The Pareto distribution is a power distribution which was initially invented to describe the distribution of income but has found application in many other areas.

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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 ...
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Theoretical/intuitive question about time-varying Generalized Pareto Distribution

I fitted the GPD to the right tail of nine log return series (I multiplied log returns by -1, so modeling the right tail equals modeling the losses) with a threshold equal to the 95% quantile. Some of ...
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How do you calculate the pareto shape parameter?

I have a set of data that shows a pareto distribution. I am trying to figure out the probability of an event within that data set. I need to know the pareto shape parameter in order to calculate this ...
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22 views

Is the variability index valid for the Pareto distribution

The Pareto distribution is defined from the CDF: $$ F^{-1}(p) = \frac{b}{(1 − p)^{1/a}},\ 0 < p < 1, $$ where, $b$ is the scale parameter and $a$ is the shape parameter. In the Gaussian ...
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36 views

Help calculating integral

I need help calculating a integral. It is a Pareto distribution with common tail $\frac{1}{(1+x)^a}$, where I assume countermononicity between two variables $X_1$ and $X_2$. $\int_0^1 ( (\frac{1}{1-x}...
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50 views

Existence of the conditional tail mean

Does the existence of the first moment of a generalized Pareto distribution with support $[0,\infty)$ imply the existence (finiteness) of the conditional tail mean -- i.e. what in risk management is ...
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19 views

How to select the thresold in Generalized Pareto distribution

I'm using generalized Pareto distribution to fit the tail data, I want to know is there any computational way to estimate the threshold parameter as we do in estimating the sigma et shape using MLE? ...
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10 views

How exactly does a hyperbolic distribution show scaling phenomenon like fractals?

I am reading a paper that applies the concept of fractals in modeling precipitation. It states that a random function $X(t)$ is "scaling" at all times if: $$ \Delta X (\lambda \Delta t) \overset{d}{=...
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27 views

what is Pareto distribution for all the hyper parameters are 0?

I am studying Bayesian probability and have a question about the Pareto distribution. My question pertains to problem 2(a) shown here. I've seen its form in the wiki. The Pareto distribution has ...
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54 views

Derivation of mode of Pareto distribution

How do we derive the mode of a pareto distribution? Any hints would be appreciated. Should KKT conditions be used?
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34 views

Preventing Pareto smoothed importance sampling (PSIS-LOO) from failing

I recently started using Pareto smoothed importance sampling leave-one-out cross-validation (PSIS-LOO), described in these papers: Vehtari, A., & Gelman, A. (2015). Pareto smoothed importance ...
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21 views

Generating a Churn Curve forecast

I have monthly retention data for customer cohorts that look something like this: ...
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37 views

How to estimate Pareto shape parameter with bayesian estimation?

I want to estimate the shape (alpha) parameter for a Pareto distribution. (We assume that we know the scale parameter =1 ). The prior is alpha = 2 (and maybe we have always to assume a distribution ? ...
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1answer
76 views

How to fit a Pareto distribution via Bayesian estimation (with a Pareto prior)?

I don't know Bayesian statistics very well, so I don't know if the question makes sense. Let me give an example. We assume that the income distribution of a country is a Pareto distribution (the ...
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18 views

Is there a distribution that covers Pareto's law?

Is there a distribution where (for example) 80% of the results come from 20% of the inputs? (i.e. Pareto's law). Hmm... there's a tag for Pareto Distribution ... is that the answer?
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419 views

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 ...
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20 views

Buy Till You Die(BTYD) - Individual LTV scores

I'm using the Buy Till You Die(BTYD) package in R to predict LTV (using Pareto/NBD), and I've been able to produced expected transactions by week, but is there a way to predict the dollar value of ...
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1answer
129 views

Bias of method of moments estimator for Pareto distribution with known scale parameter

Let $x$ be a Pareto distribution with a known scale parameter $m>0$, i.e. $x\sim f(x|a)=\frac{am^a}{x^{a+1}}, x>a, a>0$ $\mathrm{E}\left[X\right]=\frac{am}{a-1}$ Using method of moments ...
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55 views

How should I solve the following simultaneous equations?

I have the set of simultaneous equations below from the paper Parameter estimation for 3-parameter generalized pareto distribution by the principle of maximum entropy (POME) by VP Singh and H Guo: $$ ...
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108 views

How do I use NLOpt (C++) to solve for Generalized Pareto Distribution Parameters?

For some research I need to find the parameters of the Generalized Pareto Distribution. I plan to do this by minimizing the following function: $$ -\sum_{i=1}^{n_u} ln((1/\sigma)(1+\xi((x_i - u)/\...
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22 views

Invalid returned matrix for Density function in Pareto Survival Analysis

I want to perform a survival analysis using survreg procedure from survival library in R for a pareto distribution (I based my ...
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1answer
183 views

Have MLE estimators for Generalized Pareto Distribution. Given a known value of $c$, how do I calculate $a$ and $b$ using the provided estimators?

I am doing research into the three parameter Generalized Pareto Distribution $$ f(x|a,b,c) = \frac 1 b\left(1+a\left(\frac{x-c}{b}\right)\right)^{\big(-1-\frac 1 a\big)} $$ for finding VaR and CVaR. ...
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63 views

Posterior of alpha parameter (Shape) of Pareto Distribution

Im trying to generate the posterior distribution of $\alpha$ parameter of Pareto Distribution. I did all the job correctly on paper, but when i go to implement in R i have some problems.I have a ...
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151 views

R- get qq plot for Pareto distribution

I'd like to use R to do a qq plot on my data for a Pareto distribution. I've been able to do this for the lognormal (my code is below) but fitdistr() doesn't support the Pareto. I've tried a similar ...
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Fitting a power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
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246 views

How to fit a function to a CDF in R?

I've been given a dataframe that contains data for a CDF. The column X contains the 250 $X$ values, and the column P contains $p(...
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547 views

Cramer Von Mises - How to use this test correctly?

I had a problem when I tried to test the fitting of my data with the generalized Pareto distribution. I used the MLE to estimate the two parameters 'shape' and 'scale' and I generated a vector of ...
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110 views

Pareto 2 distribution

I am bit confused between Pareto and Pareto Type II distribution. In the actuar library (page 64), details of pareto distribution are given. However, in the end, it is mentioned that "Distribution ...
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1answer
103 views

Weighting observations and measurement uncertainty in bayes

I am working on using MCMC (via STAN) to estimate model parameters for a bunch of observations with measurement uncertainty. I'm having problems with weighting each observation, and have reduced the ...
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259 views

Hyperprior Noninformative Beta Binomial Model

I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model ...
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56 views

What is the name of this distribution family?

I am trying to identify this probability density function so I can read up on it to find confidence intervals for $\theta$: $$f(x;\theta,v)=\frac{\theta v^\theta}{x^{\theta+1}}I_{[v,\infty)]}(x);\ \...
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405 views

Pareto two-tailed GLM regression

How can I perform a Pareto two-tailed GLM regression? Any reference to link functions and code in R?
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175 views

What distribution results in adding two Pareto distributions

I'm wondering what distribution results in adding two (or more) type-one Pareto distributions of the form $x^{-\alpha}$. Experimentally, it looks like a two-mode power-law, asymptotic to the ...
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99 views

Evaluate goodness-of-fit of estimation of Pareto-like distribution

I would like to evaluate the goodness-of-fit of the following (Pareto-like) distribution: $$ f(r) = \sigma \centerdot r^{-\rho} $$ The function estimates the population of cities given the rank $r$ in ...
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278 views

In a “bursty” dataset, how do you filter for the few important values that make up the bulk of the information?

Note sure if there is an existing stats concept for this but I have a dataset that consists of mostly small data points with a few large ones. e.g. 1 2 1 3 1 2 87 3 2 1 1 1 1 3 1 2 1 1 1 99 How can ...
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1answer
249 views

Kolmogorov-Smirnov for Pareto distribution on sample

I want to use the Kolmogorov-Smirnov test to test if a sample is drawn from a Pareto distribution. Unfortunately, the only way to estimate the distribution's parameters is from the sample. Does ...
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2answers
187 views

Is there a package for three parameter inverse gaussian or lognormal distributions in C++?

I want to generate random numbers from one of the following distribution in C++. I haven't been able to find any libraries though. Do they exist? In order of preference: Three parameter inverse ...
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662 views

Moments of the two-parameter generalized Pareto distribution (GPD) needed

In this thread the first two moments of the two-parameter GPD are given, where the distribution might be defined as $G(y)= \begin{cases} 1-\left(1+ \frac{\xi y}{\beta} \right)^{-\frac{1}{\xi}} & ...
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358 views

Required: Method of moments fitting routine for the two-parameter generalized Pareto

I am currently using the evd package which fits a two-parameter GPD by maximum likelihood. Since in small samples the MOM is superior to the ML estimation I'd like ...
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Long tail of pareto vs. pearson IV distribution

I am told that wealth follows a Pareto distribution, and that IQ follows a Pearson IV distribution (http://www.abelard.org/burt/burt-ie.asp). Both Pareto and Pearson IV distributions have long tails. ...
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Does it make sense to fit a Pareto Curve to sales data?

The Pareto principle is used surprisingly widely in business. I'm wondering how correct it's wide use is. It seems as though it is often used without empirical verification as if the situation fits a ...
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84 views

Kullback-Leibler divergence Pareto Distribution

What is the Kullback-Leibler divergence for a Pareto Distribution? Given $p(x)$ = $ \alpha$ $\frac{x^{\alpha}_{min,1}}{x^{a+1}}$.
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298 views

How should I interpret these strange density and mixing plots when fitting a generalised pareto distribution using MCMC with JAGS?

I'm trying to fit a generalised pareto distribution to a simulated dataset using JAGS and runjags. When doing so, I get very strange density and mixing plots for the mu parameter. The sigma and xi ...
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159 views

Pearson 5/Inverse Gamma/ Double Pareto

My current research deals with landslide size frequency distribution which has been fit to a three-parameter inverse gamma function and a double Pareto function by previous researchers. I am trained ...
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1answer
152 views

How to derive the $\alpha$ for the Pareto rule

Suppose we have the CDF for the Pareto Distribution given by: $$ P(X \leq x) = 1-\left(\frac{x_m}{x}\right)^\alpha \;\;\;\;\;\;\;\;\;\; x \geq x_m$$ What is the intuitive way to find the alpha for ...
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Why is it that it is self-defeating to use the posterior mode as the bayes estimator in this case?

I am reading through an applied statistics book and in it, it makes a very luminous statement for a posterior case where the likelihood was taken from $X_1,...,X_n$ iid random variables from a ...
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276 views

How to get Pareto IV parameter estimates

I have a serie with 850 observations, and I need to fit the Pareto IV distribution. How could I do this in R? I read the guide VGAM, however, I'm not able to run it. If anyone knows, please provide ...
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Posterior distribution

Suppose $X1,..,X4$ be iid from pdf $f(x|\theta)=\frac{1}{\theta}$ ,for $0<x<\theta$. The prior distribution is $\pi(\theta)=\frac{2}{\theta^3}$ , for $\theta>1$ I have to obtain: a)...
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77 views

Is it appropriate to estimate a Pareto regression's α by maximising R-squared?

Fitting a power law regression. What are the downsides of estimating α by maximising the R-squared of (a) the theoretical values output by the regression with the candidate α and (b) the empirical ...
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110 views

Expectation of the Pareto distribution [closed]

I would like to know if my understanding of the following is correct. This has been tripping me up for a long time now. Compute $\lim_{x\rightarrow \infty}x^{1-\beta}$. This is part of a homework ...