Stable densities in four different parameterizations:
S(α,β,γ,δ;0) parameterization (top left),
the "standard" S(α,β,γ,δ;1) parameterization (top right),
S(α,β,γ,δ;2) parameterization (bottom left),
S(α,β,γ,δ;3) parameterization (bottom right).
The values of α are indicated on the
plots, skewness is indicated by color:
β=0 (black), β=0.25 (red), β=0.5
(green), β=0.75 (yellow), β=1 (blue).
In all cases, scale γ=1 and location δ=0.
Note the discontinuity in the standard 1-parameterization near alpha=1.
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Proper use of this material
I put these papers and programs
up for free access; please do not abuse this.
I have come across examples where people use text from the papers below and/or
the programs below without giving proper acknowledgement. If you quote something from my work
(or from anyone), you should state it as a quote and give a reference. If you avoid a quote
by paraphrasing, you should still give an acknowledgement.
Likewise, if you use one of these programs, acknowledge it in your publication. Something like
"The program STABLE is available from J. P. Nolan's website: academic2.american.edu/~jpnolan".
Papers on stable distributions
An
introduction to stable distributions (PDF format, 0.4 mb).
This is a draft version of Chapter 1 in a forthcoming book on stable
distributions. It describes one dimensional stable distributions in a
non-technical way, explaining the two main parameterizations of stable laws.
I would appreciate any comments you have on this chapter - what you find
useful, what's missing, spelling/grammar mistakes, etc.
(NOTE: THIS BOOK IS STILL BEING WRITTEN.
Even though amazon.com
lists the book, it is not available yet, do not order it from them (or anyone else).
I will put information on the book here when it is complete.)
If you want to refer to this document, use the following reference (in BibTeX format)
@book{nolan:2010,
author = {J. P. Nolan},
title = {Stable Distributions - Models for Heavy Tailed Data},
year = {2010},
publisher = {Birkh\"auser},
address = {Boston},
note = {In progress, Chapter 1 online at academic2.american.edu/$\sim$jpnolan}
}
StableBibliography.pdf
A list of papers and books that relate to stable distributions. It is in PDF format and approximately
0.4 mb. If you have corrections or additions to this, please let me know. BibTeX format is preferred
for all additions, required if you want to add more than a few entries.
A projection approach to multivariate stable laws
These are the slides from a talk at the XXVIII International Seminar on Stability Problems for Stochastic Models, Zakopane, Poland,
May 31 - June 5, 2009. (0.9 mb)
Advances in nonlinear signal processing This is a recent paper on using stable distributions in signal processing. It shows that significant improvement can be made in the presence of heavy tailed noise. (0.54 mb)
Modeling financial data with stable distributions This is a 2002 paper on using stable distributions in finance (with some later corrections). It describe basic analysis of financial data and examines some real data for the presence of heavy tails. (0.57 mb)
Multivariate elliptically contoured stable distributions: theory and estimation This is a 2006 paper on multivariate stable distributions that are elliptically contoured, including radial symmetry. The examples are in finance - analyzing multiple heavy tailed assets, but can also be used in engineering problems. (0.32 mb)
mle.pdf
Moderately technical document that describes a program for maximum
likelihood estimation of parameters for general (non-symmetric) stable
distributions. (0.4 mb)
DataAnalysis.pdf
Analyzing data using stable distributions - includes estimation, diagnostics, examples.
Paper from Heavy Tails Conference (50 pages with many graphs, 0.7 mb).
overview.pdf
Overview article on multivariate stable distributions.
tails.pdf
Technical paper that examines the tail behavior of stable distributions. In
particular, the issue of when the asymptotic Pareto behavior is an accurate
estimate of the density or distribution function is examined. It is shown
that the degree of accuracy is highly dependent on the parameters alpha and
beta and unless alpha is small, say less than 1.2, the Pareto behavior does
not occur until very far out on the tail. This implies that tail estimators
for alpha are likely to be of limited value for stable distributions. Also
included are some results on the mode of stable distributions and quantiles
are discussed (see the data file quantiles.dat below). (1 mb)
spectral.pdf
Estimation of stable spectral measures, by Nolan, Panorska and
McCulloch.
STABLE program for Windows
The program is "stable.exe", instructions for
use are in "stable.txt". If you have problems running the program, you may need to
download the program again. (Depending on your browser settings, you may have
to right click on the links and use the "Save as..." or "Save target as..."
menu choices.) If you run the program under Windows 2000 or Windows XP
Professional, you will need to have write access to the directory in which you
are running stable.exe.
stable.exe
- calculates stable densities, cumulative distribution functions and
quantiles. Also includes stable random number generation and maximum
likelihood estimation of stable parameters using a fast 3-dimensional cubic
spline interpolation of stable densities. (Version 3.14.02, February 2005, 1.4 mb)
stable.txt
- description of stable program and instructions for use. You can read this
ascii file online, or use the "Save as..." feature of your browser to save
it to your local disk (16 KB).
stablec.exe
- a command line version of stable.exe with no windows and no graphics.
MVSTABLE program for Windows
(Version 2.0, 15 March
2000). Calculates multivariate stable densities, simulates random vectors and
fits multivariate data.
mvstable.txt
description of the mvstable program. You can read this online or use the
"Save as" feature of your browser.
Stable quantiles
quantile.dat
Table of stable quantiles (0.7 mb). Use "Save as" feature of your browser
to save to local disk if desired. This extensive table lists quantiles of a
general stable distribution for alpha=0.1,0.2,...,1.9,1.95,1.99,2.0 and
beta=0,0.1,0.2,...,0.9,1. (Symmetry can be used for negative beta). The
quantiles are tabulated for p-values of 0.00001 to 0.00010 (step 0.00001),
0.0001 to 0.0010 (step 0.0001), 0.001 to 0.010 (step 0.001), 0.01 to 0.10
(step 0.005), 0.10 to 0.90 (step 0.01), and upper tail similar to lower tail.
This table was computed with modifications to the program STABLE that increase
the accuracy significantly.
mathestate
Web site for working on quantitative real estate problems, including some sections on stable distributions.
For learning and theoretical experimentation purposes, choose
Tools -> Tutorial Tools -> Tool #4 “Risk, Variation, and Tail Behavior”.
For estimating stable parameters from your own data set,
choose Tools -> Hands On Tools -> Tool #7 “Stable Data Analysis”.
Obtaining financial data.
The easiest place to get free financial data is
Yahoo Finance.
Go to the "Investing" tab and click on "stocks". On that page,
click on "historical quotes" under the Research Tools heading.
On this page you can pick a stock (you will need to know the stock's
abbreviation, e.g. GE=General Electric, MSFT=Microsoft, etc.).
You can get historical data over a time period by selecting
start and stop dates. If you click on "Get data", you will see the data
on the screen and can cut and paste.
You can also download the data to a file by clicking on "Download to a spreadsheet";
the data will be saved as a .csv file, which you can edit with notepad
or some other editor. Suggestions: Always use the last column, which is adjusted closing price.
And always compute the return or log return before analyzing the data; don't just look at price.
For questions, please use the form above.
