[question] How to derive the MGF of pareto distribution?

Does it even exist? I think in one website, it said it doesn't. Been trying and search the internet for days now. Sorry if this question is not appropriate for the subreddit. Let me know I'll remove it.

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π°︎ r/statistics
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π€︎ u/pvnrt24
π︎ Mar 25 2021
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What underlying processes causes power distributions (most notably, the Pareto distribution) to appear everywhere?

The Pareto principle, or the 80/20 rule, is a concept that states that a small fraction of the causes (e.g. 20%) causes a large fraction of the outcomes (e.g. 80%). Once you start looking around with this rule, it appears absolutely everywhere: I wear 20% of my clothes 80% of the time, 20% of the landowners own 80% of the land, 10% of the books I read have 90% of the impact, etc, etc.

The common appearance of this rule is attributed to power distributions, most notably Pareto distribution.

It appears then, that power distributions have a widespread occurrence in nature. However, I can find no suitable explanation for why power distributions are so common. Is there an underlying process or principle, which appears in many different forms, that causes power distributions to appear everywhere?

I'd love to hear what you think. The whole topic is quite intriguing.

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π€︎ u/RexBox
π︎ Feb 15 2021
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[Q] Generalized Pareto Distribution (GPD) Estimation From Scratch

I'm trying to fit a GPD to a set of log-return data from scratch, however, my function's output is totally different from ismev's gpd.fit's output. the treshold in my case is u = 0.028 and the sample over treshold is x = c(0.02878043, 0.02935221, 0.02944324, 0.03304120, 0.03573743, 0.03740743, 0.03818290, 0.03899707, 0.03908334, 0.03917351, 0.03917758, 0.03920981, 0.03921063, 0.03921497, 0.06043317).

The log-likelihood function for the GPD if ΞΎβ 0 is defined as:

\log \mathcal L(\xi, \sigma|x) = -n \log \sigma - \left(\frac{1}{\xi}+1\right) \sum_{i = 1}^n \log\left(1+\frac{\xi}{\sigma}x_i\right).

My code from scratch:

negativeloglikelihood <- function(theta, x){

xi <- theta[1]

sigma <- theta[2]

n <- length(x)

L <- -n * log(sigma) - (1/xi + 1) * sum(log(1 + (xi/sigma) * x))

return(-L)

}

optim(par = c(0.1, 0.1), negativeloglikelihood, x = x, method = 'Nelder-Mead')

The estimated parameters I get are xi = -1.77271345413718 and sigma = 0.107130700733396.

Now using ismev's gpd.fit:

gpd.fit(sort(log_returns), 0.028)

And I get xi = -0.24676795 and sigma = 0.01206783.

Can anyone please point me to the error I committed?

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π€︎ u/AegeusZerium
π︎ Jan 04 2021
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Suppose the income distribution in a country obeys the Pareto principle: the richest 20% of people receive 80% of the income. What does this imply about its Gini coefficient?

Definition of the Gini coefficient at Wikipedia.

Not too difficult, but I was amused to find that you can derive somewhat nontrivial rigorous conclusions here.

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π€︎ u/HarryPotter5777
π︎ Oct 11 2020
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The Pareto distribution states that the square root of the total amount of people in any creative domain does half the productive work (a total of 10 people = 3 that do 50% of the output). If that applies to members of Showerthoughts, then the square root of 21 million = 4582 insightful people.
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π€︎ u/trycoconutoil
π︎ Sep 20 2020
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How to generate numbers from a Pareto distribution?

Hi, I need to generate numbers from a Pareto distribution (https://en.wikipedia.org/wiki/Pareto_distribution) for a game I'm making. Ideally with xm=10, and alpha = 2, but close enough is OK. Is there anyway I can do this? I want to do it an "analogue" way (like with dice, cards, etc.) Thank you.

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π€︎ u/Lagrange_Vector
π︎ Aug 14 2020
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Pareto Distribution and Identity Politics at work
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π€︎ u/Skippy_the_clown
π︎ Sep 11 2020
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Have we found any scalable solutions to the fractal-style Pareto distribution of rewards? (50% of gains to the top 1%) Power laws seem almost immutable to me after my most recent reading of Taleb's work. complexityexplorer.org/exβ¦
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π€︎ u/TheSn00pster
π︎ Aug 02 2020
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ELI5: The Pareto Distribution

As far as I understand it, and for those unaware, the Pareto Distribution is a theory of probability that posits the base claim that the square root of a population receives half of any distributable item.

It's sometimes referred to as the 80-20 rule (90-10?). It was originally used to give reason to income inequity and wealth distribution but has been observed throughout our reality.

From Wikipedia, this theory can be seen in file size distribution of internet traffic, the failure rate of hard drives, sizes of sand particles, Tinder (80% of females compete for the top 20% of the most attractive males), the amount of time people play their games on Steam (a few games get played a ton, most never get played).

I can't even read all of those logarithmic type equations and functions on the page. It flies right over my head, so if someone can simplify those I would appreciate it.

What I also need help understanding is why or how this is even an observable phenomenon. Is it something that is simply a natural law, like gravity? If so, how does it happen so consistently? Or is it something we actively attempt to achieve as humans (as it relates to tangible, actionable things like wealth as opposed to natural things like sand particles)? If so, why do we do this?

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π€︎ u/Tongbulgyo
π︎ Sep 23 2020
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Dynasty Fantasy Football & The Pareto Distribution - The Valuation of Player Values - and Making Trades (a simple version of a heady topic).

So there was a trade post a few hours ago that was taken down but the basic question was - how much do you give up in a trade to acquire a star player? Since the comments in that thread are probably going to fall through the cracks - i thought i'd throw this up as I mentioned the pareto distribution, and that has given me a ginormous advantage in my ff dealings.

Here's a simplified explanation of what it is -In a group of people (say employees), you can take the square root of that group size and those people will produce 50% of the company's output. So for example, in a 10 person company, 3 people create 50% of the output. But in a company of say 10,000 people just 100 people do half the work. It's a painful observation when you apply it to global economics and social stuff... but is really well supported and when applied to fantasy football, i believe this approach can give you a real heads up as you try to trade for players.

So what's this geeky stuff mean for your team?okay - well imagine you're in an 8-league and two owners agree to the following trade:

• You receive: Dalvin Cook, Courtland Sutton, DJ Chark, and 2 1st round picks. (i'm not postulating that this is or is not fair, i'm just saying let's use it as an example).

What the pareto distribution will tell you is that league and roster size matters. MOST PEOPLE don't realize this. So maybe you're in an 8-team league, and you are cool with that trade then great. Most people would then think the price for CMC in a 12-team league should be the same. (hint - it shouldn't be) - you can use this to your advantage.

If you're in a league and I have CMC, it's going to be prohibitively more expensive in a 12-team league than an 8-team league to the point that you'll strip out your depth and offset the immense value you're going to gain from getting CMC off me. :-)

TL;DR - star players value goes up as league size increases but the value of depth goes up more. use that knowledge to your advantage.

EDIT: FORMATING & adding to the TL;DR

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π€︎ u/dynastyuserdude
π︎ Jun 02 2020
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The "Matthew Principle" and the "Pareto distribution"

What is the Marxist solution to these? That is something I haven't had answered.

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π€︎ u/ace2354
π︎ May 04 2020
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Has Zizek ever discussed the Pareto distribution ?

I see that this βnatural lawβ is appropriated into many arguments as a be all and end all to the way of life and how we need to accept it as a given. Iβm no philosopher or have any background in the subject but I have recently been exploring Zizekβs thinking and Iβm thoroughly interested in they way he interprets the world. Iβve also noticed how Jordan Peterson brutalizes his others with his ideas by referring to the Pareto curve. Who can I read to get a firmer understanding on the appropriate ways to apply the law. My friends are always falling back on this argument as a way to back up capitalism and how we need it because it develops hierarchy of competence. It frustrates me because I see this as an easy way out.

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π€︎ u/inflated_nepals
π︎ Nov 10 2019
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Using the Pareto Distribution which 20% of statistics concepts count for 80% of the results?

Just getting into Statistics. I'm sure people have different ideas of what gets used the most which makes the question all the more interesting.

Which 20% counts overall for 80% in general and in your field?

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π︎ Feb 29 2020
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Pareto Distribution

I recently learned that 20% of a language accounts for 80% of its usage. This is very inspiring.

What are the most common Japanese words?

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π︎ Mar 12 2020
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[Statistics] How does the base of the logarithm affect the MLE of a Pareto distribution

Hi. If our Pareto distribution is f_Y(y) = t*y^(-(t + 1)) if y &gt;= 1 else 0 and we want to find the MLE from a sample {Y_1, Y_2,...,Y_m} we would write the likelihood function as

L(t) = product from i=1 to m: {f_Y(y_i))}. If I'm correct, we now can take a logarithm of the likelihood function as the logarithm's and L(t)'s maximas are at the same point. After taking the logarithm and taking a derivative of the logarithmic likelihood funtion and finding where l'(t) = 0, I arrive that t = m / (sum from i=1 to m: {ln(y_i)}).

This is where I get confused as the choice of what logarithmic base we use will determine what our estimate will be. More specifically, we can choose what the estimate will be by taking an appropriate base for the logarithm.

How does that make any sense?

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π€︎ u/wabhabin
π︎ Dec 06 2019
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[OC] The market caps of S&P500 companies follows Pareto's Law/Distribution
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π€︎ u/kmh4321
π︎ Dec 29 2019
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Nonprobabilistic quota sample with a pareto distribution and sample size of 500. Can I run parametric tests?

Stats noob here and I am really confused because I have a non-probabilistic quota sample with a sample size of 500 and I want to run parametric tests on it. I am confused if I can due that due to the central limit theorem, or if that theorem doesn't apply because I don't have a random sample.

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π€︎ u/PartyPope
π︎ Sep 08 2019
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CMV: The Pareto Distribution paints a troubling picture for western society.

The Pareto Distribution has also been called the 80/20 rule.

>This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, the Pareto distribution only produces this result for a particular power value, {\displaystyle \alpha } \alpha (Ξ± = log45 β 1.16). While {\displaystyle \alpha } \alpha is variable, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.

This is obviously pretty complicated so I'm sure there are factors I'm not considering, but this seems to paint a very discouraging picture for those in society who want to rise from a low or middle-class position in society to a high-class position. Maybe this is partially why we have such a scourge of depression, anxiety, and other personality disorders. People are unable to achieve great success, and they suffer psychologically because of it. Maybe if we had a more collectivist society vs. the individualistic one that we have, people would be more content with a lower station in life.

In particular this seems to apply to people in creative hobbies, like music. I was reminded of this because there are a lot of hip hop songs where the message is that they started from nothing and rose to the top. On the surface there doesn't seem to be anything wrong with this kind of message, but if the Pareto Distribution is to be believed, there is an 80% chance that they will fail to be financially successful in music. While this can obviously be influenced by effort and dedication, I think the odds are really stacked against an aspiring artist, and I think many of them are deeply inspired by the message of "just keep trying and you'll make it". But for most of them, this will not be true, and I think failure can really psychologically impact them when they realize this.

But this is only one example. The dating world has been shown to reflect this as well. OkCupid posted an infamous now-deleted blog post confirming the 80/20 rule. The original blog post they deleted is very difficult to find, but [here's the text](https://www.mmo-champion.com/threads/2395078-Your-Looks-and-Your-Inbox-(dating-sites) at least.

there was a [study using Tinder that confirmed the same rule as well](https://medium.com/@worstonlinedater/tin

... keep reading on reddit β‘

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π€︎ u/MrEctomy
π︎ Sep 19 2018
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Pareto Distribution: Peterson's Own Goal

There are a couple of ways to counter Peterson's use of Pareto distribution.

You could challenge his reference to a finding from social science that is supposed to be terminally corrupted by neo-marxists.

You could point to the Economics field's notoriously generous use of the word 'law' (I'm looking at you the law of supply and demand).

You could point out that he is misapplying it to production or creativity (his twitter thing about ants and how his description flip-flops between wealth, productivity and creativity) in a way that isn't supported in any relevant literature. The idea of wealth following a Pareto distribution, while not proven, is un-controversial because the idea that wealth begets wealth is un-controversial, but that is radically different to the idea that productivity is Pareto distributed, which requires the fairy-tale ability to measure productivity in some universally applicable way and to believe the bizarre statement that "productivity begets productivity". I'm not sure how you can square that idea with the observation that a billion humans spend every day being completely productive in order to survive only to earn pennies. Is survival not productive? Plus you canβt acquire someoneβs productivity. I mean, productivity canβt be distributed. Anywayβ¦

I think, though, that he could tap-dance his way around these points by picking the right fuzzy sentences.

But, there is a very important lesson to take away from Peterson's use of the Pareto distribution by simply looking at Pareto distributions themselves. I'll describe one:

A good example is to build a graph. A graph being a set of nodes connected with edges. You define the rules about how the graph grows. To make things easy we'll just dump a million nodes in to start with, so the nodes are always there, all unconnected just floating around. Then we pick two nodes at random and join them by an edge. Over time this would create a random graph... for obvious reasons. Random graphs are not Pareto distributed. Try that again but this time instead of picking at random we make the probability of picking a node increase as the node has more friends. If you tried this (made some software simulation or something) you would create what's called a scale-free graph (or a small-world graph). This type of graph has a Pareto distribution of edges; a small number of highly connected nodes and huge number of barely connected nodes.

Read this next sentence and let it sink in:

**Every elemen

... keep reading on reddit β‘

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π€︎ u/Many-Worlds-Music
π︎ Feb 14 2019
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Pareto Distribution of the time I spend on my games in hours (Steam and Origin) [OC]
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π€︎ u/1llumawaffle
π︎ May 02 2019
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Peterson's eternal truths vs. economists about the so-called Pareto distribution

One of Jordan Peterson's recurring apologetic arguments in defense of capitalism and against socialism is the idea that social inequality is a function of some sort of eternal natural inequality, or what he calls "The Principle of Unequal Distribution". More specifically, in various videos, Peterson seems to believe that invoking the Pareto principle is a brilliant way to refute Marx. Sometimes he says we have no idea why unequal distribution takes place because he loves mysticism, sometimes he refers to the principle of preferential attachment, which takes various forms and names, like "the rich get richer" principle.

I'm not comfortable enough in mathematics to offer my own understanding about the underlying cause of the recurrence of power laws in nature and society. I would argue it's rooted in the laws of probability. There are also a number of logical arguments one could make about the manner in which Peterson invokes eternal natural laws to buttress his flimsy and reactionary worldview. But for now, I want to focus on the empirical aspect of this question. Let me state my bias outright ; I'm a socialist. But I will simply present the views of two very mainstream liberal economists, who are both hostile to the method and politics of Marxism.

From Peterson's book:

>[The principle] can be modelled using an approximately L-shaped graph, with number of people on the vertical axis, and productivity or resources on the horizontal. The basic principle had been discovered much earlier. Vilfredo Pareto (1848β1923), an Italian polymath, noticed its applicability to wealth distribution in the early twentieth century, and it appears true for every society ever studied, regardless of governmental form.

This, of course, is tabloid-level nonsense. And here, in the words of two of the most respected social inequality researchers, is why:

Branko Milanovic takes a favorable view of Pareto's contribution to economics and writes:

>It is just a minor simplification to say that Pareto thought that there was an iron law of income distribution, namely that inequality did not change whatever social system was in power. It gave consistency to his theory of the circulation of the elites, because whatever elite be in power (land-owning, capitalist or bureaucratic), income distribution would be the same although the people who would be rich or poor would be different. It was a serious critique of the

... keep reading on reddit β‘

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π€︎ u/Canchito
π︎ Feb 23 2018
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Pareto distribution at work
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