All Questions
774 questions
3
votes
0
answers
47
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
29
views
X-Pareto Distribution
I have an assignment in which I have to show that Weibull Pareto belongs to exponential family and then mean and variance of term $a(x)$.
$g(x)=\frac{\beta c}{x}\{\beta \log (\frac{x}{\theta})\}^{c-1}\...
- 21
0
votes
0
answers
36
views
On the finiteness of moments of a distribution
Consider a continuous random variable $X\equiv\log(Y)$. Assume that
$$
E(\exp(\alpha X))< \infty \quad \text{ for some $\alpha>0$}
$$
I would like to understand what does this assumption imply ...
- 935
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
87
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
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
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
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
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
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
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
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
7
votes
1
answer
412
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
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
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
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
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 ...
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
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
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
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
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
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
4
votes
2
answers
171
views
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
4
votes
1
answer
87
views
Bayes estimator of possion distribution with Pareto prior
Consider a random sample of size $n$ following the possion distribution with parameter $\ln \theta$, that is
$$
f(x|\theta)=\frac{(\ln\theta)^x}{\theta x!}, x=0,1,2,\cdots
$$
and the prior of the ...
- 43
1
vote
1
answer
26
views
Two-sample test of difference in probability mass at the extremes of the empirical distributions
I am running an experiment that will generate a dependent variable (DV) in two treatments, T1 and T2.
One of the hypotheses I want to test is whether the distribution of the DV in T1 has more mass at ...
1
vote
0
answers
39
views
How should I best to use reported stats on the Tippy-top?
Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
- 3,247
8
votes
4
answers
1k
views
Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193
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
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
69
votes
9
answers
8k
views
Taleb and the Black Swan
Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual ...
- 43.2k
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
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
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
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
3
votes
1
answer
253
views
Expected value of a Pareto distribution between two values
I try to calculate the expected value of a Pareto distribution.
Suppose that we have a Pareto distribution for $x \ge \theta$:
$$f(x;k;\theta)= \frac{k\theta^k}{x^{k+1}} $$
We can calculate $ E[X]$ ...
- 250
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
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
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
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
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 ...
1
vote
3
answers
310
views
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
21
votes
5
answers
2k
views
Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?
If it is impossible, what is the proof?
- 542
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 ...
1
vote
0
answers
24
views
Applying Tangent Lines to Log-Scaled Data for Outlier Detection: Seeking Statistical Theories and Models
I've analyzed the view counts for a YouTube channel's videos (just for example), sorting them by views (on the left) and drawing a tangent line to approximate the central trend on a logarithmic scale (...
- 121
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
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
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
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