All Questions
Tagged with minimum or extreme-value
196 questions with no upvoted or accepted answers
7
votes
1
answer
125
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Measurement error in maximum counts
I'm familiar with the concept of a mean value of data and the variation around the mean. Is it possible to quantify variation around maximum values?
For example, take the below data collected across ...
- 14.6k
6
votes
0
answers
681
views
Gradient-informed global optimization
I am looking for a review or comparison of global optimization techniques where the gradient of the function is available and utilized to speed up search, like the following:
A hybrid descent method ...
- 1,033
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
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
4
votes
0
answers
223
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How to prove that the prior for which Bayes rule is also the minimax rule, is the least favorable prior?
I have read in the book Mathematical Statistics: A Decision Theoretic Approach by Thomas Ferguson that The prior for which the Bayes rule is also minimax rule, then that prior is Least favorable prior....
- 51
4
votes
0
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66
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Brownian bridge to unknown via extremum
Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period?
I guess it's not ...
- 62.3k
4
votes
0
answers
1k
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Maximum Prediction in Gaussian Process
A Gaussian process (GP) is defined as a collection of random variables with a joint Gaussian distribution (Rasmussen 2006). It is well known that given observations $\left \{ \mathbf{x},\mathbf{y}\...
- 2,214
4
votes
0
answers
94
views
Estimating the mean from knowing the first n largest values
There is a sample of n values that are the first n largest values of a population.
Is there a way of getting any statistic such as mean or dispersion from such piece of information provided that the ...
- 1,087
4
votes
0
answers
79
views
Non-Analytic extrapolation
I have some samples of a stable real-world process. Its is polymodal, and does not cleanly fit any of the "textbook" analytic distributions. I need to make very accurate estimates of the maximum ...
- 9,853
4
votes
0
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114
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fitting the tail of a distribution in a regression tree
I have 3 integer valued time series $a_t$, $b_t$ and $y_t$ with $k$ observations. I want to fit $y_t$ with the 2 first, and for that purpose I use a regression tree like this:
test all combinations ...
- 141
3
votes
0
answers
46
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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
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 ...
3
votes
0
answers
555
views
Convergence rate of the maximum of Weibull random variables to a Gumbel distribution
Given a sequence of iid samples $X_1, \dots, X_n,$ where each $X_i$ comes from a Weibull distribution with shape parameter $k$ and scale parameter $\lambda$. Then it is a well-known result that the ...
- 93
3
votes
1
answer
61
views
Problem with two correlated random normals
Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
- 131
3
votes
0
answers
153
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Computing the CDF of the minimum of particular dependent random variables
For each $i=1,\dots,n$ let $Z_i\sim\text{Poisson}(\lambda_i)$, and suppose $\{Z_i\}$ are independent. Also for each $i=1,\dots,n$, let $\{Y_{ij}\}_{j\in\mathbb{N}}$ be an infinite sequence of iid ...
- 133
3
votes
1
answer
92
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Minimum of Poissons
Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,...
3
votes
0
answers
418
views
Expected value of maxima of dependent random variables
I don't know if such theorem exists, but what I am looking for is a closed-form solution for $$E[\max(X_1, ..., X_N)]$$
where $X_1, ..., X_N$ is a sequence of dependent identically distributed ...
- 492
3
votes
0
answers
685
views
What's the use of the expected fisher information matrix over the hessian in the Newton Raphson approach to finding the MLE?
This may be a naive question, but I'm looking at the Newton Raphson iterative approach ( i.e. using the formula $\boldsymbol{\theta }^{(j+1)} = \boldsymbol{\theta }^{(j)} + \textrm{Hess}_{-\ell}(\...
- 31
3
votes
0
answers
85
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An arithmetic mean preserves normal distributions, maximum preserves Frechet/Gumbel/Extreme Value distributions, but what about all other power means?
Let the $k$-power mean of two numbers $x$ and $y$ be defined as $M^k(x,y) = \left(\frac{x^k+y^k}{2}\right)^{1/k}$.
For the case $k=1$, we have that if $X,Y$ are independently normally distributed, ...
- 1,594
3
votes
0
answers
194
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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$?...
- 802
3
votes
0
answers
128
views
Extreme value theory: GPD larger expected value than average
We're using extreme value theory to model tail risks on our portfolio. After we choose the threshold, we fit generalized Pareto distribution to our data over the threshold. The expected value of GPD ...
- 371
3
votes
0
answers
149
views
Predictive modeling of an complex panel of heavy-tailed data
I am struggling to develop a sensible strategy or protocol for the predictive modeling of a complex set of data. Apologies in advance for the indeterminate nature of some of this description but it’s ...
- 10.9k
3
votes
0
answers
249
views
Extremal serial dependence
As part of my analysis of heavy-tailed time series of company returns, I would like to check whether extreme returns exhibit serial dependence, i.e. if extreme events are followed by extreme events.
...
- 161
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 ...
- 1,271
3
votes
0
answers
107
views
Repairable system and the sum of GEV random variables
We know that $X\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ and $Y\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ then $X+Y\sim {\mathrm {Logistic}}(2\alpha ,\beta )$.
I am wondering, what will be $X+Y+Z$ ...
- 328
3
votes
0
answers
671
views
Conceptual or mathematical motivation for the three extreme value distribution types?
What motivates, justifies, gives rise to the differences between the Gumbel, Fréchet, and Weibull distributions? Glen_b's comment indicates that they are distributions for extreme values generated by ...
- 1,108
3
votes
0
answers
1k
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Confidence intervals for extreme value distributions
I have wind data that i'm using to perform extreme value analysis (calculate return levels). I'm using R with packages 'evd', 'extRemes' and 'ismev'.
I'm fitting GEV, Gumbel and Weibull distributions,...
- 951
3
votes
0
answers
142
views
Distribution of variable
How to find the distribution of $$\sum_{i=1}^n (X_i - X_{1:n}),$$ where $X_i$ are i.i.d. random variables and $X_{1:n} = \min(X_1,X_2,...,X_n)$?
I need to find the distribution in a particular case, ...
- 31
3
votes
0
answers
352
views
Correct variance for minimum detectable difference
I have a question regarding variance, paired testing and minimum detectable difference (MDD).
Paired samples:
$$
MDD (δ) = \sqrt{ \frac{σ^2}{n} (t_{(α/2,n-1)} + t_{(1-β, n-1)})}
$$
I have a set of ...
- 31
3
votes
1
answer
1k
views
How to minimize Chi-Square using the CDF instead of the PDF?
Suppose one has data that is suspected to obey a normal distribution. One computes a histogram of the data, and performs Pearson's Chi-Squared Test. To perform this test, one must compare the observed ...
user146123
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
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
2
votes
0
answers
84
views
Calculating confidence Interval for a return time curve, via non-parametric bootstrapping
I have some precipitation data (yearly extremes), which I have fit with a Gumbel distribution (CDF), from which I have calculated a return time distribution. I want to calculate the 95% confidence ...
- 21
2
votes
0
answers
133
views
Is there any intuitive explanation for MoM in estimating parameters?
I found from some literature that when we use the method of moments to fit the Gumbel distribution, the estimated
(On page 24) A comparison of the variance formulas in (1.66) with the CramBr-Rao ...
- 747
2
votes
1
answer
248
views
Extreme value theory for detrended series
I'm reading "An Introduction to Statistical Modeling of Extreme Values" by Stuart Coles, and using the pyextremes package for exploring the data which is time to return (in days). After ...
- 21
2
votes
0
answers
177
views
Limit distribution of the joint distribution of maximum and minimum of a sequence of random variables
Assume we have a sequence $\mathsf{X}_1,\mathsf{X}_2,\mathsf{X}_3,...$ of iid random variables. Then the Fisher-Tippet-Gnedenko theorem shows that
$$ \mathbb{P}\left(\frac{\max\{\mathsf{X}_1,\mathsf{X}...
- 167
2
votes
0
answers
76
views
Tail-equivalence implying same domain of attraction
Suppose two distributions F and G that have the same extreme point ($x^F = x^G$) and
$$\lim_{x \to x^F}\frac{\bar{F}(x)}{\bar{G}(x)} = c \in (0, \infty)$$
Show that F and G belongs to the same domain ...
- 21
2
votes
0
answers
2k
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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
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
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
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
2
votes
0
answers
55
views
Extreme Value Theory - Determining the positive normalising constant in the Extremal Types Theorem
I am working through the following question and cannot seem to work out how the final result is obtained from the last inequality involving $a_n$. Can someone shed some light?
- 309
2
votes
0
answers
300
views
Find the limiting distribution of $(bn)^{-\frac{1}{\alpha}} X_{(n)}$
let $\{X_n\}_{n\geq 1}$ be a sequence of i.i.d random variables with common distribution $F$, and write
$X_{(n)}=\max\{X_1,\cdots , X_n\}$ , $n=1,2,\cdots$
(a) for $\alpha >0$ , $\lim_{x\rightarrow ...
- 1,339
2
votes
0
answers
62
views
Normal equations issue
Let $A\mathbf{x}=\mathbf{b}$ be an overdetermined system, with $A$ being an $n \times m $ full-column rank rectangular matrix.
Are these minimization problems equivalent?
$$ 1) \;\underset{\mathbf{x}...
- 437
2
votes
0
answers
1k
views
Calculating minimum detectable effect from sample size and conversion rate
I have a function for calculating the required sample size based on four inputs: baseline conversion rate, minimum detectable effect, confidence and statistical power:
...
- 131
2
votes
0
answers
135
views
Interpret the result of a fitted non-stationary Gumbel model
I have a dataset on wildfires that I fitted to a Gumbel distribution with a set of covariates (using the gevrFit function in the eva package in R). The result of ...
- 21
2
votes
0
answers
237
views
Why does this sequence of random variables converge in distribution?
Given iid random variables $X_1, \dots, X_n$ with common density: $$ f(x) = 1\{ x > 0 \} \cdot \frac{1}{(x+1)^2} $$ it is supposed to be the case that $\frac{\max_i X_i}{n}$ converges in ...
- 6,479
2
votes
0
answers
29
views
Problem computing population quantiles with survey micro data
All the major federal surveys come (American Community Survey, Current Population Survey, others) come with survey weights, such that the individual household observations times the population weights ...
- 3,247
2
votes
0
answers
119
views
Is Var(sample min) decreasing in sample size?
Suppose $z_n=\min\{x_1,\dots,x_n\}$ where $x_i$'s are i.i.d. according to CDF $F$ over $[0,1]$.
Is it true that $Var(z_n)>Var(z_{n+1})$? What conditions would I need to ensure this monotonic ...
- 329