All Questions
Tagged with extreme-value extreme-value or
96 questions
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
$$P(\max X_i \leq x) = P(...
- 1,271
73
votes
4
answers
191k
views
How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?
Given the random variable
$$Y = \max(X_1, X_2, \ldots, X_n)$$
where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
- 871
83
votes
3
answers
105k
views
How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
- 941
24
votes
2
answers
11k
views
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)$?
- 2,295
11
votes
1
answer
4k
views
Using bootstrap to obtain sampling distribution of 1st-percentile
I have a sample (of size 250) from a population. I do not know the distribution of the population.
The main question: I want a point estimate of the 1st-percentile of the population, and then I want ...
- 69.5k
69
votes
9
answers
8k
views
Taleb and the Black Swan
Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual ...
- 43.2k
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 ...
- 355
12
votes
3
answers
32k
views
Calculating distribution from min, mean, and max
Suppose I have the minimum, mean, and maximum of some data set, say, 10, 20, and 25. Is there a way to:
create a distribution from these data, and
know what percentage of the population likely lies ...
- 123
26
votes
6
answers
53k
views
Can mean plus one standard deviation exceed maximum value?
I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.
My professor asks me how can mean plus one standard deviation exceed the maximum.
I showed her ...
- 261
6
votes
1
answer
435
views
Confidence interval for GLM or the maximum of a function?
Imagine I have a set of (xi,yi) measures.
I can show it on a scatter plot
I want to choose the value of x that maximizes y,
or I could fit a function and find the values of the parameters that ...
- 1,094
4
votes
1
answer
2k
views
How to find the $(a_n,b_n)$ for extreme value theory
In the solution to this question (Extreme Value Theory - Show: Normal to Gumbel), the OP asked for the sequence $(a_n, b_n)$ such that $\Phi(a_nx+b_n)$ converges to the Gumbel CDF. Not only did I not ...
- 569
14
votes
4
answers
1k
views
Unbiased estimator for the smaller of two random variables
Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$
I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$?
The simple estimator ...
- 141
11
votes
2
answers
4k
views
Asymptotic distribution of maximum order statistic of IID random normals
Is there a nice limiting distribution of $\max( X_1,X_2,...,X_n) $ as $n$ goes to $\infty$, assuming that they are iid normal distributions with variance $\sigma^2$.
This is almost certainly a well ...
- 1,511
2
votes
2
answers
25k
views
Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables
Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval $(-1,1)$.
Determine the density for $U=\min(X,Y)$ and for $W=\max(X,Y)$
- 23
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
19
votes
2
answers
12k
views
What is the variance of the maximum of a sample?
I'm looking for bounds on the variance of the maximum of a set of random variables. In other words, I'm looking for closed-form formulas for $B$, such that
$$
\mbox{Var}(\max_i X_i) \leq B \enspace,
$$...
- 273
9
votes
1
answer
442
views
Intuition about the coupon collector problem approaching a Gumbel distribution
The coupon collector's problem
Let there be $n$ different types of coupons and we try to collect all of the types.
We do this by independent random draws of coupons in which each type of coupon has an ...
- 86.5k
3
votes
1
answer
1k
views
Boundaries on correlation coefficient given five other correlations
Is there a general formula for the boundaries of a correlation coefficient given a set of other correlation coefficients? I have seen the formula for three random variables where two correlations are ...
- 31
3
votes
1
answer
870
views
$\mathbb{E}$ and Variance of the maximum of independent $\mathcal{N}(\mu_i, \sigma_i^2)$
I am interested in the expectation and the variance of the maximum of several independent, normal distributed variances. That is, given a set of $I$ different RVs with $X_i \sim \mathcal{N}(\mu_i, \...
- 14k
3
votes
1
answer
353
views
How does the maximum distance between adjacent values vary for increasing $n$
That is, when is the $\underset{n \to \infty}{\lim} \max (X_i-X_{i-1})\rightarrow 0$, where $1<i\leq n$, and $X_i\geq X_{i-1}$ and when is the limit $\neq 0$? The question supposes that the ...
- 13.3k
2
votes
1
answer
259
views
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
24
votes
6
answers
48k
views
Why doesn't k-means give the global minimum?
I read that the k-means algorithm only converges to a local minimum and not to a global minimum. Why is this? I can logically think of how initialization could affect the final clustering and there is ...
- 589
24
votes
5
answers
5k
views
Why use extreme value theory?
I'm coming from Civil Engineering, in which we use Extreme Value Theory, like GEV distribution to predict the value of certain events, like The biggest wind speed, i.e the value that 98.5% of the wind ...
- 1,285
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
1
answer
9k
views
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
9
votes
3
answers
2k
views
Extreme value theory for count data
I am aware of extreme value theory for continuous distributions. I need to fit an extreme value distribution to the maximum observation of number of events on a day, per month. This seems to be the ...
- 205
8
votes
3
answers
1k
views
Variance of Minimum and Maximum of 2 iid Normal
Let $X$ and $Y$ be iid $\sim Normal(0,1)$
Let $A=max(X,Y)$ and $B=min(X,Y)$
What are $Var(A)$ and $Var(B)$?
From simulation, I get $Var(A)=Var(B)$ approximately 0.70.
How do I get this ...
- 1,347
8
votes
1
answer
1k
views
Required: Method of moments fitting routine for the two-parameter generalized Pareto
I am currently using the evd package which fits a two-parameter GPD by maximum likelihood.
Since in small samples the MOM is superior to the ML estimation I'd like ...
- 1,082
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
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
6
votes
2
answers
2k
views
Generalized Pareto distribution (GPD)
I would like to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. My questions are:
what is the rationale for replacing $z$ with $\frac{x-\mu}{\sigma}...
- 591
4
votes
3
answers
42k
views
What's the minimum sample size required to do a time series analysis?
I'd like to know the minimum number of monthly data points required to do time series analysis with the seasonality effect in forecasting.
I read some articles & they were saying that 50 or 60 ...
- 65
4
votes
1
answer
122
views
How to simulate variability (errors) in fitting a gamma model to survival data by using a generalized minimum extreme value distribution in R?
As shown below and per the R code at the bottom, I plot a base survival curve for the lung dataset from the survival package ...
- 827
4
votes
1
answer
6k
views
Cdf of minimum of two iid random variables
I am struggling with the following sentence:
Using the fact that the cumulative distribution of the
minimum of two i.i.d. random variables can be expressed as $1 - (1 - F(x))^2$....
Can anyone ...
- 694
3
votes
1
answer
5k
views
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. ...
- 133
2
votes
2
answers
4k
views
Maximization of Output based on Input
What I want to do is find the values for $X = $ { $x_j$ } that will produce the maximum $y$.
I'm currently trying to maximize my output $y$, based on my inputs $X$.
Say there are inputs, $X = $ { $...
- 121
1
vote
1
answer
323
views
Correctly simulating an extreme value distribution for survival analysis?
In the image and per the code at the bottom of this post, I plot survival curves for the lung dataset from the survival package using a fitted exponential ...
- 827
21
votes
5
answers
2k
views
Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?
If it is impossible, what is the proof?
- 542
15
votes
1
answer
10k
views
Finding local extrema of a density function using splines
I am trying to find the local maxima for a probability density function (found using R's density method). I cannot do a simple "look around neighbors" method (where ...
- 373
14
votes
1
answer
1k
views
Any example of (roughly) independent variables that are dependent at extreme values?
I am looking for an example of 2 random variables $X$, $Y$ such that
$$\newcommand{\cor}{{\rm cor}}|\cor(X,Y)| \approx 0 $$
but when consider the tail part of the distributions, they are highly ...
- 143
12
votes
3
answers
1k
views
Classes of distributions closed under maximum
Let $Q_p$ be a class of probability distributions on non-negative reals parametrized by $p$, so that
$$
Q_p([0,\infty)) = 1.
$$
I wonder which known classes of distributions are closed under ...
- 473
12
votes
2
answers
3k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, and tried to ...
- 696
10
votes
1
answer
2k
views
Extreme Value Theory: Lognormal GEV parameters
Lognormal distribution belongs to the Gumbel maximum domain of attraction, where:
$F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$,
$F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{...
- 1,271
10
votes
3
answers
433
views
If $Z_i =\min \{k_i, X_i\}$, $X_i \sim U[a_i, b_i]$, what is the distribution of $\sum_iZ_i$?
Assume the following set up:
Let $Z_i = \min\{k_i, X_i\}, i=1,...,n$. Also $X_i \sim U[a_i, b_i], \; a_i, b_i >0$. Moreover $k_i = ca_i + (1-c)b_i,\;\; 0<c<1$ i.e. $k_i$ is a convex ...
- 60.8k
10
votes
1
answer
4k
views
What's the difference between Bayesian Optimization (Gaussian Processes) and Simulated Annealing in practice
Both processes seem to be used to estimate the maximum value of an unknown function, and both obviously have different ways of doing so.
But in practice is either method essentially interchangeable? ...
- 459
9
votes
2
answers
189
views
What is the most powerful result about the maximum of i.i.d. Gaussians? The most used in practice?
Given $X_1, \ldots, X_n, \ldots \sim \mathscr{N}(0,1)$ i.i.d., consider the random variables
$$ Z_n := \max_{1 \le i \le n} X_i\,. $$
Question: What is the most "important" result about ...
- 6,479
9
votes
1
answer
2k
views
Approximating the mathematical expectation of the argmax of a Gaussian random vector
Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$.
$I$ ...
user193726
9
votes
1
answer
6k
views
MAP estimation as regularisation of MLE
Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this:
It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
- 1,059
8
votes
1
answer
330
views
Interpretation of Constraint in Maximum Entropy Derivation of Cauchy distribution
As per Wikipedia:
The Cauchy distribution is the maximum entropy probability distribution for a random variate $X$ for which
$$
{\displaystyle \operatorname {E} [\log(1+(X-x_{0})^{2}/\gamma ^{...
- 153
8
votes
1
answer
533
views
Expected minimum distance from a point with varying density
I'm looking at how the expected minimum Euclidean distance between randomly uniform points and the origin changes as we increase the density of random points (points per unit square) around the origin....
- 183