All Questions
Tagged with minimum or extreme-value
606 questions
3
votes
0
answers
46
views
Is there an analytical solution to the distribution of a sum of observations drawn from a Frechet distribution?
Let $X_i$ be an iid draw from a Frechet distribution. Let $\alpha_i \in \mathbb{R}$.
Is there an analytical expression of the distribution of $\alpha_1X_1 + \alpha_2X_2 + \alpha_3X_3$? That is, can I ...
- 31
1
vote
0
answers
22
views
Convert units, get different results when fitting extreme value distribution with extRemes
I am using the fevd() and lr.test() functions to examine precipitation using the extRemes R ...
- 11
1
vote
1
answer
78
views
Expectation of the minimum of random variables (Exponential + Erlang)
Consider the following random variable
$$
Z=\min_i\{X_i+Y_i\}
$$
for $-n\leq i\leq n$, where $X_i\overset{\mathrm{iid}}{\sim}\text{Exp}(\lambda)$, $Y_i\overset{\mathrm{iid}}{\sim}\text{Erlang}(|i|,\...
- 150
3
votes
1
answer
85
views
Maximum of two independent gamma variables
Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$.
Are there known results for the distribution of $X$? Actually I only need to know $\...
- 1,191
0
votes
0
answers
25
views
Linearity of and pointwise equality in expectation of min() function
Consider the expressions $f = c + s*E[min(a/s, X)]$ and $g = E[min(c + a, c+sX)]$ where
c >= 0
0 < s <= 1
a >= 0
X ~ Poisson($\lambda$/s)
I'd like to think that $f = g$, reasoning as ...
1
vote
0
answers
37
views
How can I measure Monte Carlo convergence in distribution with heavy tails?
I'm performing a Monte Carlo study on a simple agent based simulation, and I'm trying to formulate a heuristic for the number of MC samples to use. I'm able to measure convergence of statistics like ...
0
votes
0
answers
36
views
Fitting a regression line which passes through the anchor point
In our setting, we have data $X_1, \ldots, X_n$, which can be ordered as $X_{1,n}\leq X_{2,n}\leq \ldots \leq X_{n,n}$ and we have the points $(-\log (1-\frac{i}{n+1}), X_{i,n})$ for $i=1,\ldots,n$.
...
- 656
1
vote
1
answer
58
views
Identify maximum in quadratic regression
I am looking for a way to find the maximum in a quadratic regression.
Specifically, I have two variables X and Y. Y is a discrete and commonly used scale representing the severity of a disease, ...
- 433
5
votes
2
answers
346
views
What is the median of the minimum or maximum of multiple samples?
Suppose I have a variable with a known distribution, and suppose I sample that variable k times and record the minimum. If I repeat this many times, will the median of the minimum converge to a ...
- 55
2
votes
1
answer
39
views
analytical asymptotic approximation of the expected maximum, mean, and minimum distance of nearest neighbours in unit ball
Say I uniformly at random distribute $x = n^3$ (independent identically distributed) points in a ball of radius $r=1$ in $\mathbb{R}^3$.
What can be said about the expected maximum, minimum, and mean ...
- 277
5
votes
1
answer
216
views
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
0
votes
0
answers
17
views
Declustering impact, stationarity and discretization
I have a seasonal time series, and I am considering declustering (before any other preprocessing steps) it using runs declustering. If I observe an extremal index of 1, can I claim that my data is i.i....
- 1
0
votes
0
answers
55
views
Does the mean of the maxima of a set of distributions converge?
This question is related to a recent one I posted. In that question I ask what statistic might best represent the central tendency of the true discrete distribution of a property for a sample for ...
- 165
3
votes
3
answers
125
views
What statistic best estimates the sample mean in case of missing data in a distribution?
I have samples of particles and am interested in the particle lengths. The problem is that the samples are assessed using image analysis. As the particles overlap, the measurements are incomplete and ...
- 165
1
vote
0
answers
103
views
How to deal with outliers in panel data? [closed]
When we have cross-sectional data, we can easily detect and remove outliers. But how should one approach outliers when we are dealing with panel data? Since we have $i$ entities and $t$ times periods, ...
- 345
0
votes
0
answers
51
views
How to understand intuitively the CDF formula for the maximum statistic of three iid rv’s? [duplicate]
Given that all three iid rv’s ($X_1, X_2, X_3$) have CDF $F(x)$, the formula for the CDF $G(y)$ of the largest rv ($Y=X_i$) among the three is:
$G(y)=P(X_1 \leq y) \cdot P(X_2 \leq y) \cdot P(X_3 \leq ...
1
vote
1
answer
53
views
Derivation of a dynamical Generalized Pareto distribution
I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution ...
1
vote
2
answers
149
views
Distribution of a random variable conditional on its being a maximum or not
Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let
$$
Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
- 935
1
vote
0
answers
30
views
Multinomial Logit Extension
The derivation of the multinomial logit probabilities depends on the difference of two Type 1 extreme value (Gumbel) random variables following a logistic distribution. We say the unobserved utility ...
- 31
1
vote
1
answer
87
views
How do you determine an appropriate block length for calculating "block maxima" for GEV distribution?
I have some time series data spanning 30+ years and I am trying to do some extreme value analysis. Major disclaimer: I am not a statistician so I feel that I am wading into waters beyond my area of ...
- 925
0
votes
0
answers
31
views
Validity of bootstrapping for estimation of annual maxima distribution
I am working with a large timeseries (millions data points) spread across 5 years from which I would like to estimate the annual maxima distribution and subsequently a quantile of this distribution.
...
- 1
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\...
- 473
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
1
vote
0
answers
82
views
Bootstrapping moderately extreme quantile regression
Let $(Y_1, X_1), \dots, (Y_n, X_n)$ be iid sequence drawn from $F$. For a fixed $q\in (0,1)$, consider the linear q-quantile regression $Q_Y(q|x) = \beta_qx$, where $Q_Y(\cdot\mid x)$ is the ...
- 429
1
vote
1
answer
26
views
Two-sample test of difference in probability mass at the extremes of the empirical distributions
I am running an experiment that will generate a dependent variable (DV) in two treatments, T1 and T2.
One of the hypotheses I want to test is whether the distribution of the DV in T1 has more mass at ...
0
votes
0
answers
97
views
How to do hypothesis testing for Minimum value?
I have a sample with a size of n=100, and I want to show that the minimum value of the underlying distribution is not less than a certain threshold, with a confidence level of 95%.
The distribution of ...
- 101
1
vote
2
answers
90
views
Finding the temperature value that gives optimal value
I'm trying to analyze some sleep data from kaggle (this example data does not have correct temperature data but the actual data I will use in the future will have precise temperature) to try to find ...
- 11
1
vote
0
answers
87
views
Threshold choice for Peaks-Over-Threshold
I'm trying to estimate equivalent performances at different events, using Peaks-Over-Threshold from Extreme Value Theory. The challenge is to find the threshold and preferably with same number of ...
0
votes
0
answers
35
views
Extreme Value Analysis - Nonrandom/Preferential Sampling
I am doing an extreme value analysis (EVA) but there is a nuance in my problem that I believe is not addressed in extreme value theory. I have not been able to find information about this in textbooks ...
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
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
views
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
11
votes
1
answer
236
views
Distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d $\text{Exp}(1)$
Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$.
Simulation suggests the ...
- 11.6k
1
vote
1
answer
32
views
How to find an "upper margin" for data on visits
For a handful of store locations, I have data on each entrance and exit time.
I have counted the total num of people at a store at any given minute.
I am trying to find out the values for which the ...
- 111
2
votes
0
answers
158
views
Definition of exponent measure (extreme value theory)
Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$.
In my textbook, there is the ...
- 656
5
votes
0
answers
235
views
Running maximum of $\sum_{1\leq k\leq n} X_i$ for Cauchy random variables $X_i$
Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's and let $S_n=X_1+\cdots+X_n$ be their sum. Does an expression exist for the CDF of the running maximum up to an index $1 \leq k \leq n$?
Edit:
...
- 61
2
votes
0
answers
72
views
$1-F$ is rapidly varying if and only if there exists $b_n$ such that $\frac{\max X_i}{b_n} \to 1$ in probability
The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick.
We will say $1-F$ is rapidly varying as $x \to \infty$ if $\lim_{t \to \infty} \frac{1-F(tx)}{1-F(t)} =...
- 656
0
votes
0
answers
57
views
Empirically estimating extremal coefficient using minima of Fréchet margins
I recently came across a paper which uses the following formula to empirically estimate the extremal correlation coefficient $\chi_{ij}$ between two variables $x$ and $y$ as follows:
$$ \chi_{xy} = \...
3
votes
1
answer
72
views
Does this approach to simulation for survival analysis, of breaking the analysis into deaths versus survivors, appear reasonable?
I've spent last several weeks learning about survival analysis, see one of the last posts at How to simulate variability (errors) in fitting a gamma model to survival data by using a generalized ...
- 827
1
vote
0
answers
39
views
Statistical assessment of block size for bootstrapped distribution fitting
I have a set of intensities from unordered independent events (with no date or timestamps), many of which constitute extremes, and I want to generate an extreme value distribution. The only ...
- 111
1
vote
2
answers
137
views
Gumbel distribution conditional on exceeding a threshold
In Heffernan and Tawn's 2004 paper, they describe a procedure to sample multivariate data, conditional on one variable ($Y_i$) being extreme. The idea is that $Y_i$ is extreme if it exceeds some ...
1
vote
1
answer
83
views
If $F^n(b_n x) \to e^{-x^{-\alpha}}$, $b_n x \to x_0$ where $x_0 = \sup \{x \colon F(x) < 1 \}$
Let $X_n$ be i.i.d with common df $F$. Let $M_n = \max (X_1, \ldots, X_n)$. Suppose $P(b_n^{-1} M_n \leq x) = F^n(b_n x) \to e^{-x^{-\alpha}}$ weakly, where $x > 0$ and $\alpha > 0$.
Let $x_0 = \...
- 656
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
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 ...
- 233
2
votes
1
answer
52
views
How to assign reasonable scale parameters to randomly generated intercepts for the Weibull distribution?
This is a follow-on to post Correctly simulating an extreme value distribution for survival analysis?, as I work towards adaptation of that code to the Weibull distribution. In the below code I ...
- 827
1
vote
0
answers
144
views
Resource recommendation for extreme value theory
I'm look to learn about extreme value theory, starting from univariate case and then moving onto the multivariate case.
I have tried the textbook by de Haan, but I'm constantly lost trying to read the ...
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
0
votes
0
answers
41
views
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
42
views
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
4
votes
1
answer
289
views
Is every probability distribution also the distribution of the maximum of several i.i.d. random variables?
I found the following result used in this paper, but it was just claimed without proof and it seems extremely strong to me, so I would like a proof, or at least a reference, of a proof.
Let $D$ be ...
- 243