Questions tagged [pareto-distribution]

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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82 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 \...
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Doubt about distribution function of continuous distribution with (Generalized) Pareto tails

In this paper: https://www.sciencedirect.com/science/article/pii/S0167947315003163 They proposed an estimation method for the parameters of the Generalized Pareto Distribution. Defining: $$ F_n(x) \...
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Pareto distribution with adjustable inflection point

How do you generate a Pareto distribution without a given dataset? For example I have 1,000 Widgets and 10 groups. I would like to distribute them among the groups according to the Pareto principle ...
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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 ...
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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 ...
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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(...
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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 \ ...
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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 ...
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How do I interpret parameters from a generalized Pareto distribution?

I fit a generalized Pareto distribution using the function "fitdistcens" from the package "fitdistrplus": ...
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80 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 ...
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Grouping most strongly contributing items between different samples

I am performing an analysis by using Python's collections.Counter to obtain a scale of frequency x rank. I obtained the same statistics for multiple datasets and ...
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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 ...
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Pareto chart focusing on the trivial many

Pareto chart is mostly focusing on the vital few. But I have heard that nowadays, some are focusing first on solving the trivial many, believing that it is easier to solve and may support to lessen ...
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Pareto Chart: Determining the 80% and 20% part

Which one is true? What part is really be 80% and 20%? Edit: Clarification and additional details to the problem. According to the pareto principle, it states that loosely 80% of the effects come from ...
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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 ...
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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}...
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How to compute the joint survival function of a Multivariate distribution?

Suppose I have a multivariate Pareto distribution with cdf, $$ Prob(Z_{1}<z_{1},\dots,Z_{n}<z_{n}) = H(\textbf{z}) = 1 - \left( \sum_{i=1}^{n} (T_{i}z_{i}^{-\theta})^{\frac{1}{1-\rho}} \right)^{...
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In a power-law distribution, does the maximum value of a measure increase with the number of measured items?

This question is motivated by the following claim in this article: There is increasing evidence that many object-oriented software size metrics are characterized by scale-free, power-law ...
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Minimum of Multivariate Pareto

Suppose I have a multivariate Pareto distribution with cdf, $$ Prob(Z_{1}<z_{1},\dots,Z_{n}<z_{n}) = H(\textbf{z}) = 1 - \left( \sum_{i=1}^{n} (T_{i}z_{i}^{-\theta})^{\frac{1}{1-\rho}} \right)^{...
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127 views

Residuals in Generalized Pareto Distribution

I'm learning generalized Pareto distribution for fitting extreme value data. I came across an R package evir that is able to plot residuals. Residuals from a GPD ...
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Marginal distributions given the distribution of range

I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution. The distance of a point from the diagonal line y = x is $\frac{\sqrt{(x_0-y_0)^...
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R: Getting Wrong Profile Likelihood Confidence Interval Estimates

I am trying to estimate the profile likelihood confidence interval (CI) of the parameters ($\xi$, $\sigma$) of the Generalized Pareto Distribution (GPD). However, the lower estimate (left CI) of $\xi$ ...
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Number of ten millionaires in a country [closed]

Given that I know the number of billionaires B and millionaires M in a society and that wealth is distributed in as a Pareto, how do I figure out the number of people with more than 10 million dollars?...
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Generalized Pareto Distribution (GPD) Estimation From Scratch

I'm trying to fit a GPD to a set of log-return data from scratch, however, my function's output is totally different from ismev's ...
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Extending the 80/20 rule

I have seen comments on the web that because the 80/20 rule is fractal, it applies to the sub groups. In other words, if the top 20% of causes drive 80% of outcomes, then the top 4% of causes must ...
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If I have interval (binned) data from a D.G.P. with a truncated Pareto distribution, can I estimate the truncation point?

Suppose I have high-income data that I believe to be reasonably approximated by a Pareto distribution above some income level. I have mean and total income for several income ranges, including the &...
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229 views

Chebyshev's inequality for Pareto distribution (3 sigma rule)

According to the Chebyshev's inequality, if we take any distribution, we get >88.8889% of data in +-3 sigma interval. For a normal distribution it is 99.97%. How to calculate the interval for a ...
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192 views

Neyman-Pearson Lemma for Pareto Distribution [duplicate]

I have the following problem. Let $X_1, ..., X_n$ represent a random sample taken from a population with CDF given by $$ F(x;\beta) = 1 - \frac{\beta}{x}, ~~ x \geq \beta > 0. $$ Based on the this ...
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Programming inverse-transformation sampling for Pareto distribution

I am having trouble deriving a formula, and running a simulation with its distribution. The Pareto distribution has CDF: $$F(x) = 1 - \bigg( \frac{k}{x} \bigg)^\gamma \quad \quad \quad \text{for } x \...
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Confidence Interval Pareto Dist

Let $X_1,...,X_n$ be iid random variables from Pareto distribution with the following distribution $\theta a^{\theta} x^{-(\theta+1)}$, $x>a, \theta >1, a>0$ I have to find a $100(1-a)\%$ CI ...
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How to implement a GLM with Pareto family?

My data fits a Pareto Distribution very well, better than normal exponential, gamma, lognormal, etc. I am trying to find a way to implement a GLM with a Pareto family flag. Ideally I can then run ...
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Discerning differences in groups with millions of datapoints: is a GLM even valid?

I have a MASSIVE dataset of 22 million shrubs from a basin in the southwest US. I have selected 2 response variables which are both positive continuous variables: Shrub Canopy Volume (cubic meters) &...
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Measuring Pareto like metrics for series or dataset [closed]

Pareto rule states that 20% of records accounts for 80% of total. Actually it's just a special case for a certain dataset. If we use a series: from 1 to 10 we can easily see that Pareto rule ...
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CDF that combines properties of Pareto and Exponential

Let $Y$ be a random variable defined on the domain $[1;\infty)$ that is distributed according to the cdf $G_Y(y)$. A Pareto distribution, $$ G_Y(y) = 1 - y^{-\theta}$$ has the property that $$ P(Y&...
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Method of moments and MLE estimates for Lomax (Pareto Type 2)

I have this dataset, on which I am supposed to fit Lomax distribution with MM and MLE. Lomax pdf is: $$f(x|\alpha, \lambda) = \frac{\alpha\lambda^\alpha}{\left(\lambda+x\right)^{\alpha+1}}$$ For MM, ...
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226 views

Understanding Hypothesis tests for a Pareto distribution

I'm writing an essay that's looking at the presence of the Pareto Principle in data. Unfortunately, as a consequence of interest, I've picked a topic that involves statistical analysis well above what ...
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MLE of truncated Pareto distribution

I am trying to apply truncated Pareto distribution to a dataset, but I am not able to find a close form MLE for the shape parameter alpha. Could anyone help me with this: how to estimate the shape ...
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Is there a named distribution with the property $P(X>10^k) = p^k$?

If I'm doing my math correctly, the exponential distribution has the property $P(X >k) = p ^ k$ (with $p$ conventionally written as $e ^ {-\lambda}$). I'm wondering if there is a different ...
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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 ...
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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....
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Obtaining cdf from pdf when pdf is defined on limited region/support

This is a very simple question, but I want to make sure I am doing it correctly. I have the pdf from a Pareto distribution: $$f(x) = 160 x^{-6}, \ \ 2 \leq x < \infty$$ and want to obtain the ...
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What are the gamma-Pareto convolutions and how have they been used?

The Pareto distributions, i.e., density functions (pdf), are types I through IV and the type II variant; the Lomax distribution. This makes for a number of possible gamma-Pareto convolutions (GPC; ...
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Finding Quantiles and Sampling from a Mixture of Distributions

I am trying to replicate a result in this paper, specifically I am trying to implement the mixture distribution on page 9 of the document. The authors describe a hazard function: $$ h(x):=h_{1}(x) \...
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Compare return levels of fitted GPD using MLE in different R packages

This question is related to this post: Different quantiles of a fitted GPD in different R packages? I want to constraint "potvalues" data to be in a period of 6 years, this is, 16 observations per ...
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Error while fitting Pareto type 2 distribution

I am trying to fit a pareto type 2 distribution using the following code: ...
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Why Pareto distribution has pointy head? How its symmetrical version looks like?

I want to use Pareto distribution for the stock price probability distribution. They say (B. Mandelbrot and N. Taleb Mild vs Wild Randomness) it represents the price changes, especially the tail ...
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Robust sum of non-independent random variables

What approach could be used to sum non-independent variables? I have probability distributions of stock prices and want to calculate the probability distribution of the portfolio price (sum of some ...
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Fitting data to a Pareto with small alpha

I have a few data points (12) that look like a pareto distribution. When I try to find the MLE parameters in R, (using How do I fit a set of data to a Pareto distribution in R?), the parameter $\alpha$...
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How to Select the Largest Y Values for {X,Y} Pairs, for a Pareto-Distributed Dataset, for a Meaningful Fit?

First, apologies for the inelegant question. Second, on to the question: Background Information: I study impact craters, and the size-frequency distribution (number vs diameter) of impact craters ...
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1answer
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Balkema-de Haan-Pickands theorem, generalized Pareto and lognormal

On the wikipedia page on the Balkema-de Haan-Pickands theorem, en.wikipedia.org/wiki/Pickands-Balkema-de_Haan_theorem, it is said the "for a large class of underlying distribution functions",...