All Questions
774 questions
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
$$ \mathbb{P}\left(\frac{\max\{\mathsf{X}_1,\mathsf{X}...
- 167
1
vote
0
answers
282
views
Confidence interval for population maximum
I am sampling an arcsine distribution, with probability density function
$F(x) = \frac{1}{\pi\sqrt{(x - a)(b-x)}}$
which is defined between $a<x<b$. I want to estimate $a$ and $b$, that is, the ...
- 11
2
votes
0
answers
76
views
Tail-equivalence implying same domain of attraction
Suppose two distributions F and G that have the same extreme point ($x^F = x^G$) and
$$\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
0
votes
0
answers
142
views
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
vote
0
answers
256
views
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 ...
- 11
1
vote
0
answers
57
views
Existence of Moments for Linear Regression With Pareto Error
Suppose I have the following model linear regression model:
$y = \beta_0 + x_1i\beta_1 + x_2i\beta_2 + e_i$ with $e_i \sim Pareto(k,\alpha)$
Now if $1< \alpha < 2$, I would suppose that the ...
- 65
2
votes
1
answer
97
views
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
0
votes
0
answers
319
views
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
votes
2
answers
1k
views
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
votes
0
answers
128
views
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 ...
2
votes
1
answer
213
views
Knowing the sum, the n(), and the bound parameters of a truncated-Pareto distributed variable, how I identify the alpha (shape) parameter?
I know that there would be a fancy command on R to do the estimation of $\alpha$ given the inputs, but I am also curious about the relationship between $\alpha$ to $...
- 182
1
vote
0
answers
51
views
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
vote
0
answers
71
views
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
votes
1
answer
2k
views
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
votes
1
answer
171
views
$N \sim \text{Po}(\lambda)$ 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
$$\text{Pr}(Z_N \leq x) = F(x)^N$$
...
1
vote
0
answers
72
views
Joint distribution of top order statistics of two independent random samples of Pareto distribution
Suppose $X_1,...,X_n$ and $Y_1,...,Y_n$ are all independent copies of a standard Pareto random variable. For each of the 'two' random samples we can denote the order statistics $X_{n:n} \geq X_{n-1:n} ...
- 829
0
votes
1
answer
156
views
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
5
votes
1
answer
446
views
Deriving the limiting distribution of a sum of Pareto distributed variables
For a series of independent and identical Pareto distributed variables $X_i$ with $\alpha > 2$, their sum $S_n = \sum_{i=1}^{n} X_i$ has a normal distribution as limiting distribution for $n\to \...
- 86.5k
0
votes
0
answers
39
views
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
vote
1
answer
70
views
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
votes
2
answers
237
views
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
1
vote
0
answers
83
views
Literature on Noninformative Priors for GPD
I am starting to do some work using the Generalized Pareto Distribution (GPD), and was hoping someone might be able to point me in the direction of literature (or just general recommendations) on ...
- 99
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:
$$\forall x > m, f(x) = 0 \text{ ...
- 2,629
10
votes
1
answer
642
views
Distribution of argmax of beta-distributed random variables
Let $x_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. Let $j = \operatorname*{argmax}_{i \in I} x_i$ (ties broken arbitrarily). What is the distribution of $j$ in terms of $\alpha$ and $\beta$? ...
- 1,033
0
votes
0
answers
76
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Normalization of $M_{n} = \max(U_{1}, ... , U_{n})$
Let $M_{n} = \max(U_{1}... , U_{n})$ be the maximum of a sample size $n$ from $U(0 , 1)$ distribution.
In my statistics textbook it says that $M_{n}$ normalized is equal to $n(1 - M_{n})$ but I'm not ...
- 107
9
votes
1
answer
443
views
Intuition about the coupon collector problem approaching a Gumbel distribution
The coupon collector's problem
Let there be $n$ different types of coupons and we try to collect all of the types.
We do this by independent random draws of coupons in which each type of coupon has an ...
- 86.5k
5
votes
2
answers
1k
views
CDF of maximum of $n$ correlated normal random variables
The maximum of $n$ normal i.i.d. random variables
$$Y=\max\{x_1,...,x_n\},$$
$$x_i \sim N[0,1]$$
has the CDF
$$P(Y\le y)=\Phi(y)^n $$
but how does the CDF look like, if the variables are identically ...
- 1,250
1
vote
1
answer
2k
views
Return level plots for GEV-distribution
I was reading An Introduction to Statistical Modeling of Extreme Values by Stuart Coles, and I ran into a problem whilst trying to replicate a basic return level graph in R. For context, I first ...
- 79
3
votes
1
answer
544
views
The random variable $log(\frac{X}{x_0})$ has an exponential distribution with parameter $\alpha$
It is said that a random variable $X$ has a Pareto distribution with parameters $x_0$ and $\alpha$ for $(x_0 > 0)$ and $(\alpha > 0)$ if $X$ has a continuous distribution for which the p.d.f. $f(...
- 351
2
votes
1
answer
943
views
Method of moments estimate of Pareto Distribution
The Pareto distribution has the following $cumulative \ distribution \ function$ :
$$F(x;\alpha ,\Theta ) = \left\{\begin{matrix}
1 - (\frac{\alpha}{x})^{\theta}\ \ if \ \alpha \leq x\ & \\ 0 \ ...
- 423
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
2
votes
0
answers
208
views
Beta distribution with a priors as Uniform and Pareto Distribution
I am working on a bayesian programming problem which involves a Beta Posterior, which has mean (location) parameter coming from Uniform Distribution [U(0,1)] and concentration (kappa) coming from ...
- 85
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
727
views
Fitting distributions to censored and uncensored data in R
I need to fit lognormal, Pareto, and generalized Pareto distributions to some empirical data that is a combination of censored and uncensored data. I tried using the function ...
- 11
0
votes
0
answers
48
views
Calculating representative sample of pareto distribution
I have a Pareto-distributed population of size N.
If I wish to be 99% confident, with 0.75% margin of error, and empirically 35% made a good sample - what will be the formula to derive the sufficient ...
- 101
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:
$$p(\xi \geq x) = e^{-(...
- 378
3
votes
1
answer
299
views
Method of collecting and comparing outliers from sets of sets of populations
Background
I am a PhD student co-supervising a Master's student in our lab. I am mostly familiar with discrete mathematics, signal processing, and programming simulations. My statistics background ...
4
votes
1
answer
353
views
How to extract the shape parameter of a Fréchet fitted model using the R SPREDA package?
I'm trying to follow this post, which fits a Frechet distribution to some wind measurements as follows:
...
- 26.9k
3
votes
1
answer
914
views
Fat tails equal higher probability of non-extreme values according to Nassim Taleb?
I just came across the following passage written by Nassim Taleb Link:
The fattest tail distribution has just one very large extreme deviation, rather than many departures form the norm. [...] if we ...
- 1,129
4
votes
1
answer
421
views
Why is a Fréchet distribution slowly varying, and what is the intuition behind it?
The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0 & & x\leq 0,\\[6pt]e^{-x^{-\alpha}} & & x>0,\end{cases}$$
is regularly varying as stated here (page 19):
It is not ...
- 26.9k
4
votes
1
answer
486
views
Student's t as a power law distribution
I'm currently reading about power laws and I have came across an answer stating:
The density function of a Student's t-distribution with $n$ degrees of freedom is:
$$f(x) \sim (1 + x^2 / n)^{-(n+1)/2}...
- 750
4
votes
1
answer
319
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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 ...
