# Questions tagged [power-law]

A power-law is a function that increases proportionally to a power of its argument (ax^b). Often seen in fitted relationships or in densities (power-law distributions).

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### Convergence of Percentile in Power Law

I have a probability distribution, that in its tail follows a power law. I've noticed, while I was simulating samples, and determining parameters experimentally, that as I increase the value of a ...
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### Maximum likelihood estimator for power law with negative exponent

Background I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration): Now I ...
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### Exponentially distributed bins for viewing and analysing the tails of a histogram

Is there a rule for visualising and analysing density estimates of heavy tailed observations on a log-log scale? For example, equispaced bins and exponentially distributed bins: ...
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### Optimal new feature selection that reduces uncertainty on a fit

I have a dataset that contains a couple of hundred results from a computational task related to plasma physics. The computation time of the task is, due to the high complexity of the underlying ...
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### Correct GLM or NLS to model exponential model with response variable with positive and negative values

I have been struggling to find the right way to model this dataset, this is a Data Frame with the dataset: ...
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### What are the theoretical assumptions of a power curve versus an exponential or quadratic one?

I have a set of bivariate biological data that has a clearly non-linear distribution. I have found that a power curve best fits the data under several metrics (e.g., AIC, residuals versus fits plot, ...
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### Do stock price changes follow Pareto Distribution? [closed]

I calculated the distribution of the stock price daily ratios. The ratios are multiplicative, $d_t=p_{t} / p_{t-1}$. As far as I know the distribution should look like Power law distribution (Pareto ...
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### How do I calculate a proper sample size to get a suitable power of Kolmogorov-Smirnov test with an underlying Zipf distribution?

I have been trying to develop to calculate the sample size to maximize the power of KS test (±0.8) on an underlying Zipf distribution. I have tried estimating the power by performing simulations: <...
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### $p$-value and $R^2$ of a Power Regression Model

I'm trying to perform a power regression model to fit my data. I used this script to find the angular coefficient and exponent of my power regression. ...
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### Using AIC vs Likelihood Ratio test for comparing Lognormal and Powerlaw distributions

I am interested in comparing whether a lognormal or a power law are a better fit for a given set of data. Both distributions have been fit using MLE, with $x_{min}$ determined using KS-minimization a ...
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### Reading on "Economic nowcasting" for a probabilist

I am about to attend an interview for a postdoc in "economic nowcasting". I am work mostly in probability, particularly complex networks and random graphs. I am looking for some good reading,...
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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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### How can this CDF be decreasing? (power-law)

I am reading the first vignette of the powerRlaw R package. It uses the moby dataset, which is an array of word frequency. Each element is a word, and the value if ...
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### Difference Between SciPy.Stats.powerlaw and powerlaw package

I am trying to simulate random variables that are power law distributed based on my understanding of the definition in this Wikipedia article and several other resources where the consensus is that a &...
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### Sample size determination for power law

I am looking for a method to estimate a mean when the underlying samples are drawn from a skewed distribution. Concretely, I have a population $P$ whose pdf largely follows a power law, where only a ...
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### What is the mathematical meaning of when two variables retain a non-linear relationship even after log transformation?

I am working with a set of biological data drawn from a sample of species, specifically a skeletal measurement and body size, with the intent of calculating a regression line to try to predict the ...
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### Is Prices law a consequence of Benford's law?

Background Price's law says this: In an company, half of all value is created by the square root of the people. I'm sure Dr. Price looked at more than the median, but I suspect this was how the ...
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### Fit a function whose asymptotics is known

Consider the experimental blue curve in the figure that follows It turns out that I know the expression for the tails (-they are power laws- red and green curves) and I want to fit the curve using ...
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### Help with Power Curve MCMC

I'm trying to analyze some $D$ non-logistic cumulative data in a time series, bounded below by 0 and unbounded above. Splitting data into $W$ time windows of $d$ days, I know each window can be ...
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### Sampling distribution of the mean of the discrete-power law distribution

For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the ...
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### In the poweRlaw package, is the location parameter estimated for a lognormal distribution the $median=\exp(\mu)=\theta$ or $\mu$?

I have a set of graphs (each with the same nodes but with edges' weights defined by different research subjects) for which I would like to report statistics. One of these statistics is the betweenness ...
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### Why do poweRlaw and fitdistrplus differ in fitted lognorm parameters

I am trying to evaluate whether a power-law fit is appropriate for some data of lake areas that we have, and whether the theoretically supported alternative of the log-normal, at least at the tails, ...
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### Appropriate Data Analysis when Criterion has Heavy-Tailed Distribution

I have a data set where my independent variables (i.e., personality assessment scores) are continuous and follow a normal distribution. The criterion, sales performance, is heavy-tailed and follows a ...
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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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### Why Are Bell Curves Used for Performance Appraisals?

Why do organizations that stack-rank employees force-fit employee rankings to a bell curve? Is there any theoretical or emperical evidence to support a normal or near-normal distribution of ...
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### Cut off long tail of distribution (no matter how long is the tail) [closed]

One of the algorithms that we are working on is designed to report the time it takes for people to finish the onboarding process in an app. We need an algorithm to eliminate the long tail of a ...
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### How to perform the Kolmogorov-Smirnov test on the truncated power-law distributed data?

Recently, I read papers that perform power-law fitting on their empirical data (estimate the alpha), some of them report corresponding p-value for the Kolmogorov-Smirnov test, but many of them do not. ...
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### Powerlaw Distribution and old nodes [closed]

I am working on the size of discussions (number of replies, replies to the replies) on Twitter data, and I am observing Power-law distribution for small and medium-size discussion but it is not valid ...
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### Rank-size log-log plot inflection: drooping tail of power law

Most of my data seems nicely to fit a power law but with a "drooping tail", which I believe is quite common, although in this case the drop-off is quite steep. I have two related questions if I may ...
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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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### Random Samples from Spliced Distribution

I am studying Clauset, Shalizi, and Newman, Power Law Distributions in Empirical Data (preprint available here) in R. Packages used: ...
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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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### Finding the exponents of a multiple power law: is linear regression valid?

I want to fit a multiple power law equation of the form $y = {x_1}^{\alpha_1} {x_2}^{\alpha_2}$ where I have many examples of $y, x_1, x_2$. (Note there is no intercept.) Is it possible for me to ...
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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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### How are these two power law fitting glm models different?

I have some data that I thought I'd try fitting with a power law (in R). ...
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### Using log-log graph to find equation of power law relationship?

I have a set of data that I think forms a power law relationship, however I am struggling to work out the equation of the relationship. Here is a subset of the data I am working with: ...
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### Sum of powers of standard normal random variables

Context: While trying to teach the Central Limit Theorem I thought it would be a good idea to show a case where it breaks down. Question: Consider the sum of increasing powers of standard normal ...
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### How well is a power law distribution described by the first four moments?

For a normal distribution, the first two moments (mean and variance) are sufficient statistics for the entire distribution. Suppose I have a power law distribution, and I have data on the first, ...
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### Is the truncated power law a heavy-tailed distribution?

A heavy-tailed distribution is often defined as a distribution with a tail that is not exponentially bounded. A truncated power law (or power law with exponential cut-off) is a distribution that ...
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### Most accurate way to express the income of the top 1% in a power distribution?

I have an array of 50,972 household incomes for a small metro area, and I want to measure how much the so-called top "one-percent" make (e.g., the 99th percentile). As you'd expect, it's a power curve:...
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### Is KS test really appropriate when validating a power law/estimating power law parameters?

I'm attempting to find out whether some highly skewed data are drawn from a power law distribution, following the popular paper by Clauset, Shalizi and Newman, 2009. Clauset et al. use the Kolmogorov-...
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### Connection between logit model and power law distribution

I am wondering whether it is appropriate to make a connection between the use of a logit model and a power law distribution. My dependent variable is categorical and ordinal, therefore I am fitting a ...
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### Calculate Zipf-Mandelbrot parameters from distribution

I am fetching trending topics from social media where the frequency of likes is said to follow a Zipf-Mandelbrot distribution; i.e., some of the posts will have a high number of likes and some other ...
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### Determining the number of observations within the tail of a given distribution

I am wondering how to determine the number of observations that fall within the tail of a distribution. I am reading a paper and the authors use the assumption that 50 observations need to fall into ...
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