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Page 1: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

Maximum likelihood estimation and

diagnostics for stable distributions

John P� Nolan

ABSTRACT A program for maximum likelihood estimation of general

stable parameters is described� The Fisher information matrix is computed�

making large sample estimation of stable parameters a practical tool� In

addition� diagnostics are developed for assessing the stability of a data

set� Applications to simulated data� stock price data� foreign exchange rate

data� radar data and ocean wave energy are presented�

� Introduction

Stable distributions are a rich class of distributions that include the Gaus�sian and Cauchy distributions in a family that allows skewness and heavytails� The class was characterized by Paul L�evy ������ in his study of nor�malized sums of i�i�d� terms� The general stable distribution is describedby four parameters an index of stability � � �� ��� a skewness parame�ter �� a scale parameter � and a location parameter �� There are multipleparameterizations for stable laws and much confusion has been caused bythese di erent parameterizations� The lack of closed formulas for densitiesand distribution functions for all but a few stable distributions �Gaussian�Cauchy and L�evy� has been a major drawback to the use of stable distribu�tions by practitioners� This paper shows that the computational problemshave now been resolved and it is feasible to �t stable models to data andto use diagnostics to assess the goodness of �t�

Stable distributions have been proposed as a model for many types ofphysical and economic systems� There are several reasons for using a stabledistribution to describe a system� The �rst is where there are solid theoret�ical reasons for expecting a non�Gaussian stable model� e�g� re�ection o a rotating mirror yielding a Cauchy distribution� hitting times for a Brow�nian motion yielding a L�evy distribution� the gravitational �eld of starsyielding the Holtsmark distribution� see Feller ������ for these and other

Page 2: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

examples� The second reason is the Generalized Central Limit Theoremwhich states that the only possible non�trivial limit of normalized sums ofi�i�d� terms is stable� It has been argued that many observed quantities arethe sum of many small terms � the price of a stock� the noise in a commu�nication system� etc� and hence a stable model should be used to describesuch systems� The third argument for modeling with stable distributionsis empirical many large data sets exhibit heavy tails and skewness� Thestrong empirical evidence for these features combined with the GeneralizedCentral Limit Theorem is used by many to justify the use of stable models�Examples in �nance and economics are given in Mandelbrot ������� Fama������� Embrechts� Kl�uppelberg� and Mikosch ������� Cheng and Rachev������� McCulloch ������� in telecommunication systems by Stuck andKleiner ������� Zolotarev ������� Willinger� Taqqu� Sherman and Wilson������� and Nikias and Shao ������� Such data sets are poorly describedby a Gaussian model� but possibly can be described by a stable distribu�tion� Several recent monographs focus on stable models Zolotarev �������Christoph and Wolf ������� Samorodnitsky and Taqqu ������� Janicki andWeron ������� and Nikias and Shao ������� The related topic of modelingwith heavy tailed distributions is discussed in the books by Embrechts�Kl�uppelberg and Mikosch ������ and Adler� Feldman and Taqqu �������

Skeptics of stable models recoil from the implicit assumption of in�nitevariance in the non�Gaussian stable model and have proposed other modelsfor observed heavy tailed and skewed data sets� e�g� mixture models� timevarying variances� etc� Such models can have very heavy tails� see x��� ofEmbrechts� Kl�uppelberg and Mikosch for a discussion of the heavy tailedbehavior of ARCH and GARCH models with normal innovations� The samepeople who argue that the population is inherently bounded and thereforemust have a �nite variance� routinely use the normal distribution � withunbounded support � as a model for this same population� The variance isbut one measure of spread for a distribution� and it is not appropriate forall problems� From an applied point of view� what we generally care aboutis capturing the shape of a distribution�

We propose that the practitioner approaches this dispute as an agnostic�The fact is that until now we have not really been able to compare data setsto a proposed stable model� In this paper we show that maximum likelihoodestimation of all four stable parameters is feasible� even for large data sets�And perhaps just as important� it is now feasible to use diagnostics to assesswhether a stable model accurately describes the data� In some cases thereare solid theoretical reasons for believing that a stable model is appropriate�in other cases we will be pragmatic if a stable distribution describes thedata accurately and parsimoniously with four parameters� then we accept

Page 3: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

it as a model for the observed data�This paper is organized in the following way� The remainder of this sec�

tion describes some parameterizations for stable distributions and somebasic properties� then we discuss previous work on methods of estimatingstable parameters� Section � describes our program to do maximum likeli�hood �ML� estimation for all four stable parameters� In addition� the Fisherinformation matrix is computed for a grid of parameter values� so large sam�ple con�dence interval estimates for the parameters can be made� Section� discusses diagnostics for assessing whether a data set is stable or not�The program STABLE will perform the estimation and diagnostics and isavailable on the Web at http���www�cas�american�edu��jpnolan� andclicking on the link to stable distributions� Examples of stable ML estima�tion for several data sets are given in Section �� Finally� we give a discussionof our results in Section ��

��� Parameterizations and basic properties

There are at least half a dozen di erent parameterizations of stable dis�tributions� All involve di erent speci�cations of the characteristic functionand are useful for various technical reasons� The parameterization mostoften used now� e�g� Samorodnitsky and Taqqu ������� is the followingX � S��� �� �� ��� �� if the characteristic function of X is given by

E exp�itX� ��������

exp����jtj�

h�� i��tan ��

���sign t�

i� i��t

�� �� �

exp���jtj

h� � i� �

��sign t� ln jtj

i� i��t

�� � ��

The range of parameters are � � � �� �� � � � �� scale � � � andlocation �� � R� �We prefer not to use � for the scale parameter� sincevariances do not exist unless � � �� and even when � � �� the standardstable scale parameter is not the standard deviation� Likewise� we prefernot to use � for the location parameter� because means do not always existand even when they do� the location parameter and the mean di er in someparameterizations��

A more useful parameterization in appplications is a variation of the�M� parameterization of Zolotarev we will say X � S��� �� �� ��� � if thecharacteristic function of X is given by

E exp�itX� �

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-3 -2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

alpha=0.50alpha=0.75alpha=1.00alpha=1.25alpha=1.50

FIGURE �� Stable densities in the S��� �� �� ��� � parameterization� � ����

� �� �� �� and � as indicated��������

exp����jtj�

h� � i��tan ��

���sign t����jtj���� � ��

i� i��t

�� �� �

exp���jtj

h� � i� �

��sign t��ln jtj� ln ��

i� i��t

�� � ��

The value of this representation is that the characteristic functions �andhence the corresponding densities and d�f�� are jointly continuous in all fourparameters� Accurate numerical calculations of the corresponding densitiesshow that in this representation � and � have a much clearer meaningas measures of the heaviness of the tails and skewness parameters� seeFigure �� In contrast� in the standard parameterization� the mode of X �S��� �� �� ��� �� with � �� tends to �sign ��� as � � �� is near �� when� � �� and tends to ��sign ��� as � � ��

The parameters �� � and � have the same meaning for the two param�eterizations� while the location parameters of the two representations arerelated by �� � �� � ��tan ��

��� if � �� �� �� � �� � � �

�� ln � when � � ��

The particular form of the characteristic function was chosen to makethe S��� �� �� ��� � parameterization a location and scale family if Y �S��� �� �� ��� �� then for any a �� � b� aY � b � S��� �sign a��� jaj�� a�� �b� �� We will base the likelihood calculations below on the S��� �� �� ��� �parameterization because it is the simplest scale�location parameterizationwhich is jointly continuous in all four parameters� Some authors sidestepthe discontinuity at � � � by saying that the probability that � � � iszero therefore you can ignore it� Buckle ������ assumes that you know be�forehand that either � � � or � � � and restricts his prior for � to theappropriate interval� The shape of the data is what we really care about�and that is similar when � is near or at �� the standard parameterizationssimply masks that with a shift� Clearly it is preferable to let the data de�termine what � is and not make assumptions about the parameters� even if

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� is not near �� Finally� the use of the S��� �� �� ��� � parameterization hasthe technical advantage of reducing the correlation between the parameterestimates� especially when � is near �� More information on parameteriza�tions� modes of stable densities and generalizations to multivariate stablelaws can be found in Nolan �������

Basic properties of stable distributions can be found in Samorodnitskyand Taqqu ������� Some of the prominent properties are heavy tails thatare asymptotically Pareto �see equation ��� below�� possible skewness ofthe distributions� and smooth unimodal densities with no closed formula�Let f�xj�� �� �� ��� be the density of a S��� �� �� ��� � distribution� Knownfacts about stable densities in the standard parameterization show thatf�xj����� �� ��� � f��xj�� �� ������ and

support f�xj�� �� �� ��� �

�����

��� � � tan ������ � � � and � � �

���� �� � � tan ���

� � � � and � � ��������� otherwise�

Note that for a totally skewed �� � ��� distribution when � � �� the �niteendpoint of the support goes to �sign ��� as � � �� It can be shown thatthe mode and most of the distribution stays concentrated near ��� so thatonly a very small probability is far out on that tail� In fact� the light tailin the totally skewed cases decays faster than Pareto by ����

� Other estimators of stable parameters

Several methods have been proposed for estimating stable parameters� Forthe index of stability �� the earliest approach is just to plot the empiricaldistribution function of observed data on a log�log scale� It is well known�e�g� Samorodnitsky and Taqqu ������� pg� ��� that the asymptotic tailbehavior of stable laws is Pareto when � � �� i�e�

limx��

x�P �X � x� � c��� � ���� ���

�For this reason� the phrase stable Paretian is sometimes used for non�Gaussian stable distributions�� Thus the tail of the empirical distributionfunction on a log�log scale should approach a straight line with slope ��if the data is stable� While simple and direct in principle� this method isunreliable in practice� The main problem is that it has not been knownwhen the Pareto tail behavior actually occurs� McCulloch ������ showsthat using the generalized Pareto model suggested by DuMouchel ������ orthe Hill estimator on stable data when � � � � � leads to overestimates of

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�� McCulloch points out that several researchers have used such misleadingtail estimators of � to conclude that various data sets were not stable� InFofack and Nolan ������� it is shown that where the Pareto behavior startsto occur depends heavily on the parameterization� and that even when weshift so that the mode is near or at zero� the place where the power decaystarts to be accurate is a complicated function of � and �� In particular�when � is close to �� one must get extremely far out on the tail before thepower decay is accurate�

A second approach to estimating stable parameters is based on quantilesof stable distributions� Fama and Roll ������ noticed certain patterns intabulated quantiles of stable distributions and proposed estimates of ��scale � and location � for symmetric �� � � stable distributions� Mc�Culloch ������ extended these ideas to the general �nonsymmetric� case�eliminated the bias and obtained consistent estimators for all four stableparameters in terms of �ve sample quantiles �the �th� ��th� �th� ��th and��th percentiles�� If the data set is stable and if the sample set is large�this last method gives reliable estimates of stable parameters�

Since closed forms are known for the characteristic functions of stablelaws� several researchers have based estimates on the empirical character�istic functions� Press ������ seems to have been the �rst to do this� Sev�eral modi�cations have been made to this approach� see Paulson� Holcomband Leitch ������� Feuerverger and McDunnough �����a� and �����b��Koutrouvelis ����� and ������� Kogon and Williams �������

For symmetric stable distributions �� � � � �� Nikias and Shao ������estimate parameters using fractional and negative moments� They also de�scribe a method based on log jXj�

Maximum likelihood estimation has been done in certain cases� Whilenot easily accessible� DuMouchel ������ gives a wealth of information onestimating stable parameters at a remarkably early date� An approximatemaximum likelihood method was developed based on grouping the dataset into bins� and using a combination of means to compute the density�the fast Fourier transform for central values of x and series expansionsfor tails� to compute an approximate log likelihood function� This functionwas then numerically maximized� See also DuMouchel �����a�� �����b�������� and ������� For the special case of ML estimation for symmetricstable distributions� see Brorsen and Yang ����� and McCulloch �������Finally� Brant ������ proposes a method for approximating the likelihoodusing the characteristic function directly�

A Bayesian approach using Monte Carlo Markov chain methods was pro�posed by Buckle ������� The papers by Akgiray and Lamoureux ������ andKogon and Williams ������ give simulation based comparison of several of

Page 7: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

the methods described above�

� Maximum Likelihood Estimation

As stated above� we will use the S��� �� �� ��� � parameterization in whatfollows� To simplify notation in this section� we denote the parameter vectorby � � ��� �� �� ��� and the density by f�xj��� The parameter space is� � �� �� ���� �� ���� ������� The log likelihood function for ani�i�d� stable sample X�� � Xn is given by

���� �nX

i��

log f�Xij��

Of course the di�culty in evaluating this is that there are no known closedformulas for general stable densities� Zolotarev ������ details the much ofwhat is known about stable densities� When � is a special kind of rationalnumber� there are expressions for densities in terms of special functions�Fresnel integrals� MacDonald functions� Whittaker functions�� Puttingaside the computational di�culties of evaluating the required special func�tions� knowing densities at isolated values of the parameter space is nothelpful when one is trying to maximize a likelihood� Ho man�J�rgensen������ expressed the general density in terms of what he called �incom�plete hypergeometric� functions and Zolotarev ������ expressed the gen�eral density in terms of Meijer G�functions� Unfortunately� neither of theserepresentations are practical for actually evaluating stable densities�

The program STABLE described in Nolan ������ gives reliable compu�tations of stable densities for values of � � � and any �� � and ��� Thatprogram was improved to give more accurate density calculations on thetails� which we found to be necessary for accurate likelihood calculations�The program now includes a fast� pre�computed spline approximation tostable densities for � �� routines for maximum likelihood estimation ofstable parameters� and diagnostics for assessing the stablity of a data set�

The time required to �t a data set of n � �� points with a stablemodel is approximately three seconds on a ��� MHz Pentium II computerusing an approximate gradient based search to maximize the likelihood�The quantile estimator of McCulloch ������ is used as an initial approxi�mation to the parameters and then a constrained �by the parameter space�quasi�Newton method is used to maximize�

Page 8: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

��� Asymptotic normality and Fisher information matrix

DuMouchel ������ and �����a� show that when �� is on the interior ofthe parameter space �� the ML estimator follows the standard theory� soit is consistent and asymptotically normal with mean �� and covariancematrix given by n��B� where B � �bij� is the inverse of the � � Fisherinformation matrix I� The entries of I are given by

Iij �Z�

��

f

�i

f

�j

fdx

We have written a program to numerically compute these integrals� Itcomputes the density f to high accuracy� then numerically computes thepartials� The resulting values for the integrands are then numerically inte�grated�

When � is near the boundary of the parameter space the �nite samplebehavior of the estimators is not precisely known� Intuitively� the distribu�tion of the estimator gets skewed away from the boundary� When � is onthe boundary of the parameter space� i�e� � � � or � � ��� the asymptoticnormal distribution for the estimators tends to a degenerate distributionat the boundary point and the ML estimators are super�e�cient� See Du�Mouchel ������ for more information on these cases�

The general theory shows that away from the boundary of �� large sam�ple con�dence intervals for each of the parameters are given by

��i � z������ipn�

where ���� � � ���� are the square roots of the diagonal entries of B� Thevalues of ���i � i � �� � � have been computed and are plotted in Figure

� when � � � and �� � � The correlation coe�cients �ij � bij�qbiibjj�

have also been computed and are plotted in Figure �� These values aretabulated in the Appendix on a grid of �� � values� When � � � thestandard deviations are the same as for j�j and the correlation coe�cientsare expressed in terms of the � � case as ����i�j�ij� For a general scale� and location ��� ���� ��� and �ij are unchanged� but ��� and ���� are � timesthe tabulated value�

The grid values were chosen to give a spread over the parameter spaceand show behavior near the boundary of the parameter space � � � andj�j � �� �Because of computation di�culties� these values have not beentabulated for � � ����� We note that when � � � stable densities aresymmetric and all the correlation coe�cients involving � are � When � ��� ��� � and all the correlation coe�cients involving � are unde�ned�

Page 9: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0

02

46

810

beta=0beta=.5beta=.9beta=1

0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

FIGURE �� Graphs of the standard deviations of the estimators ��j as a function

of �� � indicated by line type� � � and �� �� Upper left is ���� upper right is

���� lower left is ��� � lower right is ���� �

Page 10: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

��

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

FIGURE �� Graphs of the correlation of the estimators �ij as a function of ��� indicated by line type �as in previous �gure� � � and �� �� Upper left

plot is ����� upper right plot is ���� � middle left plot is ����� � middle right plot

is ���� � lower left plot is ����� � lower right plot is ����� �

Page 11: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

��

The correlation coe�cients directly tell how the parameter estimates arerelated� and they are useful in estimating the information matrix for asubset of the parameters express the full covariance matrix in terms of the��i s and �ij s� invert to get the full information matrix� delete the rowsand columns corresponding to the known parameters� and reinvert to getthe covariance matrix for the remaining parameters�

A few of these values have been given in DuMouchel ������ pg� ���� For� not near �� most of the values given there agree with our results� Thatauthor uses the S��� �� �� ��� �� parameterization� so near � � �� one wouldexpect di erent values and numerical problems�

Some general observations about accuracy of parameter estimates cannow be made� The parameter of most interest is usually �� Twice thestandard error of ��� �SE���� � �����

pn� is plotted in Figure � for � �

� � �� n��� � and � and � � � � � �� �The graphs for �j�j � � are between the given ones�� Unless � is close to �� it is clear that alarge data set will be necessary to get small con�dence intervals� e�g� when� � �� and � � � sample sizes of �� � and � yields SE���� s of����� �� and ���� respectively� Since no other estimation method isasymptotically more e�cient than ML� any other method of estimating �will likely yield larger con�dence intervals� In contrast� when � � �� SE����approaches � Similar calculations of standard errors for ��� �� and ��� alsoshow that large samples will generally be necessary for small con�denceintervals� As an extreme� as � � �� SE� ��� � �� In practice� this is oflittle import because � means little as � � ��

� Diagnostics for assessing stability

In principle� it is not surprising that one can �t a data set better withthe � parameter stable model than with the � parameter normal model�The relevant question is whether or not the stable �t actually describes thedata well� Any procedure for estimating stable parameters will �nd a �best�t� by its criteria the maximum likelihood approach maximizes ���� di�rectly� the quantile methods try to match certain data quantiles with thoseof stable distributions� the characteristic function based methods �t theempirical characteristic function� All will give some values for parameterestimates� even if the shape of the observed distribution is not similar tothe �tted distribution� e�g� the data is multi�modal� has gaps in its support�etc� It is necessary to have some means of assessing whether the resulting�t is reasonable�

The use of a diagnostic depends on what you are planning to do with a

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��

alpha

2*SE

(alph

a)

0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

beta=0beta=1

n=100

n=1000

n=10000

FIGURE �� Graph of twice the standard error of �� as a function of � for various

sample sizes�

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��

data set� For testing residuals from a regression analysis� departures fromnormality around the center of the distribution are usually not important�outliers are important because they can a ect the validity of normal theoryconclusions� In the re�insurance �eld� one is only concerned with extremeevents and there one wants to estimate tails of the claim distribution asaccurately as possible� In a model of stock prices or exchange rates� onemay be interested in the shape of the whole distribution�

While non�Gaussian stable distributions are heavy�tailed distributions�most heavy�tailed distributions are not stable� One can try to �t a heavy�tailed data set with a stable distribution� but it is inappropriate in manycases� As DuMouchel ������ points out� making a statement about the tailsis quite distinct from making a statement about the entire distribution� Weamplify this point by an example similar to one used by DuMouchel� De�nefor � � � �� x� �

g�x� � g�xj�� x�� �

�c�e

�x��� jxj � x�c�jxj������ jxj � x��

where c� and c� are chosen to make g continuous andRg�x�dx � � c� �

c���� x�� � �p

����!�x�����������x� exp��x��������� c� � c� exp��x�����x���� � A random variable X with density g has a normal density in the

interval �x� � x � x�� a Pareto tail� and has fraction p � P �jXj � x�� �c�p

����!�x�� � �� in the normal part of the density and � � p on thePareto tails� For any �nite x�� this density has in�nite variance and is inthe domain of attraction of a symmetric stable distribution with index ofstability �� Suppose we observe a sample of size n from such a distributionand try to �t it with a stable distribution� If �� � p�n is small� we willlikely have few observations from the Pareto part of the distribution andwe will not be able to detect the heavy tails� Any reasonable estimationscheme would lead to an �� � �� On the other hand� if �� � p�n is large�then one would get an �� intermediate between the true � and �� becausethe central part of the data is coming from a non�heavy tailed density� Anincorrect model is being �t to the data� so it is no surprise that we get the�wrong� �� DuMouchel s argument to let the tails speak for themselves issound� though his suggestion to use the upper �" of the sample to �tthe tail is generally not appropriate� see McCulloch ������ and Fofack andNolan ������� We mentioned above that the eventual power decay on astable tail may take a long time to occur� for an arbitrary distribution�there is no general statement that can be made about what fraction of thetail is appropriate� �For a recent summary of work on tail estimation� seeBeirlant� Vynckier and Teugels ��������

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��

The diagnostics we are about to discuss are an attempt to detect non�stability� As with any other family of distributions� it is not possible toprove that a given data set is or is not stable� We note that even testing fornormality is still an active �eld of research� e�g� Brown and Hettmansperger������� The best we can do is determine whether or not the data are con�sistent with the hypothesis of stability� These tests will fail if the departurefrom stability is small or occurs in an unobserved part of the range�

The �rst step we suggest is to do a smoothed density plot of the data� Ifthere are clear multiple modes or gaps in the support� then the data cannotcome from a stable distribution� If the smoothed density is plausibly stable�proceed with a stable �t and compare the �tted distribution with the datausing EDA �Exploratory Data Analysis� techniques�

We note a practical problem with q�q plots for heavy tailed data� Whileusing q�q plots to compare simulated stable data sets with the exact cor�responding cumulative d�f�� we routinely had two problems with extremevalues most of the data is visually compressed to a small region and thehigh tail variability leads one to doubt the stability of the data set� Toillustrate this point� we simulated a stable data set with n � �� valuesusing the Chambers� Mallows and Stuck ������ method� The values of theparameters used were � � ��� � � �� � � �� and �� � �� �This samedata set is �t using maximum likelihood in the next section�� Figure � �left�shows a standard q�q plot of the data vs� the known stable distribution�On the tails there seems to be an unacceptably large amount of �uctuationaround the theoretical straight line� For heavy tailed stable distributions�we should expect such �uctuations for the following reason� If X�i� is the ith

order statistic from an i�i�d� stable sample of size n� p � �i����n and andxp is the pth percentile� then for n large� the distribution of X�i� is approx�imately normal with EX�i� � xp and Var�X�i�� � p��� p��nf�xp�

�� Figure� �left� also shows pointwise ��" con�dence bounds around the expectedvalue� A heavy tailed distribution should show much larger extremes than anormal sample� e�g� values in the hundreds for this example� Furthermore�the standard errors for the extreme values are also very large� The max�imum in this sample was X������ � ������ the corresponding populationquantile is x����� � ������ with ��" con�dence interval ������������ Insum� q�q plots will generally appear non�linear on the tails� even when thedata set is stable�

One technique we tried to lessen this e ect was to use a �thinned� q�qplot on large data sets� We illustrate the idea in Figure � �right�� whereonly � values �of the �� in the sample� are plotted� This gives a pointat every �" of the data� eliminating the most extreme values� While thismethod eliminates the worst behavior on the tails� information is lost and

Page 15: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

o

o o

o

-500 0 500

-5000

500

o

o

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-100 -50 0 50 100 150 200

-100-50

050

100150

200FIGURE �� Diagnostics for simulated stable data set with n �� ��� data points

and � ���� � ���� � � and �� ��� Left graph is a q�q plot for data vs�

exact� right is a thinned q�q plot obtained by using every ��th value of theoriginal�

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-40 -20 0 20 40

0.00.01

0.020.03

0.040.05

0.06

exactdatanormal

FIGURE �� Simulated stable data set with n����� Left graph is a stabilized

p�p plot of data vs� exact S����� ���� �� �� distribution� right graph shows the

smoothed data density �dots and �tted density �solid and normal �t �dashes�

Page 16: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

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the con�dence intervals for sample quantiles can still be huge� especiallywhen � gets small�

Our proposed solution is to use a modi�ed p�p plot� Standard p�p plotstend to emphasize behavior around the mode of the distribution� wherethey have more variation� and necessarily pinch the curve near the tails�In Michael ������� a �stabilized� p�p plot was de�ned that eliminates thisnon�uniformity by using a transformation� �The word stabilized refers tomaking the variance in the p�p plot uniform� and has nothing to do withstable distributions�� The result is that the acceptance regions for a p�pplot become straight lines spaced a uniform distance above and below thediagonal� A stabilized p�p plot for the simulated data is shown in Figure ��left��

For density plots� we smoothed the data with a Gaussian kernel withstandard deviation given by a �width� parameter� We found that the com�monly suggested width of ��inter�quartile range�n��� works reasonablywhen the tails of the data are not too heavy� say � � ��� but works poorlyfor heavier tailed data� For such cases� we used trial and error to �nd awidth parameter that was a small as possible without showing oscillationsfrom individual data points� The density plots give a good sense of whetherthe �t matches the data near the mode of the distribution� but generally isuninformative on the tails where both the �tted density and the smootheddata density are small� Figure � �right� shows the smoothed data densityfrom the simulated data� the exact population density and a normal �t �us�ing the sample mean and sample standard deviation�� We note that boththe skewness and the leptokurtosis �a higher thin peak and heavy tails� inthe data set are poorly described by a normal �t�

Finally� we tried comparing distribution functions� but did not �nd itvery helpful� Because of the curvature in the distribution functions� it ishard to compare the �tted and empirical d�f� visually� especially on thetails�

� Applications

��� Simulated stable data set

A stable data set with � � ��� � � �� � � � and �� � � and n � �� values was generated using the method of Chambers� Mallows and Stuck������� The quantile estimators of the parameters are �� � ���� �� � ������ � ���� and ��� � ����� The maximum likelihood estimates with ��"con�dence intervals are �� � �������� �� � ������� �� � ������������ � ���� � ���� The stabilized p�p plot and smoothed density are

Page 17: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

country � � � ��Australia ������ ����� ��� � ��� � ������ � ������ ������� � �������Austria ���� ����� ������ � ����� ����� � ������� ������� � ������Belgium ����� ����� ������ � ��� � ������ � ������� ������� � �������Canada ����� ����� ����� � ���� ������ � ������� ������� � �������Denmark ���� ����� ������ � ����� ������� � ������ ������� � �������France ��� � ����� ������ � ���� ������ � ������ ������ � ������Germany ����� ����� ���� � � ��� ������� � ������ ������� � ������Italy ������ ����� ����� � ����� ������� � ������� ������� � �������Japan ����� ����� ����� � ��� � ����� � ������� ������ � �������Netherlands ������ ����� ������ � ��� � ������� � ������ ������� � ������Norway ��� ����� ������ � ��� ����� � ������ ������ � ������Spain ����� ����� ������ � ��� ������ � ������ ������� � �������Sweden ����� ����� ���� � � ��� ������ � ������ ������� � ������Switzerland ����� ����� ������ � ����� ������ � ������� ������� � �������United States ���� ����� ���� � ��� ������ � ������� ������� � �������

TABLE �� Exchange rate analysis� Parameter estimates and ��� con�dence

intervals with sample size of n �� ��

visually indistinguishable from the ones in Figure �� so new diagnostics arenot shown�

��� Exchange rate data

Daily exchange rate data for �� di erent currencies were recorded �in U�K�Pounds� over a �� year period �� January ��� to �� May ������ The datawas transformed by yt � ln�xt���xt�� giving n � �� ��� data values� Thetransformed data was �t with a stable distribution� results are shown inTable �� The data are likely non�stationary over such a time period andthere are questions about the dependence in the values� nevertheless wewill do a naive �t here to illustrate the method�

Figure � shows a stabilized p�p plot and smoothed density for the Ger�man Mark data set� The data sets are clearly not normal the heavy tails inthe data causes the sample variance to be large� and the normal �t poorlydescribes both the center and the tails of the distribution� The granular�ity at the center of the graph is from the days where the exchange ratewas unchanged on successive days� As another measure of non�normality�the ratio of the stable �t log likelihood to the normal log likelihood wascomputed for each currency� The ratio of the log likelihoods for the MLstable �t to the normal �t were computed and the values ranged from ���to ����

Plots for the other currencies were similar� showing that the stable �tdoes a good job of describing the exchange rate data� We note in passingthat the currency with the heaviest tails ��� � ����� was the Italian Lire�while the one with the lightest tails ��� � ���� was the Swiss Franc�

Page 18: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

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ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-0.02 -0.01 0.0 0.01 0.02

020

4060

80100

120

FIGURE � Stabilized p�p and density plots for the German mark exchange rate

data� n�� ��

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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

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0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-20 -10 0 10 20

0.00.02

0.040.06

0.080.10

FIGURE �� Stabilized p�p plot and densities for the CRSP stock price data�

n����

��� CRSP stock prices

McCulloch ������ analyzed forty years �January ���� � December ����� ofmonthly stock price data from the Center for Research in Security Prices�CRSP�� The data set consists of �� values of the CRSP value�weightedstock index� including dividends� and adjusted for in�ation� The quantileestimates were �� � ����� �� � ��� �� � ���� and ��� � ���� McCul�loch �unpublished� used ML with an approximation for symmetric stabledistributions to �t this data and obtained �� � ����� �� � � �� � ����and ��� � ���� Our ML estimates with naive ��" con�dence intervalsare �� � ���� � ��� �� � ���� � ���� �� � ���� � ��� and��� � ���� ���� The diagnostics in Figure � show a close �t�

We note that the con�dence interval for �� is close to the upper boundof � for � and the one for �� is large and extends beyond the lower boundof ��� so the naive con�dence intervals cannot be strictly believed�

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method � � � ��quantile ����� ����� ��� �� ������MCMC ���� ��� ������� ����� �ML ��� � ���� ���� ��� ���� � � ������ ������� � ����

TABLE �� Abbey National share price parameter estimates� n ���

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0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-0.03 -0.01 0.01 0.03

010

2030

40

FIGURE �� Stabilized p�p plot and densities for Abbey share price data� n���

��� Abbey National share price

Buckle ������ listed a small data set of stock price data� The price forAbbey National shares was recorded for the period �� July ���� through� October ����� The return was de�ned as �xt���xt�� �� yielding n � ��data points� which were �t with a stable distribution� In the Monte CarloMarkov chain �MCMC� approach used by Buckle� the means of the pos�terior distributions were given� Table � lists these MCMC parameter esti�mates �transformed to the S��� �� �� ��� � parameterization�� the quantileestimates� and the ML estimates with naive ��" con�dence intervals�

The quantile method �t is essentially a normal distribution with � ������ yet highly skewed� This is likely caused by the small sample sizewith n � ��� the �th percentile is found by interpolating between the secondand third data point� It is hard to detect heavy tails when there is virtuallyno tail� The MCMC method and ML method reach similar estimates� Wetried the diagnostics on this data set and got mixed results� see Figure ��The data are concentrated on a subset of values and it is not clear how gooda stable model is for this small data set� In particular� healthy skepticismis called for when making statements about tail probabilities unless a largedata set is available to verify stable behavior�

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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-4 -2 0 2 4

0.00.2

0.40.6

0.8

FIGURE ��� Stabilized p�p plot and densities for the in�phase component of sea

clutter radar noise� n�������

��� Radar noise

This is a very large data set with n � ��� pairs of data points� Thetwo values correspond to the in�phase and quadrature components of seaclutter radar noise� We focus on the in�phase component only in this paper��Unpublished work shows that the other component is similar� and thatthe bivariate data set is radially symmetric�� The parameter estimates are�� � ����� � ��� �� � �� � ���� �� � ��� � �� and ��� ��� � ���� �The quantile based estimators are �� � ����� �� ���� �� � ���� and ��� � ���� With this large sample size� thecon�dence intervals for the ML parameter estimates are very small� Againthe correct question is not how tight the parameter estimates are� butwhether or not the �t accurately describes the data� The p�p plot anddensity plots in Figure � show a close stable �t� Because ��� datapoints add little to the p�p plot� we actually show a thinned p�p plot with�� values�

��� Ocean wave power

Pierce ������ proposed using positive ��stable distributions to model in�herently positive quantities such as energy or power� One example he usesis the power in ocean waves� which is proportional to the square of thewave height� Pierce used a National Oceanographic and Atmospheric Ad�ministration �NOAA� data set with hourly measurements of sea wave� Weused the same data set� edited out invalid numbers ����� and had ���values for the wave height variable WVHT� Pierce compared the data withan ������ � � � distribution �it is not indicated how these values areobtained�� Our analysis gave quantile estimates of �� � �������� �� � ��

Page 21: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

0 2 4 6 8 10

0.00.1

0.20.3

0.40.5

FIGURE ��� Stabilized p�p plot and densities for wave height squared �propor�

tional to power� n�����

�� � ������ and ��� � ������� the ML estimates with naive ��" con��dence intervals are ����� � ����� ���� � � ������� � ��� and�������� � ���� The fact that we get very di erent estimates of � isan indication that the data set is not stable� The diagnostics in Figure�� support this idea� The stabilized pp�plot and the density plot show areasonable �t around the mode� but a poor �t on both tails� As in anyproblem� it is possible that the energy in waves is stably distributed� butthat measurement of extremes �both high and low� of wave height areunreliable� leading to the discrepancies we see on the tails�

The referee kindly pointed out that there is recent work on the relatedtopic of wave heights and wind speed in de Haan and de Ronde �������

��� Simulated non�stable data

We simulated several data sets that were not stable and used our diagnos�tics to assess the �t with a stable model� The �rst is a data set consistingof a mixture of �� Gaussian random variables with scale � and ��Gaussian random variables with scale �� a �contaminated� normal mix�ture� The mixture has heavier tails than a pure normal� so one might try to�t it with a stable distribution� However� what we would really like to do isdetect that it is not a stably distributed data set� The ML estimates of theparameters are � � ����� � and � � ���� ��� Here the con�denceintervals are small because the sample size of n � �� � not because wehave a good �t� The density plot in Figure �� shows the smoothed datadensity and the stable �t� The curves show a systematic di erence thatindicates departure from a stable distribution� It is interesting to note thatin this example� the percentile estimate of � is ������ quite di erent fromthe ML estimate� This is another indication that the data is not stable if

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oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

-10 -5 0 5 10

0.00.05

0.100.15

0.200.25

FIGURE ��� Stabilized p�p plot and densities for simulated contaminated normal

mixture� n������

the distribution is stable� then all consistent estimators of the parametersshould be close when there is a large sample�

The next example is a mixture of two Cauchy distributions with di erentmodes � � �� � � � � � �� with �� � � for � data points and then�� � �� for another � data points� Simple diagnostics �not shown� showthe bimodality� so a stable model is clearly not appropriate� Still� it isinstructive to see what happens if we �t these data with a stable model�The maximum likelihood estimates are � � �� � � ����� and �� � ������ is meaningless in the normal case�� Apparently the likelihood for thisdata set is dominated by the central terms and is maximized by taking anormal curve with large variance� Even though this is a heavy tailed dataset� the use of an inappropriate stable model leads to a light tailed �t#

We brie�y mention two other experiments we did� In one experiment�� variables were generated from a Pareto distribution �F �x� � � �x����� x � �� with � � ��� The quantile and ML estimates of � were ����and �� respectively� � was essentially �� This shows that a stable �t to adata set with genuine Pareto tails will give poor estimates of the tail index�In the second experiment � Gamma��� variates were generated and �twith a stable distribution� The quantile and ML estimates of � were ����and ��� respectively� � was essentially �� This shows that the light tails ofthe Gamma distribution lead to estimates of � close to the Gaussian case�but the skewed nature of the data showed up in the estimate of ��

� Discussion

We have shown that ML estimation of general stable parameters is nowfeasible� The diagnostics show that several large data sets with heavy tails

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��

are well described by stable distributions� We also showed that stable mod�els are not a panacea � not all heavy tailed data sets can be well describedby stable distributions�

In practice� the decision to use a stable model should be based on thepurpose of the model� In cases where a large data set shows close agreementwith a stable �t� con�dent statements can be made about the population�In other cases� one should clearly not use a stable model� In intermediatecases� one could tentatively use a stable model as a descriptive method ofsummarizing the general shape of the distribution� but not try to makestatements about tail probabilities� In such problems� it may be better touse the quantile estimators rather than ML estimators� because the formertries to match the shape of the empirical distribution and ignores the topand bottom �" of the data�

We have not considered parameters that vary with time� mixture models�etc� While we do not do so here� it is possible to use an information criterialike AIC to compare a stable model to mixture models or GARCH mod�els for a data set� It seems likely that certain problems� e�g� the radar seaclutter problem� have physical explanations that make a stationary modelplausible� Other problems� particularly economic time series� may very wellhave time varying parameters that re�ect changes in the underlying condi�tions for that series� We cannot resolve this issue here� Our main purposeis to make stable models a practical tool that can be used and evaluatedby the statistical community�

We note that there are now several methods of estimation for multivari�ate stable distributions� In the multivariate setting one has to estimate ��a shift vector� and a spectral measure� For references and new work on thisproblem� see Nolan� Panorska and McCulloch ������� Nolan and Panorska������� and Nolan ������� One of those methods is based on estimation ofone dimensional stable parameters and would be improved with the quickML algorithm described here�

Acknowledgments� The data sets analyzed above were graciously provided byC� Kl�uppelberg �exchange rate data�� P� Tsakalides �radar data� and J�H� McCulloch �CRSP stock data�� R� Jernigan provided discussion andreferences on EDA techniques�

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� Appendix

Asymptotic standard deviations and correlation coe�cients for estimators�

� � � � � �� ��� ��� ���� ��� ���� ������ ��� ���� ����� �� �� ���� ���� ����� ���� ���� ��� ����

�� ���� ���� ����� ���� ���� ����� ����� ���� ��� ������� ��� ��� ��� � ����� ����� ��� ����� ���� �� � � ������ ��� ���� ����� ����� � ����� ���� � � � �

��� ��� ���� ���� ��� �� ���� ��� ���� ���� ��� ������ � ����� ����� ����� ���� ��� ���� ��� ���� ������ �� � ��� ����� ��� ���� ��� ����� ����� ���� � ����� ���� ���� ����� ���� � ��� ���� � � �

��� ��� ����� ���� ���� ����� ���� ���� ���� ���� ���� ������ ���� ����� �� � ���� ����� ���� ��� ���� ���� ������ � �� �� ���� ���� ���� ���� ���� ����� ��� ���

���� ��� ���� ����� ����� � ���� ���� � � �� �� � ��� ���� ��� ����� ����� ���� ����� ���� ���� ��� ����

�� ����� ��� ���� �� ���� ���� ���� ���� ���� ������ ��� �� � ����� ����� ��� ���� ���� ���� ��� �������� � � ���� ��� � ��� � ���� ���� � � ���

��� ��� ����� ����� ��� �� ���� ���� ���� ���� ����� ������ ����� ���� ���� ����� �� ��� ���� ���� ��� ������ ��� � ��� ��� ����� �� � ����� �� � ����� ����� ������� ���� ���� ��� ���� � ����� ��� � � �� �

���� ��� ���� ��� � ���� ��� ���� ��� ���� ���� ���� ������ ����� ���� ����� ��� ��� ��� ��� ����� ��� ������ ����� � �� ���� ��� ���� ���� �� ���� ���� ������� ����� ���� ��� ����� � ����� ��� � � ��

���� ��� ��� ����� �� �� ���� ���� ���� ���� ����� ������ ���� ����� ���� ����� ���� ��� ��� ��� ���� ������ ����� � �� ����� ���� ���� ��� ��� ���� ����� ������ ����� ���� ����� ����� � ���� �� � � � �

���� ��� ���� ����� ����� ���� ���� ���� ���� ���� ���� ������ ���� �� � ����� ��� ��� ���� �� ���� ��� ������ ��� ��� ����� ���� ���� ���� � � ���� ����� ������� ���� ���� ���� ���� � ���� ��� � � ��

��� ��� ���� �� ����� ���� ���� ���� ���� ���� ���� ������ ����� ����� ���� ����� ��� ���� �� ���� �� ��� ��� ���� ����� ���� ���� ���� ��� ���� ��� ����� �� ���� ����� ���� ���� ��� � ����� ��� � � ����

���� ��� ���� ����� ����� ���� ���� ��� ���� ���� ��� ������ ��� ���� ����� ���� ���� ���� �� ���� ���� ����� ����� ����� ����� ���� ��� ���� ��� ��� ���� �������� ���� ���� �� ���� � ���� ���� � � ����

��� ��� ��� ��� ��� � ���� ���� ��� ���� ���� ��� � ������ �� ����� ���� ��� ��� ���� ��� ���� ���� ������� ���� ���� ���� ���� ���� ��� ��� ��� ���� ������� ����� ���� ���� ��� � �� �� � � ���

���� ��� �� �� ����� ����� ���� �� ���� ���� ���� ������ ��� ���� ����� ���� ��� ��� ���� ���� ����� ������� ����� �� � ��� ����� ���� ��� ��� ��� ���� ������ ����� ���� � �� ��� � ��� ��� � � ��

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� � � � � �� ��� ��� ���� ��� ���� �������� ��� ���� ��� ��� ����� ���� ��� ���� ���� ���� ����

�� ����� ��� ���� ���� ���� ��� ��� ��� ���� ������� ���� ���� � � ���� ���� ���� �� �� � ���� �������� ����� ���� � �� ���� � ���� ��� � � ���

�� � ��� ���� �� � ���� ����� ���� ��� ���� ���� ���� ������ ��� �� � �� ����� ���� ��� ��� ��� ���� ������� ���� �� � � � ��� �� � �� ���� �� � ����� ������� ��� � ���� � �� ���� � ���� �� � � ����

���� ��� ���� ��� � � ��� ���� � ���� ���� ��� ������ ����� ��� � �� ��� � ���� �� ���� ��� ����� �� ���� ��� ��� � �� ����� ���� ��� �� ��� ���� ������� ���� ���� ���� ����� � �� ���� � � ��

��� ��� � � ��� � ���� ����� ���� ��� ���� ���� ����� ������ � � ����� �� � ���� ���� ���� ���� ���� �� � ������ �� � ���� ���� ����� ���� ��� �� ��� ����� �������� ��� ���� �� ���� � ���� ��� � � ����

���� ��� �� � ����� �� ��� ���� ���� ���� ���� ��� ������ ���� ����� ��� ��� ���� ���� ��� ��� ��� � ������ �� ����� ��� ���� ��� �� � �� ��� ���� �� ���� ��� ���� ��� ���� � ��� ��� � � ����

���� � ���� � ���� ���� � � � � � ����

� References

��� Adler� R�� Feldman� R� and Taqqu� M� �eds�� ������� A PracticalGuide to Heavy Tailed Data� Birkh�auser� Boston� MA�

��� Akgiray� V� and Lamoureux� C� G� ������� Estimation of stable lawparameters a comparative study� J� Bus� Econ� Stat� �������

��� Beirlant� J�� Vynckier� P� and Teugels� J� L� ������� Tail index es�timation� Pareto quantile plots and regression diagnostics� JASA������������

��� Brant� R� ������� Approximate likelihood and probability calcula�tions based on transforms� Ann� Stat� ���������

��� Brorsen� B� W� and Yang� S� R� ������ Maximum likelihood esti�mates of symmetric stable distribution parameters� Commun� StatSimul� ������������

��� Brown� B� M� and Hettmansperger� T� P� ������� Normal scores�normal plots and tests for normality� JASA ������������

��� Buckle� D� J� ������� Bayesian inference for stable distributions�JASA ���������

��� Chambers� J� M�� Mallows� C� and Stuck� B� W� ������� A methodfor simulating stable random variables� JASA �� �������

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��

��� Cheng� B�N�� and Rachev� S�T� ������� Multivariate stable futureprices� Math� Finance ���������

��� Christoph� G� and Wolf� W� ������� Convergence Theorems with aStable Limit Law� Akademie Verlag� Berlin�

���� DuMouchel� W� H� ������� Stable Distributions in Statistical Infer�ence� Ph�D� dissertation� Department of Statistics� Yale University�

���� DuMouchel� W� H� �����a�� On the asymptotic normality of themaximum�likelihood estimate when sampling from a stable distri�bution� Ann� Stat� ���������

���� DuMouchel� W� H� �����b�� Stable distributions in statistical infer�ence� � Symmetric stable distributions compared to other symmet�ric long�tailed distributions� JASA ����������

���� DuMouchel� W� H� ������� Stable distributions in statistical infer�ence� � Information from stably distributed samples� JASA ����������

���� DuMouchel� W� H� ������� Estimating the stable index � in orderto measure tail thickness A critique� Ann� Stat� ����������

���� Embrechts� P�� Kl�uppelberg� C� and Mikosch� T� ������� ModellingExtremal Events for Insurance and Finance� Springer Verlag� Hei�delberg�

���� Fama� E� ������� The behavior of stock prices� J� of Business ��������

���� Feller� W� ������� An Introduction to Probability Theory and ItsApplications Vol� II� Second Edition� John Wiley and Sons� NewYork� New York�

���� Feuerverger� A� and McDunnough� P� �����a�� On the e�ciency ofempirical characteristic function procedures� J� Roy� Stat� Soc Ser�B ������

��� Feuerverger� A� and McDunnough� P� �����b�� On e�cient inferencein symmetric stable laws and processes� In M� Cs�org�o� Dawson� D�A�� Rao� N� J� K� and Saleh� A� K� �Editors� Statistics and RelatedTopics� pp� �������

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���� Fofack� H� and Nolan� J� P� ������� Tail behavior� modes and othercharacteristics of stable distributions� Extremes �������

���� Haan� L� de and Ronde� J� de ������� Sea and wind multivariateextremes at work� Extremes ������

���� Ho mann�J�rgensen� J� ������� Stable densities� Theory Prob� Appl����������

���� Janicki� A� and Weron� A� ������� Simulation and Chaotic Behaviorof ��Stable Stochastic Processes� Marcel Dekker� New York� NewYork�

���� Kogon� S� M� and Williams� D� B� ������� Characteristic functionbased estimation of stable parameters� In Adler� R�� Feldman� R�and Taqqu� M� �eds�� A Practical Guide to Heavy Tailed Data�Birkh�auser� Boston� MA� pp� ��������

���� Koutrouvelis� I� A� ������ Regression type estimation of the param�eters of stable laws� JASA �� ��������

���� Koutrouvelis� I� A� ������� An iterative procedure for the estimationof the parameters of stable laws� Commun� Stat� Simul� ��������

���� L�evy� P� ������� Th�eorie des erreurs la loi de Gauss et les lois excep�tionelles� Bulletin de la Soci�et�e de France ��������

���� Mandelbrot� B� ������� The variation of certain speculative prices�J� of Business ����������

��� McCulloch� J� H� ������� Simple consistent estimators of stable dis�tribution parameters� Commun� Statist� Simul� �� ���������

���� McCulloch� J� H� ������� Financial applications of stable distribu�tions� Statistical Methods in Finance Handbook of Statistics Vol���� Maddala� G� S� and Rao� C� R� �eds�� North�Holland� NY�

���� McCulloch� J� H� ������� Measuring Tail Thickness to Estimate theStable Index Alpha A Critique� J� Bus� Econ� Stat� �� ������

���� McCulloch� J� H� ������� Linear regression with stable disturbances�In Adler� R�� Feldman� R� and Taqqu� M� �eds�� A Practical Guideto Heavy Tailed Data� Birkh�auser� Boston� MA� pp� ��������

Page 28: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

��

���� Michael� J� R� ������� The stabilized probability plot� Biometrika��������

���� Nikias� C� L� and Shao� M� ������� Signal Processing with Alpha�Stable Distributions and Applications� Wiley� New York� New York�

���� Nolan� J� P� ������� Numerical computation of stable densities anddistribution functions� Commun� Stat� Stochastic Models �� ��������

���� Nolan� J� P� ������� Parameterizations and modes of stable distri�butions� Stat� Prob� Letters �� ��������

���� Nolan� J� P�� Panorska� A� K� and McCulloch� J� H� ������� Esti�mation of stable spectral measures� to appear in Math� ComputerModelling�

���� Nolan� J� P� and Panorska� A� K� ������� Data analysis for heavytailed multivariate samples� Commun� Stat� Stochastic Models ���������

��� Nolan� J� P� ������� Fitting data and assessing goodness of �t withstable distributions� In Proceedings of the Conference on Applica�tions of Heavy Tailed Distributions in Economics Engineering andStatistics� American University� Washington� DC� June ���� �����

���� Paulson� A� S�� Holcomb� E� W�� and Leitch� R� ������� The estima�tion of the parameters of the stable laws� Biometrika �� �������

���� Pierce� R� D� ������� Application of the positive � stable distri�bution� In Signal Processing Workshop on Higher�Order Statistics�SPW�HOS ���� IEEE Press� Piscataway� N� J�� pp� �������

���� Press� S� J� ������� Estimation in univariate and multivariate stabledistributions� JASA �� ��������

���� Samorodnitsky� G� and Taqqu� M� S� ������� Stable Non�GaussianRandom Processes� Chapman and Hall� New York� New York�

���� Stuck� B� W� and Kleiner� B� ������� A statistical analysis of tele�phone noise� Bell Syst� Tech� J� �����������

���� Willinger� W�� Taqqu� M�� Sherman� R� and Wilson� D� �������Self�similarity through high�variability Statistical analysis of Eth�ernet LAN tra�c at the source level� In Proceedings of theACM�SIGCOMM ��� pp� ������

Page 29: Maxim - American Universityfs2.american.edu/jpnolan/www/stable/mle.pdf · maxim um lik eliho o d estimation of general stable parameters is describ ed The Fisher information matrix

��

���� Zolotarev� V� M� ������� One�Dimensional Stable DistributionsAmer� Math� Soc� Transl� of Math� Monographs Vol� ��� Amer�Math� Soc�� Providence� RI� �Transl� of the original ���� Russian�

���� Zolotarev� V� M� ������� On representation of densities of stable lawsby special functions� Theory Probab� Appl� ����������

John P� NolanDepartment of Mathematics and StatisticsAmerican UniversityWashington� DC ������� USA

E�mail jpnolan$american�edu