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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47 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 ...
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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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82 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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83 views

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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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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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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Fitting a lognormal-pareto distribution to empirical data distribution in R

Let's say I have data whose distribution is leptokurtic, with a lognormal peak and a pareto tail (for this question, I will generate lognormal-pareto data using package CompLognormal but I will need ...
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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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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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790 views

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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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",...
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How can I calculate the exponent (parameter or tail index) of the Pareto distribution associated to the following paperclips experiment?

The experiment is the following: take a pool of paperclips for which in each round we follow these steps: 1) choose two paperclips randomly 2) if they are not linked with each other then link them ...
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Generating two-sided pareto random numbers

I am trying to generate random numbers from a two-sided Pareto distribution. The paper I'm replicating states that the following two-sided Pareto density function is used: $$ 1-f(x) = f(-x) = \frac{...
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Pareto Variable Transformation [duplicate]

A random sample, $X_1, X_2,... X_n$ is drawn from a Pareto population with pdf $$f(x|\theta)=\frac{\theta}{x^2}I_{[\theta,\infty)}(x)$$ I've been trying to figure out the distribution of $$T=\log\Big[...
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Processes that converge to the Pareto distribution

Do any stochastic processes generate the Pareto distribution as the steady-state statistic of the ensemble? For example, $$ dS_t = f(t, S_t, W_t) $$ where in the Ito sense the p.d.f. of $ g(S_t) $ ...
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If $X\sim \operatorname{lognormal}$ then $Y:=(X-d\mid x\geq d)$ has approximately a Generalized Pareto distribution

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d\mid x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: ...
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How does the Pareto principle give the 80/20 rule?

It seems to me that the Pareto principle says that for any $n$-many people that produced $m$-many goods, $\sqrt[2]{n}$-many people would produce $\frac{m}{2}$ many goods out of the total $m$ many ...
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Discrete Pareto Distribution vs Zipf Distribution and Power Law vs Zipf Law

I need to get a simple, but clear idea of Discrete Pareto Distribution vs Zipf Distribution and Power Law vs Zipf Law. (Are they similar/ how they relate to each other.) Wikipedia definitions do not ...
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Are there any non parametric tests to check for Pareto data?

I'm in search of a non parametric test to check whether my data are Pareto distributed, but I couldn't find a proper reference for it. I'm using R to simulate these so if there's any R in built test, ...
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How can Pareto(alpha = 5, x_min = 2) be heavy-tailed where alpha is the shape parameter or the tail index?

Point 1 : It's known that, usually, when the tail-index (alpha) is between 0 and 2, of a certain data set, the distribution is considered as heavy-tailed. Point 2 : It's know that Pareto ...
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1answer
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What's the relationship between Pareto shape parameter (alpha) and exponential rate parameter (lambda)?

I'm trying to do my undergraduate research on non parametric density estimation for a heavy tailed distribution. For that, I'm with a data set, which I assumed it should be Pareto distributed with ...
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Expectation of kth order statistic of Pareto distribution

I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$ My attempt: $$...
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Training Neural Network at Decile Level

I have a simple feed forward neural network regression model that I'm training on customer data to predict their usage amount. The MAPE is above 50%. The data is heavily skewed and when I log ...
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Sum of N random variables from the same distributions [duplicate]

Given $n$ independent random variables from the same distribution, how to obtain the distribution of their sum? For example, the distribution of $n$ normal distribution is $N(n\mu, n\delta^2)$. ...
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What's the relationship between degrees of freedom of t distribution and tail exponent (alpha) of Pareto distribution?

I'm going to generate a set of data from a T distribution and truncate the body(so that we make it approximately Pareto distributed) of it and estimate the tail exponent(shape parameter) of the ...