# 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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### 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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### 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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### 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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### 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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### 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, $\sqrt{n}$-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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### 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 ...