All Questions
774 questions
1
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323
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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
2
votes
2
answers
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{...
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
votes
1
answer
713
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Verifying the statistics are complete and sufficient for two parameter Pareto distribution
Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution
with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$
$\textbf{(i)}$ Show that ...
- 203
3
votes
1
answer
73
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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
4
votes
2
answers
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 ...
- 43
73
votes
4
answers
191k
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How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?
Given the random variable
$$Y = \max(X_1, X_2, \ldots, X_n)$$
where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
- 871
1
vote
1
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83
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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
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 ...
83
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3
answers
105k
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How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
- 941
3
votes
2
answers
2k
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Fitting Pareto distribution to data example in SciPy
In docs.scipy.org there's code to sample data from a Pareto distribution and then fit a curve on top of the sampled data. I could understand most of the code snippet except the term ...
- 133
2
votes
1
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213
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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
2
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0
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72
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$1-F$ 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
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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 ...
2
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1
answer
400
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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
1
vote
2
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137
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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 ...
2
votes
1
answer
52
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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
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1
answer
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
3
votes
1
answer
299
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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 ...
2
votes
1
answer
64
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Are there conditions for which the Pareto distribution arises? Are there characterization theorems of the Pareto distribution?
There are many real-world phenomena in which a variable of a population follows the Pareto distribution. I am wondering, what are the sufficient conditions for the distribution to be Pareto? And if it ...
- 763
2
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1
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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_{...
- 21
5
votes
1
answer
5k
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Asymtotic distribution of the MLE of a Uniform
A property of the Maximum Likelihood Estimator is, that it
asymptotically follows a normal distribution if the solution is unique.
In case of a continuous Uniform distribution, the Maximum Likelihood ...
2
votes
1
answer
113
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An approximate confidence interval for the $\alpha$ parameter of a Pareto Type II distribution when $\lambda$ is known
The Pareto Type II distribution, also known as the Lomax distribution, has the following density,
$$f(x|\alpha,\lambda)=\frac{\alpha\lambda^{\alpha}}{(\lambda+x)^{\alpha+1}}, \qquad x>0,\ \alpha>...
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25
votes
2
answers
14k
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Is it possible to understand pareto/nbd model conceptually?
I am learning to use BTYD package that uses Pareto/NBD model to predict when will be a customer is expected to be back. However, all literature on this model is full of mathematics and there does not ...
- 567
1
vote
2
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2k
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Determining shape parameter for Generalized Pareto Distribution Scipy
I have a set of values to which I want to fit a Generalized Pareto Distribution. Scipy provides functions for doing so:
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats....
0
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0
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57
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Empirically estimating extremal coefficient using minima of Fréchet margins
I recently came across a paper which uses the following formula to empirically estimate the extremal correlation coefficient $\chi_{ij}$ between two variables $x$ and $y$ as follows:
$$ \chi_{xy} = \...
4
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0
answers
1k
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Pareto distribution with Gamma prior on parameter $\theta$
I want to calculate the posterior distribution of Pareto distribution with known parameter $X_m$ and unknown parameter $\theta$, with conjugate prior on $\theta$ the Gamma distribution:
My effort is ...
0
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0
answers
45
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How to esimate the mean and variance of data from a Pareto distribution
I have large sample of data that is approximately from a Pareto distribution with unknown parameters. Unfortunately the distribution is sufficiently heavy tailed that just taking the sample mean is ...
- 2,077
0
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0
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25
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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
4
votes
2
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337
views
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
15
votes
3
answers
7k
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Central limit theorem and the Pareto distribution
Can somebody please provide a simple (lay person) explanation of the relationship between Pareto distributions and the Central Limit Theorem (e.g. does it apply? Why/ why not?)?
I am trying to ...
- 181
4
votes
1
answer
540
views
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
3
votes
1
answer
1k
views
What is the distribution of max-min for a Gaussian distribution
For a process N(t), where at any instance of t=T0, the distribution of N(T0) is Gaussain with mu=0:
What is the distribution of max(N(t))-min(N(t))?
From my simulation, it has some non-zero positive ...
- 131
3
votes
4
answers
573
views
Mean of minima of $n$ random variables
I keep bouncing into the following result. Let $X$ be a random variable with a cumulative distribution function $P(X<x)$. We draw $n$ independent values from this distribution, and the minimum of ...
- 217
24
votes
2
answers
11k
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Distribution of the maximum of two correlated normal variables
Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly
normal with correlation coefficient $r$.
What is the distribution function of $\max(X_1, X_2)$?
- 2,295
5
votes
2
answers
1k
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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
10
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1
answer
643
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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
26
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6
answers
53k
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Can mean plus one standard deviation exceed maximum value?
I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.
My professor asks me how can mean plus one standard deviation exceed the maximum.
I showed her ...
- 261
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
$$\text{Pr}(Z_N \leq x) = F(x)^N$$
...
5
votes
2
answers
272
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Unbiased Estimator of Largest Mean of Two Normal Distributions
Given samples from two normal distributions:
$X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X^2)$ for $i = 1,...,n$
$Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y^2)$ for $i = 1,...,n$
How ...
1
vote
0
answers
64
views
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 $...
2
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1
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81
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A problem with the expectation of a Pareto
My course notes (3rd-year module in Bayesian Statistics, unpublished) contain the following section.
Assume we have data on the number of people queuing at an ATM at a specific hour for several
...
- 599
4
votes
1
answer
1k
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m out of n bootstrap implementation in R
I am wishing to estimate the sampling distribution of an extreme order statistic (the sample maximum). The usual nonparametric (n-out-of-n) bootstrap fails miserably in this case.
Chernick (2011) ...
- 1,649
5
votes
1
answer
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
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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
0
votes
1
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76
views
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
1
vote
1
answer
60
views
Does statistically simple algos qualify as AI algos?
We have a customer purchase transaction history data with variables like below
recency - how recently they bought?
frequency - How often they bought?
monetary - How much value did they bring to the ...
- 3,342
1
vote
1
answer
32
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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
0
votes
0
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42
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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
9
votes
1
answer
443
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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 ...
- 86.6k