All Questions
Tagged with minimum or extreme-value
606 questions
3
votes
1
answer
120
views
(Non-limit) distribution of maxima from different univariate, discrete and stationary time series
Motivation: I'm currently studying the convergence of maxima from simulated time series to max-stable distributions, and in order to do so, I want to better understand the penultimate distribution of ...
- 85
3
votes
2
answers
107
views
Estimating the blockchain mining time for $N$ nodes
I am trying to simulate a set of times for the below problem.
There are N nodes. Each node generates a random number($R$) in the range $[0,K]$ per second. Guess the time it takes by each node to ...
- 31
1
vote
1
answer
575
views
Relationship between the number of moments and Tail of the distribution?
While studying about kurtosis and extreme value theory, I came across the concept of tails of the distribution. So I wanted to ask that why is it such that distribution with higher number of moments ...
1
vote
1
answer
4k
views
How to fit distributions to data in R?
I have 6 sets of Volume(v) & Duration(d) data. I have fitted a quite few distributions to the data such as Weibull, Gamma, Log-Normal, Exponential, GEV, Pareto, Log Logistic, Poisson, and GP. This ...
- 31
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\...
- 105
7
votes
2
answers
674
views
Is power law distribution for extreme event special like normal distribution?
The power-law distribution is defined as below in Wikipedia article:
The most extreme case of a fat tail is given by a distribution whose
tail decays like a power law.
$$ \mathrm{Pr}[X>x] \sim x^{-...
- 875
1
vote
0
answers
65
views
If the density functions $f_1, f_2$ each are in domains of attraction $MDA(\xi_1)$ and $MDA(\xi_2)$, what can we say about $0.5f_1+0.5f_2$?
My question is about the maximum domains of attraction $MDA(\xi)$ from extreme value theory. I would like to be able to say statements such as "since $f$ and $g$ both are in $MDA(\xi)$, $f+g$ is also ...
- 1,594
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
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 ...
- 131
4
votes
1
answer
1k
views
Distribution of the second minimum of a set of random variables
Given $n$ i.i.d. random variables $X_1,...X_n$, what is the distribution of the second smallest value ?
I know from this question that CDF of the minimum value is $1 - (1-F(x))^n$ where $F(x)$ is the ...
- 155
1
vote
0
answers
9
views
Estimate return levels for non-annual data
Apologies if this has been asked before - I have looked but could not find anything.
I intend on estimating return levels for average annual temperature data using a number of different distributions....
- 11
0
votes
0
answers
23
views
Optimize black box function with one of the output as constraint
I have used deep learning to obtain an objective function (or black box) which I need to optimize to get max output.
So the inputs are a1,b1,b2,...,b8 while outputs are x,y and z = x^2/y.
I need ...
- 41
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 ...
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 + \...
- 2,160
2
votes
1
answer
55
views
Is there something conceptually wrong with calculating the mean of minimum values?
Assume I have distance values (e.g. how close an animal gets to a city) for each individual in my sample. So, for each individual animal, I will have a minimim distance. Next, say I have 4 categories ...
- 840
0
votes
0
answers
417
views
KL divergence for Generalized Extreme Value distribution
I have found a derivation for the Kullback–Leibler divergence between 2 Gumbel distributions here:
http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf on page 64
That document also has a ...
- 61
0
votes
0
answers
58
views
Looking for ... the Generalized Fisher-Tippett distribution
so I would like to use a parametric model that smoothly interpolates between the 3-parameter Weibull, the 3-parameter Frechet and the 3-parameter Gumbel.
Just like the Fisher-Tippett distribution (a....
- 73
3
votes
3
answers
1k
views
Expectation of the minimum of a continuous random variable $X$ and a discrete random variable $Y$
Let $X\sim Exp(1)$ and independently let $Y$ have the pmf $P(Y=k)= p$, $P(Y = \infty) = 1-p$, where $k < \infty$. I'd like to calculate $\mathbb{E}(Z)$, where $Z = \min(X,Y)$.
Usually, we tackle ...
- 309
2
votes
1
answer
804
views
Using Gumbel distribution to approximate distribution of sample maximum --- formulae for the parameters?
Suppose you have an observable sample $X_1,...,X_n \sim \text{IID } F_X$ which has a right-tail that decreases sufficiently rapidly to apply the extreme-value theorem (e.g., a normal distribution) to ...
- 133k
1
vote
1
answer
197
views
Random Sampling from Farlie-Gumbel-Morgenstern bivariate exponential distribution
I would like to obtain an algorithm for generating iid samples from Farlie-Gumbel-Morgenstern bivariate exponential distribution (as described in the book by Johnson and Kotz as Gumbel's Model II ...
0
votes
0
answers
228
views
Minimize asymptotic variance of fintely many estimates
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded function with integrable Euclidean norm on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\...
- 213
1
vote
0
answers
51
views
How error affects maximum detection?
I have a discrete function $f$ which lies over a certain domain $X$.
My goal is to find the value of $X$, $x_{max}$, for which the function is maximum.
I have opted for a simply search: using numpy ...
- 141
1
vote
0
answers
111
views
Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
- 213
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
43
views
Question about GEV
I'm doing some analysis involving rectangular pulse processes.
Suppose for each process {Xi} that X changes after equal so-...
- 413
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.$$...
- 1,209
1
vote
2
answers
2k
views
Determining shape parameter for Generalized Pareto Distribution Scipy
I have a set of values to which I want to fit a Generalized Pareto Distribution. Scipy provides functions for doing so:
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats....
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
1
vote
1
answer
120
views
how to deal with extreme values of longitudinal biomarker?
My dataset includes repeated measured longitudinal biomarker values on cancer patients. For example, every patient would take CEA (Carcinoembryonic antigen) test every 8 weeks. Now the problem is that ...
- 11
3
votes
1
answer
842
views
How to construct a function with given local minima?
I need to construct a function $f(x,y)$ in which there are 3 minima: 2 local and 1 global as given below.
Locals are: z = f(0.2,0.3) = 0.7 | z = f(0.6,0.8) = 0.8
Global is: z = f(0.85,0.5) = 0.6
As ...
- 173
1
vote
0
answers
48
views
Consecutive differences of a uniform law
Let $N>0$ be the number of considered samples. We draw $x_1, \ldots, x_n$ from a uniform distribution over $[0;1]$. We compute $y_1, \ldots, y_{n-1}$ the differences of the sorted $(x_i)_i$.
I'd ...
- 245
1
vote
1
answer
2k
views
Rewards in reinforcement learning for minimization problem
I am new to ML/DL/RL. I am looking to solve the classic travelling salesman problem (TSP), where the salesman has to visit all cities only once and finding the smallest path to do that (minimize ...
- 113
1
vote
0
answers
491
views
Obtaining the probability of exceedance corresponding a given return period
I have a time series of data (15 years). Following plots show the fitted PDF (generalized extreme value distribution) and corresponding CDF (i.e. 1 minus CDF).
The data used here is not the total ...
1
vote
0
answers
78
views
Finding Quantiles and Sampling from a Mixture of Distributions
I am trying to replicate a result in this paper, specifically I am trying to implement the mixture distribution on page 9 of the document.
The authors describe a hazard function:
$$
h(x):=h_{1}(x) \...
- 349
0
votes
1
answer
109
views
Compare return levels of fitted GPD using MLE in different R packages
This question is related to this post: Different quantiles of a fitted GPD in different R packages?
I want to constraint "potvalues" data to be in a period of 6 years, this is, 16 observations per ...
- 1
0
votes
0
answers
112
views
Robust Statistics for Finance with focus on Outliers
There's Robust Statistics with things like using median instead of mean to ignore outliers (usually considered as errors that should be ingored).
... Robust statistics are statistics with good ...
- 447
4
votes
1
answer
107
views
How to explain local minima found between two trained Neural networks?
I have trained 2 neural networks with SGD and then I have taken a linear path between their weights. Say W_0 and W_1 are the weight matrices of network 1 and network 2, respectively.
Then I compute ...
- 1,373
5
votes
1
answer
5k
views
Asymtotic distribution of the MLE of a Uniform
A property of the Maximum Likelihood Estimator is, that it
asymptotically follows a normal distribution if the solution is unique.
In case of a continuous Uniform distribution, the Maximum Likelihood ...
1
vote
1
answer
190
views
Finding maximum of quadratic function that depends on other variables
I am trying to fit a model of the following form in R:
yield = solar_rad + I(solar_rad ^ 2)
where each observation is a field and ...
- 141
0
votes
0
answers
100
views
Process of the max of a gaussian process
I know how to calculate the distribution of the max of a Gaussian process.
I am now wondering what's its process. Are its properties known? (For instance I guess that the length of constant parts ...
- 121
8
votes
1
answer
409
views
Are min$(X_1,\ldots,X_n)$ and min$(X_1Y_1,\ldots,X_nY_n)$ independent for $n$ to infinity?
Assume that we have given two continuous iid random variables $X$ and $Y$ with support $[1,c)$, where $c$ is some constant greater than one.
Now assume I have a given iid sample $X_1, \ldots,X_n$ and $...
- 81
2
votes
0
answers
48
views
Exponential Inequality For Probability of Being Close to Maximum
Given $n$ independent identically distributed random variables $X_1, X_2, \ldots, X_n$ that have $|X_i| < \lambda$ for all $i$. Let $\max(X)$ be the maximum of these $n$ variables.
Is there a ...
- 103
1
vote
1
answer
757
views
How to solve for the minimum KL Divergence when the distribution is discrete?
Say we have a simple case of $p(x,y)$ is a 3x3 matrix:
$$\begin{bmatrix}
1/6 & 0 & 0 \\
1/6 & 3/6 & 0 \\
0 & 0 & 1/6
\end{bmatrix}$$
And $q(x,y)=...
- 415
1
vote
0
answers
44
views
Back-calculation of the minimum sample size for an mechanical experiment
I want to back-calculate the minimum sample size for an experiment.
I have a known mean and standard variance of a statistical population. From the statistical population I chose two samples with ...
- 11
1
vote
0
answers
19
views
Probability of random population value being higher than sample maximum
Considering a small sample size (n < 10) from a population, I'm trying to find how likely a random population value would be greater than the maximum of the sample.
Hoping ye could help me with ...
- 11
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
4
votes
1
answer
164
views
Frechet and subexponentiality
In the context of extreme samples from distributions, I note that some subexponential distributions (such as the lognormal) are in the "domain of attraction" of the Gumbel/exponential class of ...
- 533
1
vote
1
answer
781
views
Asymptotic Distribution of Minimum Uniform Random Variables
I've been working on this problem for a while, and I've made some progress, but I'm still stuck on some parts. I was hoping to get some assistance with this!
Let $M_n = \min(X_1, ..., X_n)$ where $...
4
votes
3
answers
1k
views
Balkema-de Haan-Pickands theorem, generalized Pareto and lognormal
On the wikipedia page on the Balkema-de Haan-Pickands theorem, en.wikipedia.org/wiki/Pickands-Balkema-de_Haan_theorem, it is said the "for a large class of underlying distribution functions",...
- 533
1
vote
0
answers
88
views
How to compute largest values of random variables? [closed]
Suppose we have two discrete random variables and we want perform maximum operation to obtain the max PDF.
We know that max of two independent random variables is:
if Z = max(X,Y)
...
- 11