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
0 answers
19 views

Applying Tangent Lines to Log-Scaled Data for Outlier Detection: Seeking Statistical Theories and Models

I've analyzed the view counts for a YouTube channel's videos (just for example), sorting them by views (on the left) and drawing a tangent line to approximate the central trend on a logarithmic scale (...
Andrew Anderson's user avatar
4 votes
1 answer
76 views

Bayes estimator of possion distribution with Pareto prior

Consider a random sample of size $n$ following the possion distribution with parameter $\ln \theta$, that is $$ f(x|\theta)=\frac{(\ln\theta)^x}{\theta x!}, x=0,1,2,\cdots $$ and the prior of the ...
Javier's user avatar
  • 43
0 votes
0 answers
62 views

Asymptotic variance covariance matrix of maximum likelihood estimate of Pareto distribution

I have some problem deriving the asymptotic variance-covariance matrix for MLE of the pareto distribution and hope to receive help. Consider random variable $X$ follows Pareto distribution, where $f_X(...
ssjjiw's user avatar
  • 53
1 vote
0 answers
40 views

Threshold choice for Peaks-Over-Threshold

I'm trying to estimate equivalent performances at different events, using Peaks-Over-Threshold from Extreme Value Theory. The challenge is to find the threshold and preferably with same number of ...
Daniel Westergren's user avatar
0 votes
0 answers
40 views

How to esimate the mean and variance of data from a Pareto distribution

I have large sample of data that is approximately from a Pareto distribution with unknown parameters. Unfortunately the distribution is sufficiently heavy tailed that just taking the sample mean is ...
Simd's user avatar
  • 2,029
0 votes
0 answers
54 views

Tail of the maximum of a time-varying Poisson-GP marked process

Consider a time-varying version of the Poisson-GP marked process on the real line as commonly used in Peak Over Threshold (POT) modelling of a variable $Y$. More precisely we have a given time-varying ...
Yves's user avatar
  • 5,398
3 votes
1 answer
157 views

Expected value of a Pareto distribution between two values

I try to calculate the expected value of a Pareto distribution. Suppose that we have a Pareto distribution for $x \ge \theta$: $$f(x;k;\theta)= \frac{k\theta^k}{x^{k+1}} $$ We can calculate $ E[X]$ ...
John Smith's user avatar
2 votes
1 answer
58 views

Are there conditions for which the Pareto distribution arises? Are there characterization theorems of the Pareto distribution?

There are many real-world phenomena in which a variable of a population follows the Pareto distribution. I am wondering, what are the sufficient conditions for the distribution to be Pareto? And if it ...
Maximal Ideal's user avatar
2 votes
1 answer
98 views

An approximate confidence interval for the $\alpha$ parameter of a Pareto Type II distribution when $\lambda$ is known

The Pareto Type II distribution, also known as the Lomax distribution, has the following density, $$f(x|\alpha,\lambda)=\frac{\alpha\lambda^{\alpha}}{(\lambda+x)^{\alpha+1}}, \qquad x>0,\ \alpha>...
29703461's user avatar
0 votes
0 answers
174 views

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\...
Aniruddh's user avatar
  • 101
2 votes
1 answer
541 views

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 ...
Aleph Alpha's user avatar
0 votes
0 answers
69 views

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) = \...
SimonDude's user avatar
1 vote
1 answer
45 views

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 ...
Reza Afra's user avatar
4 votes
0 answers
847 views

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 ...
Homer Jay Simpson's user avatar
2 votes
1 answer
79 views

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 ...
mjc's user avatar
  • 589
0 votes
0 answers
134 views

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 ...
Norbert Wesolowski's user avatar
3 votes
0 answers
107 views

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 ...
andrew brdk's user avatar
1 vote
1 answer
58 views

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 ...
The Great's user avatar
  • 3,272
1 vote
0 answers
148 views

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 ...
s l's user avatar
  • 87
0 votes
0 answers
129 views

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 (...
Barbab's user avatar
  • 333
1 vote
0 answers
57 views

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 ...
SimonDude's user avatar
1 vote
1 answer
199 views

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 $...
GiulioGCantone's user avatar
1 vote
0 answers
68 views

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} ...
Joogs's user avatar
  • 809
5 votes
1 answer
374 views

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 \...
Sextus Empiricus's user avatar
1 vote
0 answers
82 views

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 ...
John Smith's user avatar
6 votes
2 answers
404 views

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 ...
John Smith's user avatar
3 votes
1 answer
501 views

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(...
Stackcans's user avatar
  • 341
2 votes
1 answer
782 views

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 \ ...
Kalvin's user avatar
  • 423
2 votes
0 answers
184 views

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 ...
maamli's user avatar
  • 85
1 vote
0 answers
604 views

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 ...
Chris J's user avatar
  • 11
0 votes
0 answers
47 views

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 ...
goidelg's user avatar
  • 101
4 votes
1 answer
337 views

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 ...
Antoni Parellada's user avatar
4 votes
1 answer
420 views

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}...
Blg Khalil's user avatar
0 votes
0 answers
66 views

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)^{...
econ_ugrad's user avatar
3 votes
1 answer
394 views

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 ...
forecaster's user avatar
  • 8,195
1 vote
2 answers
99 views

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)^...
Lewkrr's user avatar
  • 530
0 votes
0 answers
150 views

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 $\xi$ ...
Blg Khalil's user avatar
0 votes
1 answer
55 views

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?...
Joshua Snider's user avatar
0 votes
2 answers
144 views

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 ...
Guest's user avatar
  • 87
1 vote
1 answer
676 views

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 ...
Statsnewbie's user avatar
3 votes
1 answer
535 views

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 ...
Sigma's user avatar
  • 569
6 votes
2 answers
4k views

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 \...
John Huang's user avatar
2 votes
1 answer
903 views

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 ...
Guilty_Scene's user avatar
0 votes
0 answers
1k views

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 ...
jameshgrn's user avatar
0 votes
1 answer
84 views

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) &...
jameshgrn's user avatar
1 vote
0 answers
57 views

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 ...
Andrew Anderson's user avatar
6 votes
1 answer
234 views

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&...
Christian's user avatar
  • 111
3 votes
1 answer
1k views

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, ...
PK1998's user avatar
  • 151
1 vote
1 answer
645 views

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 ...
M.White's user avatar
  • 11
1 vote
0 answers
136 views

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 ...
Aragorn's user avatar
  • 111