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Estimation in Univariate and Multivariate Stable Distributions

Author(s): S. James PressSource: Journal of the American Statistical Association, Vol. 67, No. 340 (Dec., 1972), pp. 842-846Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2284646 .

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8/2/2019 Press (1972)


EstimationnUnivariatend MultivariateStableDistribution

S. JAMESPRESS*

This paper proposes several methods of estimatingparameters in stable dis-tributions. ll the methods nvolvesample characteristic unctions. ne of the

methodswhich s based uponthe method of momentss treated insome detail.Asymptotic ormaldistributions or the proposed moment stimators re pro-vided. Moreover, all methods provide consistent stimators. The estimationproblem is treated for both univariate and multivariate table distributions.

1. INTRODUCTIONND BACKGROUND

Stable distributions are of interest in many applica-tions. For example, they have been applied in astronomy

to model gravitational fieldsby Holtsmark [10] (see also[8, p. 215]), and by Chandrasekhar [3, p. 70]. They havealso been suggested in business and economics to providea model forthe probability laws governing price changesof speculative securities (see [12, 16]).

The definitionof a stable law is conveniently providedby its characteristic function representation. Let the logcharacteristic function for a scalar random variable Yfollowinga stable law be given by

Ft -log+0(t) = iat - |IeL1+if3 t w(t,)j (1 1)

where co(t,a) =tan((ra/2), for a 0 1, and w(t, a) =(2/r)

log ti, for a=1; (t/|t|)=0 at t=O. The parameter a

is called the characteristic exponent of the law (O < a < 2),>0 is called the scale parameter (although sometimes

y=ba will be called the scale parameter), a is a locationparameter, - o <a < oo, and : is a symmetry parameter(-1?< 0< 1, and 3= 0 implies a symmetric distribution).

The probabilistic theory of univariate stable distribu-tions is well known (see, e.g., [9, 8]). Moreover, earlyapplications of these laws generally assumed that all fourparameters of the model are given. Recently, however,problems of inference have begun to receive attention,although only for the univariate case, and only for a

small subset of the parameter space. The inferenceprob-lem with these laws is not straightforward; it is compli-cated by the fact that although densities exist, they arenot generally available in closed form,making it difficultto apply conventional estimation techniques. However,numerical methods have begun to offer ome hope (see

[4]). Aftera brief review of recent efforts, ome less con-ventional estimation techniques will be suggested, andattention will be focused on a moment method.

* S. James ress is associateprofessorfstatistics, raduate SchoolofBusiness,UniversityfChicago,Chicago, ll. 60637. The author sgrateful o W. Du Moucheland an associated ditor ormanyhelpfuluggestions. hisresearch,wasinancednpartbyNSF GrantGP-17592.

Parameter stimation or = 2,thenormal istribution,does not need to be reviewed here. Estimation in theCauchy distribution a = 1) has receivedsome attentionin the univariatecase (see, e.g., [17, 2, 1]). More gener-ally, problems of inference n the univariatesymmetricstable distributions ere studied by Fama [5] and Famaand Roll [6, 7] for he case of I < <a<2. In general, heirapproach has involved estimationby means of samplefractiles, nd simulation to study the properties f the

estimators.The properties f these estimatorshave notyet been fullyexplored. Mandelbrot [12] suggested agraphical arge sample procedureforestimating basedupon the sample cdf.

Several more methods of estimating parameters ofstable distributions re suggested. Moreover, t will beseen that the new methodshave the advantage that allparameters re estimablesimultaneously. urthermore,both symmetric nd asymmetricdistributions an bestudied in the same way, and the methods generalizedirectly o the multivariate ase. It will be straightfor-wardto consider ll a, 0 < a< 2 within hepurview f thetechniques.Several ofthese methodsyield implicit sti-

mators only. The moment-matching ethod,however,whichgeneralizesome work fKleinman [II ] tomultipleparameterestimation n both univariateand multivari-ate distributions, ields explicit estimators. n no caseto be described s any claim of optimality f the methodintended. A simulation of the propertiesof these esti-mators s currently nder way, and if successful,will bereported n at a later date.

Let the log characteristic unction f Y be given as in(1.1). Let n independent observationsY1i, , ynbetaken. The problem s to estimate (a, -y,a, f), where

= b0. Consistentstimatorsre developed sing ample

characteristicunctions.Denote thesample characteristic unction f +(t) by

+'(t) = l1/nEni ilyi. (1.2)

Thus, '(t) is computablefor all values of t. Note that{+(t),- o <t< oo } is a stochastic rocess, nd for ach t,|(t) I is bounded above by unity.Hence, all moments f

f(t) are finite, nd +(t), forany fixedt, is the sample

? Journalof the AmericanStatistical Association

December 1972, Volume 67, Number 340

Theoryand Methods Section

842

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EstimationnStable Distributions 843

average of independent nd identicallydistributed an-dom variables.Thus, by the law of largenumbers, (t)is a consistent stimator f +(t).

EstimationMethod (MinimumDistance).Define

g a, y, , /) sup J(t) -(t) . (1.3)

Then, the minimumdistance estimators f (a, y,a, /)

are the values of these parameters which minimizeg(a, -y, , /). It is well knownthat these estimators restrongly onsistent.

EstimationMethod I (Minimumrth-MeanDistance).Define

h(a, y, , 3) I +(t) - (t) IrW(t)dt, (1.4)

whereW(t) denotes suitable convergenceactor,uchasW(t)-_(27r)- exp {- t2/2}, or W(t)=e-1Ht1 hen theminimum th mean distanceestimators re thosevaluesof (a, -y, , 3) whichminimizeh(a, -y, , /3) ora fixedr,

r> 1. It is not clearwhichrshould be selected,

or whichweight unctions mostappropriate.However,consistentestimators are obtained in any case. Numerous othernorms of j (t) -+(t) I may be used as estimation pro-cedures lso, and without imulation, t maynot becomeclearhow to select the optimalone.

Explicit point estimatorsbased upon the methodofmoments re developed nthe next ection. ection3 pro-videsasymptotic istributions or hemoment stimatorsdevelopedin Section 2 so that large sample confidenceintervalsmay be found.The last section Section4) showshow the estimationprocedures orunivariate table dis-tributions, ased upon sample characteristic unctions,

are extendible o the case of estimation n multivariatestable distributions.

2. MOMENTESTIMATION

An analytical estimationprocedurewhich yields ex-plicitestimators nd involvesminimalcomputation s aversion fthemethodofmoments.

From (1.1), for ll a,

I+(t) -e'ltla (2.1)

Hence, -ytl -log I(t) |. Now choosetwononzeroval-ues oft; say, t1 nd t27t1 t2. Assume a .1.Then,

Y t a = -log I (th) jY t2a -log I (t2)

Solvingthese twoequations simultaneouslyor and y,and replacing (t) by itsestimatedvalue, gives

loglog log

log t11t2 (2.2)

andlog t I og[-log I $(t2)

log ~~~~-log2I log[-logI|

(t1)f . (23log

z~~~~logogit1/t223

To estimate B nd a, defineu(t) -Im [log 0(t)],whereIm ['(t) ] denotes the imaginarypart of any complexvalued function J(t).Then, from 1.1),

u(t) = at - yI t aJ-1t w(t, a).

Choose wononzeroaluesI ft; say, 3 nd 4, 37t4. Thenfora 1,

7ra U (tk)a -Y Itk 'tan - k- 3 4. (2.4)2 tk

Since

P(t) -(1/n =1 costy1) i(l/n 5> sintyj),

in polarcoordinates, (t) =p(t) exp[iG(t)],where

p2(t)= (1/nEI', ,cos ty)2 + (1/nFj 1sin yj)2

and tan 0(t)= ( sintyj)/( 1 cos tyj).Hence,

log $(t) = p(t) + io(t), i(t) = Im[log4(t)] = 0(t).

Choose theprinlcipal alues of og +(tk), k= 3, 4. That is,usingprincipalvalues,for t3, 4,

(t)= arctan( 1 sintyj)/( 1cos tyj). (2.5)

Replacing u(t) in (2.4) by its estimatedvalue, given in(2.5), and solving he two mplied inearequationssimul-taneouslyfor and a givestheestimators

11q(t3) 4(t4)

t3 t (2.6)7raA

| t4 - t3 | tan -

and

|t4 _ _ t|a-L ( 4~it3 _ __ _ _

A3 . (2.7)- I 3jl-

Case ofa= I

For a = 1, (1.1) gives

u(t) = at - 2,yft/irog t

That is,for wononzerovalues oft, ay t3 and t4, 31t4,

_2y log Itk U(tk)

a - k = 3,4. (2.8)ir tk

Solvingthetwoequations mplied n (2.8) simultaneouslyfor 3 nd a, withu(t) replacedbyu2(t), ivesthemomentestimators or = 1,

[(t3) 9(t4)

: ___- t4 j, (2.9)

2-y t4- log -

ir t3

' It 'mayhe that thesamepa.ir fvaluesfi and t,, which s usedtoestimate andyrwill lso servewell to estimate and f. Howevert his questionrequires urther

stuldy.

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844 Journal f the American tatisticalAssociation, ecember1972

and

log It ( log t3

ii t ) t4

a- ________________-- (2.10)log I t4/t3

The equations (2.2), (2.3), (2.6), and (2.7) yieldmoment

estimators or a, y,f, a) for he case ofa 1; for = 1,d and a are estimatedfrom 2.9) and (2.10), and from(2.1), y=-log I +(ti) I /I til

Symmetrictable Distributions

Symmetrictable distributionsrethoseforwhich = 0in (1.1). Suppose the distributions also centered roundthe origin.By also takinga=0, (1.1) reducesto log +p(t)= _Y tl. Hence, |+(t)| =+p(t), and (2.2) and (2.3) pro-vide moment stimators or and y.

The estimators iven n (2.5) and (2.6) are consistentsince they arebased upon c(t), which s consistent.How-

ever, the rate of convergence o the population valueswillvary,depending n thechoicesofti, . - , t4.Optimalchoices of the t/'s equire furthertudy.Fortunately,nmanyapplications, uch as with ecurity rices, bserva-tions re usually available inexpensively,o thatthepos-sibly arge samplesrequired or mall estimation rror rerealistic price changeson a dailybasis over a periodofseveraldecades maygenerally e obtainedat lowcost).

3. INTERVALSTIMATION

Confidencentervalsmaybe obtained for heparame-tersofstable distributions hen argesamplesare avail-

able. Asymptoticdistributionsre developed below forthe aseof ymmetrictabledistributionsentered roundtheorigin. he method s analogous for he moregeneralcase.

Letyi, , yn enote sampleof ndependent andomvariables, ll ofwhichfollow he law givenin (1.1) foraX1, for =,B=0.

Define he family fcomplex andomvariables {Y,(t),- 00 <t< c }, where

Yn t) = l/n E: 1=eilyj.

Next,define he real valued families frandomvariables

Un t) = 1/n -1= costyj,Vn(t)= 1/nEj=1 sintyj. (3.1)

Then,

Un(t) = 12[Yn(t)+ Yn(-t)],(3.2)

Vnt) = (1/2i) Yn t) - Yn-t)] .

Since

Yn(t) = c (t) = Un(t) + iVn(t),

s ittu 1n2not(.) ]2nd (T2. give.

log{ 2 log[Un(ti) Vn(t1)]}

A - log - 0og[Un(t2) + Vn(t2)

log tl/t2 - (3.3)

and

log t I log{ -l og[U2 t2 + v2h

- log t2 log lolog[Unti) +V(t2)]}log I = log | t2| _2_ (__ __- (3.4)

log JltThe estimators f a and yareseen to be simpledifferenti-able functionsf the components f zn [Un(tl), Un(t2),Vn(t1),Vn(t2)]'.The means,variances, nd covariancesofthese fourcomponentswillnowbe evaluated.

First, note that because the distribution f Y is sym-metric, [Vn t) = 0, for ll t.Moreover, ecauseE [Yn t)]=qO(t)=(-t) I E[U (t)] =4(t). Hence,

cov[Yn(t1), Yn(t2)] = cov[1/n jeitivi,1/n keit2Yk]

= 1/n2 j Zk ov(eitlYi, eit2Yk)

= 1/n ov(eitly, it2Y).

That is,

cOv[Yn(t1), Yn(t2)] = 1/n[4O(tit2) 4(t1)4(t2)]. (3.5)

Substitutinghe meansfrom bove gives

E(zn) = O' - [(tl) (t2) 0,0] [01o2, 03, 04]

Now define the 4X4 covariance matrix, = (O-ij)nvar(zn).The elements fX are evaluated n a straight-

forwardway by using (3.5) in (3.2). For example,since

40(t) 0(-t)n012 = n cov[Un(t1), Un(t2)] = n/4{cov[Yn(tj), Yn(t2)]

+ COV[Yn(t1),n(-t2))]+ cOV[Yn(-t1),Yn(t2)]

+ COV[Yn(-tl), Yn(-t2)]}

=4 4[(tl + t2) +-(tl - t2) -2(ti)0(t2)

The other covariancesare found n a similarway. Vari-ances are obtainedby setting l t2.

Results arenow summarized exceptfor 12,whichhasalreadybeengiven).

1=M + 4(2tj) - 242(t3)], j = 1, 2

ai ='1[I - 4(2tj)], j = 3, 4

034 = [O(tl - t2) - 0(tl + t2J,

aij =0, otherwise.

Note from3.1) thatUn(t)and Vn(t) resamplemeansofindependent nd identicallydistributed andom vari-ables, and since IUn(t) < 1 and IVn(t) < 1, theirvari-ances are finite.Hence, the central imittheoremmpliesthat forfixed j, = 1,2 if2( ) denotesprobability aw,

lim ? { ?n(z - 0)} = N(O, I). (3.6)

Now define he function

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Estimationn Stable Distribution, 845

(Z.) -wi,log 2-[Un(t1) + v (tl)

+ 102og-B og[Un(t2)+ 1n(t2)]} (

Then, from 3.3) and (3.4),

g(Zn)= a , forW1 [log tlAt2 )11, 2 = 11- W21,

and q(zn) =log y for w1= log| t2I /1og[| tl I I ]--012,

and W2 log tl /1og tl / 2 | ==]W22. From large sampletheory see, e.g., [15, p. 321]),

lim VnG& a)} = N(O,) )) (3.8)

and

lim L { n(log A- log_y) = N(0, 212) (39)nl2

where

fl.i 7j(cl), (A2) - n C'i, w2j), j = 1, 2,

and

w2(; C02j) i1 Ek=1 ailc 49OV9' (3.10)

whereg g(O| Wlj,W2j)isdefinedn (3.7) and intheremarkfollowing.t onlyremains o evaluate the derivatives n(3.10). Since

q = wilog2 _Ilog(02 + 02)} + w2log -2log(02 + 2)89 cj og 02lg 2 20

ag ~~wj= ---- ' 1j 1,2,aoj 4(ti) log4(t,)

and ag/03 = g/004= 0. Substitutinghe covariances andderivatives nto (3.10) and simplifying,3.8) and (3.9)become

lim L {/ - N(0, 1), (3.11)n-- o 77

i. -/ log - log N(0, 1), (3.12)

where

2 [1+ I?(2tj) 1 2 1 (tl) 12]

2 [ (t1) logI ?(t1) I log ?t]

[1+ (2t2) -2 1 (t2) 12] (3.13)

2 |?(t2) I og 0 (t2) log - o

[ (tl+t2) + I +(tl-t2) 1 -2 l (tl) l (t2) 1

and

2 [1+I +(2th) 1 -21 (tl) 12][log It2 1]2

2 [ (t) log (ti) I log; f;j

[1+ (2t2) -2 1 (t2) 12] [log (3 .2

+~~~~~ t 2 (3.14)2 [ (th) 10o t (th) I10o

[ ?(tl+th) + I ?>(tl-th) 1 -2 1?>(ti) t ?(t) ]

log t log t2

(t) log1 (tl) I og +(t2) [log t1

Equations (3.11)-(3.14) providethe required symptoticdistributions or a and -y.Thus, if Xe/2 denotesthe E/2significanceointfor standardnormal ariate,with on-fidence oefficient1 -e), it followsn largesamplesthat

A A- - < a<a

and

A )e/2772 < < A+XIe/2772

7 exp 7X } <?exp{XE /2J

4. MULTIVARIATEYMMETRICTABLEDISTRIBUTIONS

A majoradvantageof themomentmethodof estimat-ing parameters funivariate table distributionss thatthe methodgeneralizesdirectly o the multivariate ase.

For dependentvariables followingmultivariate ym-

metric table aws,the situation s more omplicated hanin the univariate case, but solvable, at least in largesamples. For example, suppose Y: pX1 followsa non-singularmultivariate ymmetrictable awwith ogchar-acteristic unction2

log +(t) = ia't - 1(t'At)a/2 (44)

where '= (t1, , tp), is a positivedefiniteymmetricmatrix, nd 0 <a <2. The problem s to estimate a, a,and Q, given sampleofp-variate bservations, i, ,Yn, whereall yj's are independent X 1 vectors.By an-alogywiththeunivariate ase, evaluate the samplechar-

acteristic unction+()I/n

-lneit'yi, (4.2)

and use it as an estimator f ?(t). Now theminimum is-tance, (1.3), or the minimumrth-meandistance esti-mators, 1.4), can be computed ust as in theunivariatecase. Alternatively,moment stimatorswill be found.

Select two distinctt vectors, ay, t= t3jsje, j = 1, 2,wheree denotes a p-vectorof ones, and si and S2 arescalars. Observethat

log |4(t) |= - 2(t'At)aI2. (4.3)

2

A detailed reatmentfmultivariatetable aws s given n [13] and [14].

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846 Journalf theAmericantatisticalssociation,ecember 972

Substitute = t1, nd t t2, eparately n (4.3), take logsineach equation, nd then akethe ratioofthetworesult-ing equations. Replacing + t) by +(t) in the resultgives

log log~log 4t2)

log sI/s2I1

The estimatorn (4.4) isconsistent ince (t) is consistent.Now estimate Q (wij),by selectingnew values of t.

Firstrewrite 4.3) as

t'Qt = [-2 log ?(t) I 2/a.

Sincet'Qlt s a quadratic form,

t'Qt = '=1 Ep 1titj1ij [-2 log ?(t) j 2/a.

Since there are p(p+l)/2 distinctwij's,p(p+l)/2 dis-tinct inearly ndependent onzero hoicesofthe t vectorwill be required.Denote themby ri, 12, , M where

M= p(p+1)/2. Then,

vkQk = [-2 logj (Tk) j 2/a, (4.5)

fork= 1, * , M, providesa systemofM linearequa-tions nM unknowns,which s solvable by determinants(a is assumedtobe known, .e.,preevaluated, rom4.4));thus, he solutionsof (4.5) areuniquely determined,ndprovide he moment stimator fQ, which s, ofcourse,consistent.

The location parameterof the distribution s easilyestimated onsistently y defining

u(t) _ Im[log py(t)] = a't = E ajtj.

Then, since the multivariate estimation procedure isanalogousto the univariateprocedure, t is clear from(2.5) that f

4(t) = Im[log -y(t)],

andprincipalvalues are taken,

4(t) = arctan[Ej- sin 'yj/>i cos'y= ]

Then, if ' *, , T*are distinct, inearly ndependent,nonzero alues ofthe t vector, hesystem fequations

a Tj = (Tj) j , ,p,

provides basisfor stimating. Thus, fT= (T, I T

isthep Xp matrix f ssignedtvectors, nd u(t*) denotesthe th elementofthe p X1 vectorU, the moment sti-mator fa is given explicitly y

a = U, T #0. (4.6)

The foregoing emarks bout consistency, symptoticnormality, nd efficiency f the momentestimators ofunivariate table distributions pply to the estimators f

multivariate table distributionss well. Hence, the mul-

tivariatemoment stimators re consistent, nd relativeefficiency illdepend upon thechoice oft vectors.

A generalization fthe nonsingularmultivariate ym-metric table law in (4.1) is

log ?(t) = ia't - 2 E2=1(tIjjt)aI2,(4.7)

whereQj is positive semidefiniteoreach j, l Qj ispositive definite, nd no two (Qj's are proportional. oranypreassignedm,the Qj'smaybe estimatedna manneranalogousto theway Q was estimatedform= 1, as longas sufficient bservations are available. However, theproblem n the moregeneralcase is clearlyconsiderablylarger n dimension, equiringmsystems f M simultane-ous equationstobe solvedsimultaneously.Much investi-gationon theproperties fthesimpler ases needs to bedone before the propertiesof estimators for the moregeneralcase can be evaluated.

[ReceivedMarch1971. RevisedApril 1972.]

REFERENCES

[1] Barnett,V.D., "OrderStatistics stimators f the Locationof the Cauchy Distribution,"Journal of the AmericanStatisticalAssociation, 1 (December1966), 1205-18.

[2] Bloch,D., "A Note on the Estimationof the LocationPa-rameter fthe Cauchy Distribution," ournal f theAmer-ican Statistical ssociation, 1 (September 966),852-5.

[3] Chandrasekhar, ., "Stochastic Problems n Physics andAstronomy," eviews f ModernPhysics, 5 (1943), 1-89.

[4] Du Mouchel, W.H., "Stable Distributions n StatisticalInference,"Ph.D. dissertation,Dept. of Statistics,YaleUniversity, ew Haven, Conn. (1971).

[5] Fama, E., "The Behaviorof StockMarket Prices,"Journal

ofBusiness, 8 (January 965),34-105.[6] and Roll, R., "Some Properties f SymmetrictableDistributions," ournal f theAmerican tatisticalAssocia-tion, 3 (September 968), 817-36.

[7] - and Roll, R., "Parameter EstimatesforSymmetricStable Distributions,"Journalof the AmericanStatisticalAssociation, 6 (June1971),331-8.

[8] Feller,W., An Introductiono Probability heory nd ItsApplications, ol. II, New York: JohnWiley& Sons, Inc.,1966.

19]Gnedenko, .V. and Kolmogorov,A.N., LimitDistributionsfor Sums of IndependentRandom Variables,Cambridge,Mass.: Addison-Wesleyublishing o., 1954.

[10] Holtsmark,J.,Ann. d. Physik, 8 (1919), 577.[11] Kleinman,D., "Estimating heParameters f Stable Prob-

ability Laws," Unpublished manuscript,UniversityofChicago,1965.[12] Mandelbrot,B., "The Variation of Certain Speculative

Prices,"Journal fBusiness, 6 (October1963), 394-419.[13] Press, S.J., AppliedMultivariate nalysis,New York: Holt,

Rinehart& Winston,nc., 1972.[14] , "Multivariate Stable Distributions,"Journal of

Multivariate nalysis, December 1972).[15] Rao, C.R., Linear Statisticalnferencend Its Applications,

New York: JohnWiley nd Sons, nc., 1965.[16] Roll, R., "The EfficientMarket Model Applied to U.S.

Treasury Bill Rates," Unpublished Ph.D. dissertation,GraduateSchool ofBusiness,UniversityfChicago,1968.

[17] Rothenberg, .J.,Fisher,F.M. and Tilanus,C.B., "A Noteon Estimation from a Cauchy Sample," Journal of the

American tatistical ssociation, 9 (June1964),460-3.

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