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.

Filter by
Sorted by
Tagged with
1
vote
0answers
22 views

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 ...
0
votes
1answer
39 views

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{...
1
vote
0answers
22 views

Pareto Variable Transformation

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[...
2
votes
1answer
33 views

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) $ ...
1
vote
1answer
165 views

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: ...
0
votes
0answers
12 views

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 ...
0
votes
0answers
29 views

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) ...
0
votes
0answers
43 views

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 ...
0
votes
0answers
12 views

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 ...
2
votes
1answer
36 views

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, ...
0
votes
0answers
20 views

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 ...
2
votes
1answer
43 views

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 ...
0
votes
0answers
9 views

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. ...
2
votes
0answers
64 views

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: $$...
0
votes
0answers
19 views

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 ...
0
votes
0answers
13 views

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 ...
0
votes
0answers
11 views

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. ...
0
votes
0answers
257 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 $N(n\mu, n\delta^2)$. ...
1
vote
1answer
62 views

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 ...
0
votes
1answer
101 views

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 ...
5
votes
1answer
139 views

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 ...
0
votes
0answers
27 views

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, ...
1
vote
1answer
38 views

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 ...
0
votes
0answers
9 views

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 ...
0
votes
1answer
55 views

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 ...
-1
votes
1answer
293 views

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? ...
1
vote
0answers
78 views

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, ...
2
votes
1answer
162 views

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:\...
1
vote
0answers
44 views

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 ...
0
votes
1answer
41 views

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{...
2
votes
0answers
145 views

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 ...
0
votes
1answer
236 views

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 $\...
1
vote
0answers
8 views

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 ...
2
votes
1answer
141 views

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=...
3
votes
0answers
211 views

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 ...
3
votes
3answers
64 views

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 ...
2
votes
0answers
35 views

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, ...
2
votes
1answer
43 views

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}, ...
0
votes
0answers
618 views

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 ...
3
votes
2answers
114 views

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|...
2
votes
1answer
2k views

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 ...
1
vote
0answers
171 views

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 ...
4
votes
2answers
209 views

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 ...
0
votes
0answers
67 views

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 ...
4
votes
2answers
926 views

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 ...
0
votes
1answer
454 views

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 ...
2
votes
2answers
491 views

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 ...
0
votes
0answers
376 views

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 ...
2
votes
0answers
84 views

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 ...
4
votes
1answer
158 views

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 ...