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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Formulation of two parameter Pareto distribution
So everywhere I've looked, I have seen the two parameter Pareto distribution formulated as
$\frac{\alpha\lambda^\alpha}{x^{\alpha+1}}$. The distribution we are using in our course is $\frac{\alpha\...
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TTest for discrete data with a pareto distribution
I want to compare two sets of discrete data (0 to infinite, but realistically around 450) that don't follow a normal distribution but a pareto distribution, so supossedly I cannot use the Student's ...
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Verifying the statistics are complete and sufficient for two parameter Pareto distribution
Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution
with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$
$\textbf{(i)}$ Show that ...
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Linear Combination of Bounded Pareto RVs
I am working with bounded Pareto distributions and was wondering whether I can say anything about the distributions of linear combinations of Pareto RVs? Suppose the PDF
$f(x; \alpha_i, L_i, H_i) = \...
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Estimating parameters of a Pareto-like distribution and examining its goodness-of-fit
I have developed a theoretical distribution in the form of
$$
f(x) = \frac{\beta}{\alpha}\left(1+\frac{x}{\alpha}\right)^{-\beta - 1}
$$
Where $\alpha$ and $\beta$ are parameters of the model with ...
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Fit a tapered Pareto distribution in R
Are there any R packages that I can use to fit a tapered Pareto distribution thru Maximum Likelihood? Preferably, I am looking for something that can return the confidence intervals of the parameters ...
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Pareto distribution with Gamma prior on parameter $\theta$
I want to calculate the posterior distribution of Pareto distribution with known parameter $X_m$ and unknown parameter $\theta$, with conjugate prior on $\theta$ the Gamma distribution:
My effort is ...
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A problem with the expectation of a Pareto
My course notes (3rd-year module in Bayesian Statistics, unpublished) contain the following section.
Assume we have data on the number of people queuing at an ATM at a specific hour for several
...
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Help analysing Mean Residual Life Plot for GPD
I'm trying to fit a GPD for a set of time dependant data. I have two columns, data which is a value on the negative real line where values closest to zero are considered extremes, and time. Using only ...
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GPD MLE for Multiple Samples
So this is not the exact data I have, but I was more just wondering the approach for this problem. To demonstrate the problem I have, I will give an example.
Say we are measuring Wind Speeds in a ...
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What likelihood to use to model sample means from a Pareto-like distribution?
Suppose there is a random variable with Lomax (Pareto Type II) probability density
$$
P(x; c) = \frac{c}{(1 + x )^{c + 1}}, \quad x \ge 0, c > 0.
$$
Let's draw n_samples=30000 samples of length ...
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Numerical Solution of two convoluted stable paretian random variables
I am trying to numerically compute the joint density of X and Y, where both are stable paretian distributed random variables with different alphas (1.4 and 1.7).
I can compute the PDF via inversion ...
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Does statistically simple algos qualify as AI algos?
We have a customer purchase transaction history data with variables like below
recency - how recently they bought?
frequency - How often they bought?
monetary - How much value did they bring to the ...
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Given n iid Pareto distributed random variables, find the UMP one sided test of the first moment
Given $X_1,...,X_n$ ($n\geq 2$) are iid and each have density:
$f_X(x) = \frac{c^\theta \theta}{x^{1+\theta}}\mathbb{1}(x> c)$ for known $c$ and $\theta > 1$
then we can easily find the first ...
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GPD and GEV Fitting: Maximum Likelihood vs. Least Squares
I am trying to build a model based on real world data which involves fitting generalized extreme value distributions and generalized Pareto distributions. Most literature immediately turns to the ...
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Connection between forms for Generalized Pareto Distribution
On Wikipedia (https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I–IV) one can find the relation between the different types of Pareto Distribution and the Generalized Pareto Distribution (...
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Existence of Moments for Linear Regression With Pareto Error
Suppose I have the following model linear regression model:
$y = \beta_0 + x_1i\beta_1 + x_2i\beta_2 + e_i$ with $e_i \sim Pareto(k,\alpha)$
Now if $1< \alpha < 2$, I would suppose that the ...
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Knowing the sum, the n(), and the bound parameters of a truncated-Pareto distributed variable, how I identify the alpha (shape) parameter?
I know that there would be a fancy command on R to do the estimation of $\alpha$ given the inputs, but I am also curious about the relationship between $\alpha$ to $...
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Joint distribution of top order statistics of two independent random samples of Pareto distribution
Suppose $X_1,...,X_n$ and $Y_1,...,Y_n$ are all independent copies of a standard Pareto random variable. For each of the 'two' random samples we can denote the order statistics $X_{n:n} \geq X_{n-1:n} ...
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Deriving the limiting distribution of a sum of Pareto distributed variables
For a series of independent and identical Pareto distributed variables $X_i$ with $\alpha > 2$, their sum $S_n = \sum_{i=1}^{n} X_i$ has a normal distribution as limiting distribution for $n\to \...
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Literature on Noninformative Priors for GPD
I am starting to do some work using the Generalized Pareto Distribution (GPD), and was hoping someone might be able to point me in the direction of literature (or just general recommendations) on ...
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Do you need large amounts of data to estimate parameters in extreme value distributions?
There is probably not a hard answer for this, but I am wondering if you need to collect more data when trying to estimate the parameters of generalized pareto distribution well?
The reason I ask is ...
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The random variable $log(\frac{X}{x_0})$ has an exponential distribution with parameter $\alpha$
It is said that a random variable $X$ has a Pareto distribution with parameters $x_0$ and $\alpha$ for $(x_0 > 0)$ and $(\alpha > 0)$ if $X$ has a continuous distribution for which the p.d.f. $f(...
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Method of moments estimate of Pareto Distribution
The Pareto distribution has the following $cumulative \ distribution \ function$ :
$$F(x;\alpha ,\Theta ) = \left\{\begin{matrix}
1 - (\frac{\alpha}{x})^{\theta}\ \ if \ \alpha \leq x\ & \\ 0 \ ...
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Beta distribution with a priors as Uniform and Pareto Distribution
I am working on a bayesian programming problem which involves a Beta Posterior, which has mean (location) parameter coming from Uniform Distribution [U(0,1)] and concentration (kappa) coming from ...
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Fitting distributions to censored and uncensored data in R
I need to fit lognormal, Pareto, and generalized Pareto distributions to some empirical data that is a combination of censored and uncensored data. I tried using the function ...
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Calculating representative sample of pareto distribution
I have a Pareto-distributed population of size N.
If I wish to be 99% confident, with 0.75% margin of error, and empirically 35% made a good sample - what will be the formula to derive the sufficient ...
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Why is a Fréchet distribution slowly varying, and what is the intuition behind it?
The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0 & & x\leq 0,\\[6pt]e^{-x^{-\alpha}} & & x>0,\end{cases}$$
is regularly varying as stated here (page 19):
It is not ...
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Student's t as a power law distribution
I'm currently reading about power laws and I have came across an answer stating:
The density function of a Student's t-distribution with $n$ degrees of freedom is:
$$f(x) \sim (1 + x^2 / n)^{-(n+1)/2}...
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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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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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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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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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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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Neyman-Pearson Lemma for Pareto Distribution [duplicate]
I have the following problem.
Let $X_1, ..., X_n$ represent a random sample taken from a population with CDF given by
$$
F(x;\beta) = 1 - \frac{\beta}{x}, ~~ x \geq \beta > 0.
$$
Based on the this ...
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Programming inverse-transformation sampling for Pareto distribution
I am having trouble deriving a formula, and running a simulation with its distribution. The Pareto distribution has CDF:
$$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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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}}$$
For MM, ...
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Understanding Hypothesis tests for a Pareto distribution
I'm writing an essay that's looking at the presence of the Pareto Principle in data. Unfortunately, as a consequence of interest, I've picked a topic that involves statistical analysis well above what ...
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MLE of truncated Pareto distribution
I am trying to apply truncated Pareto distribution to a dataset, but I am not able to find a close form MLE for the shape parameter alpha. Could anyone help me with this: how to estimate the shape ...
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Is there a named distribution with the property $P(X>10^k) = p^k$?
If I'm doing my math correctly, the exponential distribution has the property $P(X >k) = p ^ k$ (with $p$ conventionally written as $e ^ {-\lambda}$). I'm wondering if there is a different ...
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Fitting Pareto distribution to data example in SciPy
In docs.scipy.org there's code to sample data from a Pareto distribution and then fit a curve on top of the sampled data. I could understand most of the code snippet except the term ...
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Determining shape parameter for Generalized Pareto Distribution Scipy
I have a set of values to which I want to fit a Generalized Pareto Distribution. Scipy provides functions for doing so:
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats....
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Obtaining cdf from pdf when pdf is defined on limited region/support
This is a very simple question, but I want to make sure I am doing it correctly.
I have the pdf from a Pareto distribution:
$$f(x) = 160 x^{-6}, \ \ 2 \leq x < \infty$$
and want to obtain the ...
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