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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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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Validity of Monte-Carlo method to estimate a probability distribution which follows a power law

I am using a Monte-Carlo method to estimate a probability distribution function (pdf). Basically, I have several input parameters following known distributions, from which I can draw samples, that I ...
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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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86 views

Posterior Predictive Distribution for Uniform Likelihood and Pareto Prior

I'm trying to find the posterior predictive distribution for data $X_i, \dots X_n$ from a a $Uniform [0, \theta]$ distribution. The prior distribution for $\theta$ is a $Pareto[\alpha, \beta]$ ...
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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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Estimating tail share of apparently subexponential distributions drawn from finite population, given a finite sample

Suppose I have data on a large sample of some units of observation, where the observed quantity has meaningful differences and ratios. The sample is much smaller than the population, but both are ...
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Piecewise Pareto Distribution

What are the best practices for Piece-wise Pareto Distribution or maybe Pareto Mixture Model(?). Example: $x\in [0, 1) \Rightarrow \alpha=0.1$ $x\in [1, 10) \Rightarrow \alpha=0.5$ $x\in [10, 100) ...
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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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Extrapolation of a pareto distribution over a time period

I have wealth values for an entire population for year T, which follows Pareto distribution. And I have tail values of wealth for year T+1 (also a Pareto distribution). Is there any way to extrapolate ...
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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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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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What if my GPD fit for the data with threshold = 0 but the data doesn't fit Pareto distribution?

Please correct me if I am wrong, if data with threshold=0 fit the GPD distribution, it means Pareto distribution fits the data. But here in my case, it turned out that the Pareto doesn't fit the data. ...
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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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Extreme Value Theory - Can I apply the Hill estimator to block maxima?

I want to apply the Hill estimator for the shape parameter: $ \hat{\xi}_{k,m}^{hill}= \frac{1}{k}\sum_{i=1}^{k}\log \frac{x_{(i)}}{x_{(k)}}\quad 2\leq k\leq m$ where $\{x_{(i)},i=1,...,m\}$ are the ...
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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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Would this be a correct way to sample from intuitively correlated variables?

I have three variables and each represent some metric of people and can have values within different ranges e.g. ...
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421 views

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 ...
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A statistical test to check whether a set of data are Pareto distributed

I need to construct a statistical test which outputs the p value for the hypothesis, H0: Data are Pareto distributed Vs H1: Data are not Pareto distributed. I found a test in stack exchange, but it's ...
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Variance of unbiased estimator for the shape parameter of Pareto distribution

I'm interested in getting the error bounds of the unbiased estimator of the shape parameter ($\alpha$) using maximum likelihood method of Pareto distribution. The unbiased estimator is known to be ...
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What's the formula to estimate the shape parameter of Pareto distribution using weighted least squares method?

I'm trying to simulate my own method using R to estimate the shape parameter of Pareto distributed data by weighted least squares. I searched via several links of research papers, but I could not find ...
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two-parameter Pareto distribution with known A [duplicate]

I am trying to solve the following problem. Any help would be great: Scores are distributed as a two-parameter Pareto distribution with a=3 Scores for 3 groups are as follows: Group A has expected ...
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need explanation about the exponent parameter s in zipf distribution

I need to model the popularity of some requested files from a library with Zipf distribution and I want to simulate it in MATLAB. I don't know what's the effect of parameter s on my result. for ...
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How does MGF of Pareto distribution of first kind exist for non-positive values of t? [closed]

I have reached upto the stage shown in the attached picture. The r.v. X is always positive and its power $\beta+1$ is also always positive. Therefore, how can it be said that MGF exists for t <= 0? ...
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Prior / Reparameterization for Binomial Hierarchical Model

I am following the argument made by Gelman et al. in Bayesian Data Analysis (ref. in 3rd edition, p.109 onwards) for defining a non-informative prior for a hierarchical Binomial model (in the text, ...
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LRT for PRT both unknown

I have a random sample of ${X_1,...,X_n}$ from the following pdf: $${\theta \beta^ \theta \over {x^{\theta+1}}}$$ where $\theta>0$, $\beta>0$, $x\ge\beta$ I want to find the LRT to test $H_o:\...
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Quotient of Pareto and Gamma random variables

I cannot find an explicit formula for the quotient of a Pareto random variable divided by a Gamma random variable. The only that I found is something like, for $P(X)$ pareto's like and $P(Y)$ Gamma's ...
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Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $X_i$'s are i.i.d Pareto variables

Let $X_1,X_2,\ldots,X_n$ be i.i.d variables having a Pareto distribution with density $$f(x)=\frac{a\theta^a}{x^{a+1}}1_{x>\theta}\,,$$ where $a,\theta>0$. What is the distribution of $\sum\...
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uniform pareto system error

Let $X\sim U(0,\theta)$. Given a sample of size n, the likeliohood function is $l(\theta \mid x)=\frac{1}{\theta^n}$ Consider a pareto prior distribution $\theta\sim pareto(k,a)$ with density $\frac{...
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Confusion about Pareto Principle and Pareto distribution

The Pareto principle, applied to wealth for example, says that around 20% of the population holds 80% of the wealth. Accordingly, it is said that a person's wealth follows a Pareto distribution. I'm ...
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Estimator for Pareto Distribution & Delta Method

Assume that $Y$ has a Pareto distribution with parameters ($\theta, t$ = 1). An estimator of $\theta$ is $\tilde{\theta}$ where $\bar{Y} = \frac{\tilde{\theta}t}{\tilde{\theta} - 1}$. Solve for $\...
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How to write the set of indexes of Pareto optimal reward set in formal methods

I denote that a reward vector of an item $a$ as $r_a$. Say there is a set of items denoted as $A_t$. I want to get a set $A^\prime_t$ of items from $A_t$ that has non-dominated reward vectors. For a ...
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Are the Feller-Pareto and the generalized beta distributions really the same?

The Feller-Pareto distribution was originally is defined in terms of a transformed beta distribution. If $Y\sim \beta(\gamma_1, \gamma_2)$ then $W=\mu + \sigma\left(\left(1/Y\right) - 1\right)^\gamma=...
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How to fit newer cohorts using Pareto/NBD or Beta/Geo for CLTV

I am trying to fit the popular Pareto/NBD or Beta/Geometric models for non-contractual, continuous customer data. On top of that I then fit the Gamma/Gamma model for monetary value (using the very ...
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If the best-fitting distribution has infinite variance, should low observed variance be troubling?

Suppose you have observations which, over the observed range of outcomes, are well-fitted by some distribution like the Pareto that, for certain parameter values, has a an infinite variance. For ...
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Is there a robust estimator of the tail mean that is better than the sample tail mean?

Suppose there is a large but finite population with values drawn from some heavily skewed distributional family with finite mean. I am supplied exogenously with the value of the 95th percentile, ...
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A random variable $X$ on $(0,\infty)$ which behaves like Exp for small $x$ and Pareto for large $x$

Are there any examples of distributions which behave like Exponential for small values and like Pareto for large values. $$\ln \mathbb{P}[X>x] \sim -\lambda x, \qquad \text{ for } x \text{ small}, ...
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Is Pareto exponential?

I read on wikipedia about the exponential family (https://en.wikipedia.org/wiki/Exponential_family) that: , but later on in the same article I read that: Some distributions are exponential ...
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Is there a closed-form solution for the tail index of a GB2 distribution?

In the Generalized Beta distribution of the second kind (GB2), where a, p, and q are shape parameters and b is a scale parameter, the pdf is defined on $\mathbb{R}_+$ by: $$ GB2(y;a,b,p,q) = \frac{|a|...