0
answers
256
views
Multinomial Logistic Regression as a latent variable model
I was reading the wiki entry for multinomial logistic regression https://en.wikipedia.org/wiki/Multinomial_logistic_regression#As_a_latent-variable_model
and it states that we can view the multinomial ...
- 11
0
votes
0
answers
128
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Probability bound of the difference of order statistics for correlated and identical Gaussian random variables
Suppose, there are $n$ identical and correlated Gaussian random variables namely, $X_1, X_2, ..., X_n$ with $X_i\sim\mathcal{N}(0,\sigma^2)$ for all $i\in\{1,2, ...n\}$. The correlation coefficient ...
5
votes
1
answer
171
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$N \sim \text{Po}(\lambda)$ and $X_1,X_2,....,X_N$ are iid and independent of $N$, what is distribution of $Z_N = \max \{X_i\}_{i=1}^{N}$
I think the title covers most of my concerns. The distribution of the $X_i$ does not really matter, I am just experiencing difficulties in finding an expression for
$$\text{Pr}(Z_N \leq x) = F(x)^N$$
...
4
votes
1
answer
319
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computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$
Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
0
votes
0
answers
52
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Distribution of the minimum of the components of a multivariate normal random variable [duplicate]
Let $\mathbf{X} = (X_1, \dots, X_p)^\mathsf{T}$ be a $p$-dimensional random variable following a multivariate normal distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{...
- 789
2
votes
1
answer
693
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The probability that the minimum of a multivariate Gaussian exceeds zero
Suppose $$X \sim \mathcal N_n(\text{diag}(\Sigma), \sigma^2 \Sigma)$$
where $\Sigma$ may be allowed to be low rank, and let $Y = \min_i
> X_i$.
What can be said about $P\left(Y \geq 0\right)$?
In ...
- 20.8k
5
votes
2
answers
272
views
Unbiased Estimator of Largest Mean of Two Normal Distributions
Given samples from two normal distributions:
$X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X^2)$ for $i = 1,...,n$
$Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y^2)$ for $i = 1,...,n$
How ...
1
vote
0
answers
69
views
Variance of $k$th order statistic of normal vector [duplicate]
Let $Z \sim \mathcal{N}(0, I)$. Let $Z_{(k)}$ be the $k$th order statistic of $Z$.
Is it true that $\text{Var}(Z_{(k)}) \to 0$ as $n\to \infty$ for $1 \leq k \leq n$?
Any estimate on the rate?
What ...
- 291
0
votes
0
answers
27
views
Estimating error for a sample minimum
Let's say I'm benchmarking some computer program and, due to non-randomness in the input data, I'm interested in the minimum running time (as opposed to the average over random inputs). In addition to ...
- 101
2
votes
0
answers
106
views
How to fully estimate a probability density from only a sample of minimum values?
We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$.
By means of ...
- 1,733
7
votes
2
answers
286
views
What is the distribution of a bivariate normal component conditional on the max of the other component?
Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
- 563
1
vote
1
answer
176
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Meaning of Extreme Value distribution vs. lowest/highest Order Statistic
How exactly does the meaning of the Extreme Value Distribution differ from the distribution of the lowest/highest (extreme) order statistics?
I understand that the extreme value distribution (EVD) ...
- 111
7
votes
1
answer
243
views
Is there a random variable $X$ with positive support such that the ratio of the two smallest realizations of an iid sample goes to one?
Imagine I have given a random variable $X$ with supp$(X)=(0,\infty)$ and $\mathbb P(X \in (0,a))>0$ for any fixed $a>0$
Now given an iid sample $X_1,...,X_n$ - is it possible that
$$X^{(2)}/...
user244467
-1
votes
1
answer
210
views
Estimation of an exponential parameter
I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter.
I tried with ...
- 1
3
votes
1
answer
2k
views
Mean of maximum of exponential random variables (independent but not identical)
I am looking for the the mean of the maximum of N independent but not identical exponential random variables. I found the CDF and the pdf but I couldn't compute the integral to find the mean of the ...
- 33
3
votes
3
answers
741
views
Finding the mean of the max order statistic drawn from standard normal
Let $X=\max\{X_1, X_2, \cdots, X_N\}$, where each $X_i \sim N(0,1)$ and are independent. What is the approximate value of $X$ for large $N$.
The term "approximate" isn't defined very clearly. I'm ...
- 61
1
vote
1
answer
94
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Simple probability question (similar to birthday paradox)
If $x$ objects are randomly distributed to $n$ groups, what is the formula for working out how big $x$ needs to be for the probability that at least one of the groups gets an amount $y$ (or larger) to ...
- 41
2
votes
1
answer
1k
views
Hypothesis test for minimum/maximum
I need some kind of hypothesis test (or at least a reasonable rule of thumb) that will enable me to validate if observed minimum and/or maximum is "close enough" to the theoretical minimum/maximum. I ...
- 141k
2
votes
0
answers
119
views
Is Var(sample min) decreasing in sample size?
Suppose $z_n=\min\{x_1,\dots,x_n\}$ where $x_i$'s are i.i.d. according to CDF $F$ over $[0,1]$.
Is it true that $Var(z_n)>Var(z_{n+1})$? What conditions would I need to ensure this monotonic ...
- 329
2
votes
0
answers
87
views
When is variance of sample maximum greater than unconditional variance?
Let $X_1$,...,$X_n$ be $n$ i.i.d. RVs with continuous distribution $F$. Further let $X_{(1)}$,...,$X_{(n)}$ be the associated order statistics such that $X_{(1)}<X_{(2)}<...<X_{(n)}$.
Under ...
0
votes
1
answer
4k
views
Finding the PDF of Y, where Y = min X
Have $ X_{1},X_{2},\cdots,X_{10}$ random sample from a distribution with PDF:
$$ f(x;\theta) = e^{ - (x- \theta) },\, \theta \leq x \lt \infty $$
Know that $ \hat{\theta}_{MLE} = Y = min(X_{i},\;i=...
- 39
7
votes
1
answer
1k
views
Maximum of a probability vector distributed as a Dirichlet variate
Let $p_1, p_2, \ldots \sim \text{Dirichlet}(\alpha_1, \alpha_2, \ldots)$. What is the distribution of $\max(p_1, p_2, \ldots)$?
I have searched for the order statistics of the Dirichlet distribution ...
- 1,033
10
votes
1
answer
539
views
Expected value of maximum ratio of n iid normal variables
Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$.
How would one be able to upper bound the expected maximum of the ratio ...
- 103
2
votes
1
answer
350
views
Expectation of two identical lognormal distributions
I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions.
Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...
- 229
5
votes
1
answer
583
views
What's the probability that the next test will exceed the previous maximum?
I take a sample of 10 from a population that I suspect is non-normal (but it is continuous). I need to calculate the probability that the next sample will be larger than the maximum of the previous 10 ...
- 305
4
votes
2
answers
373
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.
...
- 347
6
votes
1
answer
2k
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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 ...
- 2,096
4
votes
1
answer
562
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 ...
- 1,531
23
votes
3
answers
3k
views
Distribution of the largest fragment of a broken stick (spacings)
Let a stick of length 1 be broken in $k+1$ fragments uniformly at random. What is the distribution of the length of the longest fragment?
More formally, let $(U_1, \ldots U_k)$ be IID $U(0,1)$, and ...
- 14.9k
2
votes
1
answer
114
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 ...
- 61
3
votes
1
answer
158
views
Joint distribution of a random variable and the sample maximum
This is one necessary part of a slightly larger problem, but this part has me stumped.
We have that $X_1, X_2, ..., X_n\stackrel{iid}{\sim} U(0,\theta)$. What is the joint density of the first ...
- 39
2
votes
0
answers
1k
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maximum gap between order statistics of normally distributed random variables [closed]
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 solution. So here I am, my questions is: ...
- 21
9
votes
1
answer
9k
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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 ...
- 60.8k
5
votes
1
answer
169
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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 ...
- 1,176
3
votes
0
answers
142
views
Distribution of variable
How to find the distribution of $$\sum_{i=1}^n (X_i - X_{1:n}),$$ where $X_i$ are i.i.d. random variables and $X_{1:n} = \min(X_1,X_2,...,X_n)$?
I need to find the distribution in a particular case, ...
- 31
3
votes
2
answers
341
views
Probability of a random variable to be the largest among others
Let us have $N$ random variables generated by uniform distribution. That is, $$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 ...
- 75
12
votes
2
answers
3k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, and tried to ...
- 696
2
votes
1
answer
209
views
The min draw from F(x), where the max is an order statistic of the max draws from different, yet overlapping distributions?
Consider $m$ independent draws from each of $n$ distributions. $X_{i,j}$ the $i_{th}$ draw from cdf $F_{j}(x)$. where $1 \leq i \leq m$ and $1 \leq j \leq n$. Therefore we have $m\cdot n$ total ...
- 1,049
9
votes
2
answers
932
views
What is the distribution of maximum of a pair of iid draws, where the minimum is an order statistic of other minima?
Consider $n\cdot m$ independent draws from cdf $F(x)$, which is defined over 0-1, where $n$ and $m$ are integers. Arbitrarily group the draws into $n$ groups with m values in each group. Look at the ...
- 1,049
9
votes
3
answers
10k
views
What is the expected MINIMUM value drawn from a uniform distribution between 0 and 1 after n trials?
Assume you draw a uniformly distributed random number between 0 and 1 n times. How would one go about calculating the expected minimum number drawn after n trials?
In addition, how would one go ...
- 597