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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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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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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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Is there a closed-form solution for ratios of order statistics?

Is there a closed-form solution for the expected value and variance of the ratios between specified order statistics drawn from a large sample from a known parametric distributional family? Actually, ...
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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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How to fit a distribution to a function of binned values

I am interested in fitting power law or Pareto distributions to a set of values. First, we assume that the abundance $A$ of an organism is related to mass $M$ with a power law with exponent -2: $A \...
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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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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|...
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How to use method of moment to find Pareto distribution estimator?

I have $f_{\alpha, \beta}(y)=\frac{\alpha}{\beta}(\frac{\beta}{y})^{\alpha +1}, y\ge\beta,\ \ \alpha,\beta\gt 0$. Both $\alpha, \beta$ unknown. To find estimators using the method of moment, we ...
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How to get the natural parameter of Pareto distribution if $\beta$ is fixed?

With $\beta$ known, $f_\alpha(y)=\frac{\alpha}{\beta}(\frac{\beta}{y})^{\alpha+1}=e^{\ln\alpha-\ln\beta+(\alpha+1)\ln\beta-(\alpha+1)\ln y}$. Now the problem is should we expand $-(\alpha+1)\ln y$ and ...
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Finding Pareto distribution's Kurtosis

I have no clue how to solve this question: Questions: Given that each of a,b,c,d and e is a digit from {0,1,2,3,...,9} and f is an alphabet from {A,B,C,D,...,Z} X has a Pareto distribution with ...
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What is a hooked powerlaw?

I happened to encounter lots of scientific/business scenarios where a Zipf/Pareto/powerlaw describes well my data. However, whenever the mean of the distribution is large enough, the fact that these ...
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How to determine the type of probability distribution for a dataset?

I have aggregated(total) youtube videos views. I have take log of that views. And calculated autoregressive koefs that can be used for the video views predictibility tests. Let say I have aggregated ...
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Checking for Pareto Distribution

I am trying to test the following statement: " Zipf's law is an observation about how often different words are used. Zipf's law predicts that in a body of text, the distribution of word frequencies ...
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Fitting a distribution on Income data

Why is the Pareto Distribution a better fit to the upper tail of Income data, and the Lognormal distribution a better fit to the lower tail? What happens if we fit the data the other way around? My ...
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Pareto Chart v.s. machine learning feature selection

A Pareto chart could be used to show which factors have the greatest impact and where attention is likely to yield the greatest benefit. Also common machine learning feature selection methods can ...
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Can we improve on the sample mean as an estimator of the true mean of a Pareto distribution, 1 < α < 2?

Suppose I have a sample drawn from a population which is approximately distributed i.i.d. according to the Pareto distribution for values of x greater than X*. Suppose, moreover, that the tail index 1 ...
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Besides the Pareto and Zipfian distributions, which distributions obey the power-law?

I need a list of distributions that obey the power-law, beside the commonly used Pareto and Zipfian distributions. A comprehensive list or a reference to a comprehensive list will be particularly ...
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is Pareto distribution exponential dispersion family and the form is unique?

I am trying to show that Pareto distribution with $$f(y;\alpha)=\alpha y^{-\alpha-1} $$ is exponential dispersion (ED) family which means that it can be rewritten as: $$f(y;\theta,\phi)={\rm exp}\{\...
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Standard error of Pietra index with Pareto assumption

I am working on this problem of income distribution. I am assuming that my income data $X_i$ is Pareto : $f(x_i;\alpha) = {\alpha \over x_0}({x_i \over x_0})^{-(\alpha + 1)}$ I found my MLE ...