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Computational aspects of stable distributions

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Abstract Stable distributions are a class of probability distributions that generalize the normal distribution. They are the only possible limits of normalized sums of independent, identically distributed terms, so sums of a large number of such terms have to approach a stable law. The non‐Gaussian stable distributions have heavy tails with infinite variance, and can be skewed. In most cases, there are no known formulas for the density or cumulative distribution function of these laws, so using them in practice requires significant computational methods. This paper explains some of the computations used to make stable laws useful in practical problems. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization Algorithms and Computational Methods > Algorithms Algorithms and Computational Methods > Maximum Likelihood Methods
Stable densities of standardized stable random variables in the 0‐parameterization (left column) and 1‐parameterization (right column) for varying α and β. Note how the mode in the 0‐parameterization is always near 0, but near α = 1 the mode in the 1‐parameterization shifts abruptly
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Linear regression of change in weekly interest rates for 10‐year US bonds (x) vs. AAA corporate bonds (y) for 2008–2009
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Histogram of n = 10,000 simulated values of a distribution. The solid line is the corresponding density, numerically computed using the method above
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Cumulative distribution functions of standardized stable random variables. The 0‐parameterization is used on the left column and the 1‐parameterization is on the right column
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Algorithms and Computational Methods > Maximum Likelihood Methods
Algorithms and Computational Methods > Algorithms
Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization

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