All Questions
Tagged with extreme-value normal-distribution
58 questions
8
votes
1
answer
3k
views
Expectation of the maximum of two correlated normal variables
I am curious what the derivation for the expectation of the maximum of two jointly normal random variables $X$ and $Y$ with correlation coefficient $\rho$.
I could start with the following but the ...
- 103
7
votes
1
answer
411
views
Estimation of a uniform distribution corrupted by Gaussian noise
Problem definition
I have a dataset composed by $m$ observations $y^{(1)},\dots,y^{(m)} \in \mathbb{R}^2$ generated as follow
\begin{equation*}\begin{aligned}
y &= z + v \newline
z & \sim\...
- 11.7k
3
votes
1
answer
1k
views
What is the distribution of max-min for a Gaussian distribution
For a process N(t), where at any instance of t=T0, the distribution of N(T0) is Gaussain with mu=0:
What is the distribution of max(N(t))-min(N(t))?
From my simulation, it has some non-zero positive ...
- 11.7k
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 ...
- 82.8k
0
votes
0
answers
25
views
Separating components of a likelihood maximization
Apologies for the naive question, but I have a problem I would like to solve.
Suppose I have a two dimensional likelihood of the form
$L \propto \exp\{-\frac{1}{2}\} \begin{bmatrix}x & y\end{...
- 1
1
vote
0
answers
51
views
Distribution of the difference between the maximum of $n$ identical and correlated Gaussian random variables and any one of them
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 ...
- 82.8k
3
votes
2
answers
289
views
Does the following distribution converge to anything?
Consider the following process for generating a random sample:
Sample $X_1, X_2, \dots, X_n \sim \mathcal{N}(0,1)$
Compute $M = \max\limits_i |X_i|$
Scale the values to get $Z_i = X_i / M$
Can we ...
- 86.5k
15
votes
2
answers
21k
views
What is the distribution for the maximum (minimum) of two independent normal random variables?
Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...
- 215
0
votes
0
answers
41
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Identifiability of a bivariate normal distribution with identified minimum
I am suffering from to understand a proof of a paper.
(Nádas, Arthur. "The distribution of the identified minimum of a normal pair determines' the distribution of the pair." Technometrics 13....
- 441
0
votes
0
answers
34
views
Find maximum of bimodal posterior pdf
can you help find the maximum (analytically) of the following posterior pdf?
$p(\theta|x) = \frac{\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta-x)^2} + \frac{1-\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta+...
- 1
12
votes
2
answers
489
views
Expected value of spurious correlation
We draw $N$ samples, each of size $n$, independently from a Normal $(\mu,\sigma^2)$ distribution.
From the $N$ samples we then choose the 2 samples which have the highest (absolute) Pearson ...
- 1,119
3
votes
1
answer
331
views
Extreme value distribution for multivariate normal
I have a series of data sets. Each data set represents a measurement in 3D space relative to a global origin. I want to model the extreme values of my data. If I were to calculate the extreme radius ...
- 58
24
votes
2
answers
11k
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Distribution of the maximum of two correlated normal variables
Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly
normal with correlation coefficient $r$.
What is the distribution function of $\max(X_1, X_2)$?
- 334k
9
votes
1
answer
2k
views
Approximating the mathematical expectation of the argmax of a Gaussian random vector
Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$.
$I$ ...
- 11
5
votes
2
answers
1k
views
CDF of maximum of $n$ correlated normal random variables
The maximum of $n$ normal i.i.d. random variables
$$Y=\max\{x_1,...,x_n\},$$
$$x_i \sim N[0,1]$$
has the CDF
$$P(Y\le y)=\Phi(y)^n $$
but how does the CDF look like, if the variables are identically ...
- 69.5k
1
vote
0
answers
36
views
Given 5 variables, all independently normally distributed, what is the probaility that variable A is lower than the other 4 variables?
Suppose variables A B C D and E are independent, normally distributed, with known variance and mean.
What is the probability that A is less than B and C and D and E?
Essentially, I have model ...
- 82.8k
1
vote
0
answers
125
views
Distribution of maximum of sample means
Let $X_1, ..., X_n$ be a sample from $N(\mu, 1)$. Fix $1 \leq m<n$ and define $$T_i= \frac{1}{m}\sum\limits_{j=i}^{i+m-1} X_j,$$ for $i \in \lbrace 1, ..., n-m+1 \rbrace$. We have the test that ...
- 111
1
vote
1
answer
256
views
Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables
Peggy Strait, 1974, Pacific Journal of Mathematics
ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES
Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&...
- 241
32
votes
3
answers
17k
views
Extreme Value Theory - Show: Normal to Gumbel
The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory.
How can we show that?
We have
$$P(\max X_i \leq x) = P(...
- 381
1
vote
0
answers
232
views
Maximum absolute from complex Gaussian distribution
Consider a random variable $Y$ with complex Gaussian distribution, i.e $Y \sim \mathcal{C N}(\mu,\sigma^2)$. We can write $Y$ as real ($Y_r$) and imaginary component ($Y_j$) as $Y = Y_r + i Y_j$. ...
- 5,706
0
votes
1
answer
50
views
Optimal way to rank candidates - concrete statement
I'd like to have some statistical/probabilistic formalisations (solutions..) of the following concrete case I have heard :
"Imagine you have a set of candidates to be interviewed for a job. You ...
- 5,730
2
votes
1
answer
693
views
The probability that the minimum of a multivariate Gaussian exceeds zero
Suppose $$X \sim \mathcal N_n(\text{diag}(\Sigma), \sigma^2 \Sigma)$$
where $\Sigma$ may be allowed to be low rank, and let $Y = \min_i
> X_i$.
What can be said about $P\left(Y \geq 0\right)$?
In ...
- 1,539
3
votes
0
answers
194
views
Distribution of the minimum of the squared Euclidean norm of a $N(\mu,\Sigma)$ random variable
Suppose that $X^n := \{x_1, x_2, \ldots, x_n\}$ is a sample of $n$ i.i.d $p$-dimensional points, where $X \sim N(\mu, \Sigma)$.
What is known about the distribution of $\min_{x_i \in X^n} \|x_i\|^2_2$?...
- 82.8k
1
vote
0
answers
142
views
Expectation of a sequence of random variables based on a set of iid Gaussian random variables
This is a rather convoluted problem: I'll my best trying to explain it. So, we have $m$ iid standard Gaussian RVs $Q_i$. We get a realization from each of them, and these values $q_1,\dots,q_m$ are ...
- 18.4k
1
vote
1
answer
456
views
Distribution of minimum distance in a iid Gaussian sample
$X_1,...,X_n$ denotes an iid sample with the same Gaussian distribution. I am interested in the distribution of the following quantity.
We first pick $i \in [n]$
Then we extract $j^* \in argmin_{j\...
- 802
5
votes
1
answer
303
views
MLE for the maximum of n values that are observed only with noise
Suppose $x_1, ..., x_n$ is a fixed set of real numbers. Let $\epsilon_1, ..., \epsilon_n \sim N(0, \sigma^2)$ be i.i.d. with known $\sigma^2$, and suppose we get to observe only $z_i = x_i + \...
- 2,299
3
votes
1
answer
600
views
Limiting distribution of maximum of i.i.d. Gaussians with decreasing variance
Consider a random vector $X^{(m)} = (X^{(m)}_1,\dots,X^{(m)}_m)$ where, for fixed $m$, the elements of $X^{(m)}$ are i.i.d. $\mathcal{N}(0,\sigma^2 / m)$.
Define $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
- 334k
2
votes
0
answers
1k
views
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: ...
- 1,622
11
votes
2
answers
8k
views
Decision trees, Gradient boosting and normality of predictors
I have a question regarding the normality of predictors. I have 100,000 observations in my data. The problem I am analysing is a classification problem so 5% of the data is assigned to class 1, 95,000 ...
- 365
2
votes
1
answer
395
views
Convergence maximum of Normal rv to gumbel through simulation (metropolis hastings)
I would like to see the convergence of an order statistic to its respective Extreme Value attractor by simulating with the Metropolis Hastings algorithm (I am self-studying MCMC algos).
I was trying ...
- 290k
2
votes
1
answer
580
views
Distribution with 3 Modes, Find the 2 In-Between Minima
Suppose I have a dataset consisting of numbers drawn from three normal distributions $\mathcal N\!(\mu_{\rm left}, \sigma_{\rm left}^2),\ \mathcal N\!(\mu_{\rm center}, \sigma_{\rm center}^2),\ \...
- 57.6k
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$. ...
- 15k
3
votes
1
answer
683
views
Variance of the maximum of linear combinations of iid normal random variables
Let $X_i$ for $i\in\{1,\dots,n\}$ be a set iid normally distributed random variables with mean $\mu$ and variance $\sigma^2$. Let $Y_j$ for $j\in\{1,\dots,2^n\}$ be a positive linear combination of ...
- 82.8k
0
votes
0
answers
51
views
Estimate true mean of the maximum of N sample means
Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with unknown mean $\mu_i$ and unknown standard deviation $\sigma_i$, $i=0,...,N-1$.
For each $i$, $M$ independent random samples ...
- 2,515
3
votes
1
answer
5k
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Expectation of max of two normal random variables
I have been reading this paper about the maximum and minimum of two normal distributed variables.
Inside the paper there is the formula for the expectation of this the maximum of the two variables. ...
- 131k
0
votes
1
answer
159
views
Interpretation of a Gumbel distribution's results
I am using (essentially) the approach outlined in the paper "Statistical-based WCET estimation and validation" (http://drops.dagstuhl.de/opus/volltexte/2009/2291/pdf/Hansen.2291.pdf) to build a Gumbel ...
- 1,571
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 ...
- 8,001
4
votes
1
answer
87
views
Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)
I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14:
$$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
- 18.4k
1
vote
1
answer
351
views
Approximation/bound to a_n and b_n in normal maxima to Gumbel
I was reading this which tries to find $a_n$ and $b_n$ such that
$$F\left(a_n x+b_n\right)^n\rightarrow^{n\rightarrow\infty} G(x) = e^{-\exp(-x)},$$
where $F$ is the cdf of a standard normal.
The ...
- 252
4
votes
0
answers
189
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Extreme value distribution for univariate normal: Derive parameters of the Gumbel [duplicate]
I have a question regarding the extreme value distribution corresponding to i.i.d. samples $X_i$ from a normal distribution, say $X_i\sim N(\mu, \sigma^2)$.
According to the theorem of Fisher-Tippett-...
- 41
13
votes
3
answers
1k
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Does there exist someone faster than Usain Bolt today?
EDIT: I am more interested in the technical issues and methodology of determining the likelihood of a "true" maximum in a given population given a sample statistic. There are problems with estimating ...
- 141k
0
votes
0
answers
54
views
Drawing the smallest value from a set of distributions
I have a specific number of normal distributions $N_D$ all with their own mean $\mu$ and std $\sigma$. Now I obtain a sample from all distributions which results in a set of $N_D$ samples. What is the ...
5
votes
1
answer
112
views
Distribution of $\dfrac{X_{i}}{\max X_{i}}$?
Suppose we have a sample of standard normal i.i.d. observations $X_{1}, X_{2}, \cdots ,X_{n}$, what can we say about the asymptotic distribution of $V_{i}$ and $W_{i}$, where:
$V_{i}=\dfrac{X_{i}}{\...
CommunityBot
- 1
1
vote
0
answers
46
views
Probability that the Maximum of Many Normal Draws from Multiple Classes is of one Class
Given a set of $N=n_i+n_j+n_k$ draws from distributions $N(\mu_i,\sigma_i^2), N(\mu_j,\sigma_j^2), N(\mu_k,\sigma_k^2)$, what is the probability that the maximum drawn value was from distribution $i,j,...
- 11
1
vote
0
answers
308
views
Could the sum of two normally distributed random variables be a GEV distribution?
I'm playing with Matlab, I have got a test statistic $T$ wich is of the form :
$$T(x)=\sum_{i=1}^{n}f_{i}(x)+\sum_{i=1}^{n}g_{i}(x)+c $$
Where $f$ and $g$ are functions of the observations $x$, $n$ is ...
- 902
1
vote
0
answers
53
views
Mean & Variance after applying a Maximum function n-times [closed]
I am trying to solve a statistics problem with very little statistics theory. I can determine a solution to my problem by writing a simulation program, but I really need to have the proper formula ...
- 11
0
votes
1
answer
191
views
How to find the maximum likelihood of this scenario occurring?
I have the equation: $ 0= 2.01106 - 0.00274(34.647+24.24a)-0.02059(45.647+21.122b)+1.37984(2.05-0.206c)-0.01176(10.588+11.963d)+0.00394(118.29-21.097e)-0.03552(92.17+2.855f)$
I have the above ...
CommunityBot
- 1
4
votes
1
answer
442
views
Using extreme value theory to estimate bounds
Suppose I have I have a random variable $X$ that I know is doubly bounded on support $[0,\theta]$ but I dont know $\theta$ (we don't know anything on the distribution of $X$, but assume it is not ...
CommunityBot
- 1
3
votes
0
answers
770
views
GEV of Normal Distribution and relationship of the parameters
My question goes on Extreme Value Theory for the Normal distribution (www.math.ethz.ch/~embrecht/RM/chap7.pdf):
Which type of GEV (Generalized Extreme Value) distribution does the Normal distribution ...
- 19.3k
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