All Questions
Tagged with extreme-value extreme-value or
606 questions
1
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48
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How to calculate Gumbel with LMoments and GEV with method of moments
I need to calculate the values for certain return periods of a flood event (up to 5000). It has to be GEV with method of moments and Gumbel with L-Moments. But I am not sure about how to calculate ...
- 11
1
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0
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64
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Definition p-value and find p-value in practice
I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $...
0
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0
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34
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Find maximum of bimodal posterior pdf
can you help find the maximum (analytically) of the following posterior pdf?
$p(\theta|x) = \frac{\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta-x)^2} + \frac{1-\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta+...
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2
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2
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221
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Calculating probability related to maximum of random variables
Let $X_1, X_2, \cdots, X_n$ be non-negative continuous iid random variables. The goal is to find the probability:
\begin{align*}
\Pr(\max_{k+1 \leq i \leq j } X_i < \max_{1 \leq i \leq k }X_i)
\end{...
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0
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75
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What can be concluded when standard deviation plus mean exceeds largest value?
The sum of the mean and standard deviation of a non-normal distribution can exceed the value of the largest sample. For a good explanation of why, see Can mean plus one standard deviation exceed ...
- 101
4
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2
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170
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Goodness-of-fit for Lomax distribution
I have some data n > 3000 https://drive.google.com/file/d/1gwB_U_TOX-IQHZJJDX-WeErLzrZZFoXu/view?usp=sharing (Third column) that I believe based on my physical theory should follow a Lomax ...
- 65
1
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1
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176
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How to choose the wanted root of the maximum likelihood function when there are multiple roots?
I need to estimate a parameter of a distribution but I don't have an explicit estimator. I decided to do a partition of the interval range for the parameter and use the newton-raphson method to find ...
2
votes
1
answer
178
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CDF of max of $n$ cauchy variates
Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf
$$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$
Define:
$$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
- 447
2
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0
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84
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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
1
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0
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49
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Is the variance of the maximum of a set of variables higher than the variance of the other variables?
Does the maximum of a set of random variables have high variance compared to the other variables in the set? If so, can someone give an intuitive explanation of why?
Some details about the motivation:
...
- 41
1
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1
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321
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Robustness of Quantile Regression
Is the 99th Quantile Regression model a robust model?
From my understanding, Quantile Regression is supposed to be robust in nature, but removing some outliers using IQR, the results obtained by 99th ...
- 41
2
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0
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133
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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
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1
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76
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How can the author get the following conclusion from the QQ plot?
In this paper: https://www.tandfonline.com/doi/pdf/10.1080/02664763.2021.1940109, the authors have two actual datasets (e.g., 59 observations showing continuous annual flood data) and the authors want ...
- 747
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0
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170
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Help analysing Mean Residual Life Plot for GPD
I'm trying to fit a GPD for a set of time dependant data. I have two columns, data which is a value on the negative real line where values closest to zero are considered extremes, and time. Using only ...
0
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1
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198
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Fitting Gumbel distribution based the maximal observation
Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution.
Question: Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (...
- 747
8
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4
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1k
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Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193
1
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1
answer
218
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How do I use MLE for non-iid actual data?
In this paper, the author try to fit the Gumbel distribution based on the r largest value of each year using the maximal likelihood estimators: the likelihood function for r largest values $X_{n1},\...
- 747
4
votes
1
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540
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Can we fit extreme value distribution by build-in package?
I try to find a package in R to fit Gumbel distribution by Block Maxima Approach using maximal likelihood function (see here)
$$
G(x; \mu , \sigma)=\exp[-e^{-\frac{x-\mu}{\sigma}}].
$$
The block ...
- 747
2
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1
answer
398
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Why does Gumbel distribution have two different expressions?
Let $X_1,X_2,\dots,X_n$ be iid random variables with distribution function $F(x)$ and $M_n:=\max\{X_1,\dots,X_n\}$. By the extreme value theorem, there exist two sequences of real numbers $a_n>0$ ...
- 747
2
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1
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248
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Extreme value theory for detrended series
I'm reading "An Introduction to Statistical Modeling of Extreme Values" by Stuart Coles, and using the pyextremes package for exploring the data which is time to return (in days). After ...
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1
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1
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266
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Why my fitted genextreme distribution have no mean/variance?
I have the following code for estimating a generalized extreme value distribution from scipy.
...
- 7,351
1
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3
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310
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Most probable value vs maximum of the distribution [closed]
Given a distribution p(x), there are two things that can be calculated.
Value of x for which p(x) is maximum.
Most probable value of x weighted over p(x).
Would these two values of x be the same?
- 113
5
votes
1
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83
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Which distribution is it?
I recently came across the following distribution
$$
\Pr(X\le x)=e^{\tfrac{1}{a}-\tfrac{1}{x}}\left(\dfrac{a}{x}\right)^{\tfrac{1}{a}},\; 0\le x< a,
$$
and the cdf is 0 for all $x\lt 0$ and 1 for ...
- 313
2
votes
1
answer
50
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Probability of sample minimum below a certain value
I have a list of 1000 songs with their bpm (beats per minute). If I were to sample 30 songs, is there a way to find the probability that the sample minimum is below a certain value like 100 bpm?
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1
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1
answer
53
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Violation of IID in Peaks over Threshold
I'm using the peaks over threshold method to answer a researchquestion. I'm working with time-series data and the observations are not entirely independent. I know that there is some methods you could ...
- 31
1
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1
answer
67
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How do I start and go about analyzing the variability of extreme rainfalls in a region?
I have a gridded dataset of monthly precipitation and would like to analyze the variability of extreme rainfall. However, aside from looking at the overview of basic statistics (mean, standard ...
- 23
2
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1
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120
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Question Concerning the Invariance of a Log-Transformed Normal Random Variable under Reciprocal Transformations
So I just started looking through through E.J. Gumbel's "Statistics of Extremes" (1958) and I came across a rather strange problem that I had never seen before. The problem is phrased as ...
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4
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2
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335
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Monte-carlo simulation and extrapolation
I am reviewing some work and the proposed solution seems to me not to be reliable. But I fail to find any references or even consistently formulate why I think this approach does not work.
Assume you ...
- 314
1
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0
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82
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Latent variables for spatio-temporal Extreme Value in R [closed]
Latent variables models are often used for spatial extremes modeling
see e.g., Davison, Padoan and
Ribatet. A typical application
use block maxima such as annual maxima of temperature, assumed to ...
- 5,716
0
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0
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25
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Find extreme values in relative frequencies
I have the relative frequencies of elements in roughly 450 samples (with varying sample sizes). These elements are organisms in fecal samples.
I am interested in finding extreme values of these ...
2
votes
1
answer
259
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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_{...
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4
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2
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1k
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What is an efficient algorithm for finding the minimum of a parabola-shaped function? [closed]
I have a continuous function f(x) that is bounded on the interval (0, N), where N is a large positive integer (~10,000,000). The function is shaped like an upwards-facing parabola, however, it is ...
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0
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0
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212
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Weighing Maximum Likelihood Estimations
I'm trying to arrive at a time series of optimized parameter values $Z_t$ that maximizes the likelihood of occurrence of a specific time series $Y_t$. There is a subsample within the sample that ...
2
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0
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177
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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
1
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0
answers
282
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Confidence interval for population maximum
I am sampling an arcsine distribution, with probability density function
$F(x) = \frac{1}{\pi\sqrt{(x - a)(b-x)}}$
which is defined between $a<x<b$. I want to estimate $a$ and $b$, that is, the ...
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2
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0
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76
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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
0
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0
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142
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Connection between forms for Generalized Pareto Distribution
On Wikipedia (https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I–IV) one can find the relation between the different types of Pareto Distribution and the Generalized Pareto Distribution (...
- 363
1
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0
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256
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Multinomial Logistic Regression as a latent variable model
I was reading the wiki entry for multinomial logistic regression https://en.wikipedia.org/wiki/Multinomial_logistic_regression#As_a_latent-variable_model
and it states that we can view the multinomial ...
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2
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1
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97
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How to build CDF when there are extreme values
I want to build a CDF for some phenomenon, say $P($storm duration $ D \leq d)$. The particularity of that phenomenon is that it has extreme values.
I understand that I can fit some PDF (I have data ...
- 21
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0
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319
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Choose best binning for binned maximum likelihood fit?
I am trying to find the strength of signal over a background using a continuous variable, whose distributions are known for the expected signal, the expected background, and the observed data, along ...
- 11
7
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2
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1k
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Distribution that doesn't belong to any maximum domain of attraction?
Question
Does there exist a (non-degenerate) distribution that does NOT belong to any maximum domain of attraction?
That is:
Does there exist any non-degenerate probability distribution function $F$ ...
- 223
0
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0
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128
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Probability bound of the difference of order statistics for correlated and identical Gaussian random variables
Suppose, there are $n$ identical and correlated Gaussian random variables namely, $X_1, X_2, ..., X_n$ with $X_i\sim\mathcal{N}(0,\sigma^2)$ for all $i\in\{1,2, ...n\}$. The correlation coefficient ...
1
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0
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51
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Distribution of the difference between the maximum of $n$ identical and correlated Gaussian random variables and any one of them
Suppose, there are $n$ identical and correlated Gaussian random variables namely, $X_1, X_2, ..., X_n$ with $X_i\sim\mathcal{N}(0,\sigma^2)$ for all $i\in\{1,2, ...n\}$. The correlation coefficient ...
1
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0
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71
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Expected value of maximum of $n$ iid exponential random variables [duplicate]
I was recently playing around with this distribution. Let $Y_n \sim \max_i X_i$ where $X_i \sim \exp(\lambda)$. Then the well-known result
$$ f_{Y_n}(y) = \lambda n e^{-\lambda y}(1-e^{-\lambda y})^{n-...
- 141
4
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1
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2k
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Live peak / trough detection (data provided)
At the bottom of this question is the data of three time series in CSV-format. All are of same length and they all contain measurements of the same event "A". But each time series is using a ...
- 91
5
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1
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171
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$N \sim \text{Po}(\lambda)$ and $X_1,X_2,....,X_N$ are iid and independent of $N$, what is distribution of $Z_N = \max \{X_i\}_{i=1}^{N}$
I think the title covers most of my concerns. The distribution of the $X_i$ does not really matter, I am just experiencing difficulties in finding an expression for
$$\text{Pr}(Z_N \leq x) = F(x)^N$$
...
0
votes
1
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156
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extreme event time series R
I'm new into time series and was wondering if there is some implementation in R for decomposing a time series into 'trend', 'extreme value', 'cyclical' and' error'.
I'm dealing with yearly weather ...
- 1
0
votes
0
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39
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Does independence implies independence conditionally on max of the data?
Let be $X_1, ..., X_n$ I.I.D. numerical random variables with contiunous density $f$.
Note $M(X) = \max(X_1, ..., X_n)$ their maximum.
Are $X_1, ..., X_n$ independent conditionally on $M(X) = x$ for ...
- 2,629
1
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1
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70
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Numerical superiority necessary to beat in $L^\infty$ a population one standard deviation ahead
Suppose $m$ independent random variables $X_i$ have the distribution $\mathcal{N}(0, 1)$, and $n$ independent random variables $Y_j$ (also independent of the $X_i$) have the distribution $\mathcal{N}(...
- 11
0
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2
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237
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Sum of squares for a Dirichlet distribution
I have some data that takes the form of vectors $(a_0,...,a_n)$ lying on the simplex $\Sigma a_i = 1$ (all $a_i$'s non-negative). I have noticed that the maximum $\max_i a_i$ is very highly correlated ...
- 3