All Questions
Tagged with minimum or extreme-value
606 questions
6
votes
2
answers
433
views
Do you need large amounts of data to estimate parameters in extreme value distributions?
There is probably not a hard answer for this, but I am wondering if you need to collect more data when trying to estimate the parameters of generalized pareto distribution well?
The reason I ask is ...
- 99
4
votes
0
answers
75
views
Estimation of the density at the bound of the support of a real random variable
Let $X$ be a random variable with real values and with density $f$.
Assume the support $f$ is bounded with supremum $m$ and has a positive value at that supremum:
$$\forall x > m, f(x) = 0 \text{ ...
- 2,629
10
votes
1
answer
642
views
Distribution of argmax of beta-distributed random variables
Let $x_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. Let $j = \operatorname*{argmax}_{i \in I} x_i$ (ties broken arbitrarily). What is the distribution of $j$ in terms of $\alpha$ and $\beta$? ...
- 1,033
0
votes
0
answers
76
views
Normalization of $M_{n} = \max(U_{1}, ... , U_{n})$
Let $M_{n} = \max(U_{1}... , U_{n})$ be the maximum of a sample size $n$ from $U(0 , 1)$ distribution.
In my statistics textbook it says that $M_{n}$ normalized is equal to $n(1 - M_{n})$ but I'm not ...
- 107
9
votes
1
answer
442
views
Intuition about the coupon collector problem approaching a Gumbel distribution
The coupon collector's problem
Let there be $n$ different types of coupons and we try to collect all of the types.
We do this by independent random draws of coupons in which each type of coupon has an ...
- 86.5k
5
votes
2
answers
1k
views
CDF of maximum of $n$ correlated normal random variables
The maximum of $n$ normal i.i.d. random variables
$$Y=\max\{x_1,...,x_n\},$$
$$x_i \sim N[0,1]$$
has the CDF
$$P(Y\le y)=\Phi(y)^n $$
but how does the CDF look like, if the variables are identically ...
- 1,250
1
vote
1
answer
2k
views
Return level plots for GEV-distribution
I was reading An Introduction to Statistical Modeling of Extreme Values by Stuart Coles, and I ran into a problem whilst trying to replicate a basic return level graph in R. For context, I first ...
- 79
2
votes
1
answer
63
views
A front-loaded Gumbel-like distribution
I'm looking for a distribution that is somewhat like the Gumbel distribution and I was wondering if anyone could help.
The parameters are a positive integer $n$ and real numbers $\mu>0$ and $\sigma&...
- 1,248
0
votes
0
answers
127
views
Estimate argmax of function that is measured at discrete points
I have gathered simulation data of a function $f(x)$, where $x$ is a continuous variable. I measure $f$ at discrete points $x_k$. Since the underlying process is stochastic, I performed Monte Carlo ...
1
vote
0
answers
45
views
Does this distribution with polynomial tails have a name?
I have $N$ random variables which are identically and independently distributed with complementary CDF:
$$Pr[X \geq x] = \frac{a}{X} + \frac{b}{X^2}$$
for $x \geq 1/2 \sqrt{a^2 + 4 b} + a/2$.
This ...
- 359
1
vote
0
answers
71
views
Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions
I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are ...
- 41
1
vote
0
answers
36
views
Given 5 variables, all independently normally distributed, what is the probaility that variable A is lower than the other 4 variables?
Suppose variables A B C D and E are independent, normally distributed, with known variance and mean.
What is the probability that A is less than B and C and D and E?
Essentially, I have model ...
- 11
1
vote
0
answers
125
views
Distribution of maximum of sample means
Let $X_1, ..., X_n$ be a sample from $N(\mu, 1)$. Fix $1 \leq m<n$ and define $$T_i= \frac{1}{m}\sum\limits_{j=i}^{i+m-1} X_j,$$ for $i \in \lbrace 1, ..., n-m+1 \rbrace$. We have the test that ...
- 111
1
vote
0
answers
37
views
How close am I to the true minimum?
This might be a trivial question but my statistics knowledge very is rudimentary:
I'm trying to measure the amount of clock cycles that my computer needs to execute a certain function. The number of ...
- 51
0
votes
1
answer
84
views
How can i find out closest lognormal distribution parameters from a GEV distributed data in R
The question is a bit weird so i'll open it up.
So i have a table of return periods for different amounts of rain. The table has been made using GEV distribution on known data and then the mean and ...
1
vote
0
answers
72
views
Extreme Value Analysis of Hurricane wind speeds
As per the theory, an EVA with annual maxima presupposes that the series is complete, i.e. all years have an event. However, hurricanes don't occur every year, and so the hurricane wind speeds in ...
- 398
8
votes
2
answers
763
views
Intuition behind Weibull distribution?
I don't understand the physical meaning of Weibull distribution's $k$ parameter. Here is a simplified formula of cumulative probability function of Weibull in the simplest form:
$$p(\xi \geq x) = e^{-(...
- 378
3
votes
1
answer
299
views
Method of collecting and comparing outliers from sets of sets of populations
Background
I am a PhD student co-supervising a Master's student in our lab. I am mostly familiar with discrete mathematics, signal processing, and programming simulations. My statistics background ...
4
votes
1
answer
353
views
How to extract the shape parameter of a Fréchet fitted model using the R SPREDA package?
I'm trying to follow this post, which fits a Frechet distribution to some wind measurements as follows:
...
- 26.9k
3
votes
1
answer
913
views
Fat tails equal higher probability of non-extreme values according to Nassim Taleb?
I just came across the following passage written by Nassim Taleb Link:
The fattest tail distribution has just one very large extreme deviation, rather than many departures form the norm. [...] if we ...
- 1,129
4
votes
1
answer
486
views
Student's t as a power law distribution
I'm currently reading about power laws and I have came across an answer stating:
The density function of a Student's t-distribution with $n$ degrees of freedom is:
$$f(x) \sim (1 + x^2 / n)^{-(n+1)/2}...
- 750
4
votes
1
answer
319
views
computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$
Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
1
vote
1
answer
256
views
Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables
Peggy Strait, 1974, Pacific Journal of Mathematics
ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES
Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&...
- 241
0
votes
0
answers
71
views
Minimum of Multivariate Pareto
Suppose I have a multivariate Pareto distribution with cdf,
$$ Prob(Z_{1}<z_{1},\dots,Z_{n}<z_{n}) = H(\textbf{z}) = 1 - \left( \sum_{i=1}^{n} (T_{i}z_{i}^{-\theta})^{\frac{1}{1-\rho}} \right)^{...
- 253
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 $Y = Y_r + i Y_j$. ...
- 121
4
votes
1
answer
453
views
Residuals in Generalized Pareto Distribution
I'm learning generalized Pareto distribution for fitting extreme value data. I came across an R package evir that is able to plot residuals. Residuals from a GPD ...
- 8,655
2
votes
1
answer
123
views
Density plot with epanechnikov with exceedance data
I'm trying to replicate empirical density plot from the paper "Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution".
The data is ...
- 8,655
0
votes
0
answers
52
views
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
0
answers
2k
views
How to interpret Hill estimate of tail index
I'm seeking a non-technical explanation of how to interpret the Hill estimate of the tail index for fat-tailed data, and, if possible, some explanation of seemingly contradictory results that ...
- 21
2
votes
0
answers
150
views
A non statistical/mathematical analogy to max vs argmax
I recently had a discussion on the topic 'usefulness/awareness of the function argmax() in non descriptive analysis'. That means areas, where you do not want to ...
- 1,658
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 ...
- 3
2
votes
1
answer
2k
views
Proving the convergence of the maximum of Uniform Distribution
I have a random sample of size $X_1, X_2, .., X_n$ following $U(0,2)$. I need to prove that $X_{(n)}$ which is the maximum ordered statistics will converge to $2$ in probability and almost surely.
I ...
- 1,048
1
vote
3
answers
557
views
How to robustly present a min and a max value?
I have a set of measurements from an air polution sensor. I want to determine the min and the max value in a period of time (let's say in a day).
The min and the max don't have to be the true ...
- 11
0
votes
0
answers
162
views
R: Getting Wrong Profile Likelihood Confidence Interval Estimates
I am trying to estimate the profile likelihood confidence interval (CI) of the parameters ($\xi$, $\sigma$) of the Generalized Pareto Distribution (GPD). However, the lower estimate (left CI) of $\xi$ ...
- 750
2
votes
2
answers
235
views
Fourth class of extreme-value distributions?
The generalized extreme-value distribution encompasses three classes of distributions:
Frechet, which are regularly varying, infinite right limit.
Gumbel, which are not regularly varying, infinite ...
- 533
1
vote
1
answer
369
views
Distributions in the Weibull max-domain of attraction
Can I please have a few examples of distributions that, when block-max sampled for extreme values, are in the max-domain of attraction of the Weibull distribution? I know the Beta distrution is, but ...
- 533
1
vote
0
answers
229
views
Gutenberg-Richter Recurrence Law: why is rate defined as probability of being exceeded?
According to Kramer (1996): Guttenberg and Richter gathered data for Southern California earthquakes over a period of many years and organized data according to the number of earthquakes that exceeded ...
- 463
3
votes
1
answer
43
views
Invariance of average output given output maximization
Assume that here are two areas $a = {1,2}$ and that $(e_{i1},e_{i2})$ is IID Gumbel location 0 scale 1 for all $i$. Assume further that
$$w_{i1} = \mu_1 + e_{i1} \\
w_{i2} = \mu_2 + e_{i2}$$
and that ...
- 5,565
1
vote
1
answer
238
views
What are arguments against using the (log-)likelihood as a loss function?
Context: My goal is to fit a GEV distribution function to data $z$, where the location parameter is parametrised as linear combination of predictor variables $\mu(\vec{x}) = \mu_0 + \mu_1 x_1 + ...$ (...
- 85
1
vote
0
answers
19
views
Estimating the number of bottles in a set [duplicate]
Say a distillery released a limited edition set of whiskey bottles but do not say how many have been released... If the bottles are individually uniquely numbered (bottle No.55, for example), is it ...
- 11
2
votes
0
answers
49
views
MLE for the number of samples given $k$ largest values
I have the views on the top 100 videos using a tag in TikTok and want to estimate the total number of videos in that tag. I know the distribution for other tags so I can make a guess as to what it is ...
- 2,608
2
votes
1
answer
546
views
Hypothesis Testing and the Generalised Extreme Value distribution
Is it correct to say that the generality of the Fisher-Tippett theorem means block-maximum data will always fit a GEV distribution? And how can we reject hypotheses on GEV parameters?
Original ...
- 533
1
vote
0
answers
77
views
Is it meaningful to regularise a GEV log-likelihood?
Situation/Data: I'd like to start with an example from climate science. Suppose you have a univariate time series $\vec{z} = (z_1, z_2, ..., z_n)^T$, where $z_t$ are block maxima of time step $t\in1,.....
- 85
1
vote
0
answers
21
views
What do you recommend to see if my data fits Minimum Extreme Value distribution?
I have the following data:
...
- 111
1
vote
1
answer
154
views
Hypothesis testing for large N small k
I've got a set of differentially expressed biomarkers that I want to check for the significance of this observation.
For a similar problem, I've seen the hypergeometric test being used, where
$k$ = ...
25
votes
2
answers
2k
views
Which distribution has its maximum uniformly distributed?
Let's consider $Y_n$ the max of $n$ iid samples $X_i$ of the same distribution:
$Y_n = max(X_1, X_2, ..., X_n)$
Do we know some common distributions for $X$ such that $Y$ is uniformly distributed $U(a,...
- 403
1
vote
0
answers
477
views
Pathological curvature
How common is the issue of pathological curvature? I have been reading a post on the internet about it, and I would like to know how common this happens with deep learning models.
Could you also ...
- 11
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 ...
- 20.8k
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
2
votes
1
answer
250
views
How to estimate the maximum value from a set of data with errors?
Say I have a set of n measurements. The measurement process has a known error. I can't assume that the true values being measured follow a normal distribution. How can I estimate the actual maximum or ...
- 23