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A Class Of Asymmetric Distributions

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Tomasz J. Kozubowski

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A CLASS OF ASYMMETRIC DISTRIBUTIONSTomasz J. Kozubowski 1Department of MathematicsThe University of Tennessee at Chattanooga615 McCallie AvenueChattanooga, TN 37403, U.S.A.Phone: 615 755 4762Fax: 615 755 4586E-mail: tkozubow@cecasun.utc.eduKrzysztof Podg�orski2Department of Mathematical SciencesIndiana University{Purdue University IndianapolisIndianapolis, IN 46202, U.S.A.Phone: 317 274 8070Fax: 317 274 3460E-mail: podgorsk@math.iupui.eduAbstract. We discuss a class of asymmetric distributions arising in a random summationscheme. We call members of the class asymmetric Laplace distributions as the standardLaplace distributions, which are symmetric, constitute a proper subclass. Among distribu-tions which are limits in random summation schemes asymmetric Laplace distributions playan analogous role to that of normal distributions among distributional limits of non-randomsums. Asymmetric Laplace laws are more \peaky" and have heavier tails than normal laws.They have stability properties and are convenient in applications, as their densities haveexplicit forms and estimation procedures are easily implemented. Anticipating increasinginterest in this class of distributions we present statistical tools which can be utilized inpractice, including algorithms for simulation and estimation. We also discuss more generalclasses of distributions, where asymmetric Laplace distributions appear as special cases.Keywords: Aggregate claims; collective risk models; compound distribution; estimation;geometric stable law; heavy tailed modeling; interest rates; insurance mathematics; Laplacedistribution; Linnik distribution; mathematical �nance; maximum likelihood; Paretian-stablelaw; random summation; risk theory; �-stable distribution.1Research partially supported by UC Foundation Faculty Research Grant No. R04 105249.2Research partially supported by the Purdue Research Grant, 1998.1

1 Introduction and notationProbably the most widely known and used theorem of the probability theory is the CentralLimit Theorem (CLT). This theorem in its most often used form gives necessary and su�cientconditions for the convergence of sums of independent and identically distributed (i.i.d.)random variables to the normal law. Consequently, many scientists and practitioners believethat, provided the number of summands is large, their sum can always be approximated bya normal distribution. This, however, may not be the case. If the summands have in�nitevariance, then the sum may converge to a stable law (see for example Samorodnitsky andTaggu [20]). Moreover, even if the variables are independent and normally distributed, thesum of their random number may not be distributed according to the normal law.In Figure 1, we compare two histograms, each obtained for 5000 observations of thesums of i.i.d. random variables. For the one on the left, the observations were generatedas non-random sums of 1000 independent normal random variables with a non-zero mean.On the right hand side we generated sums of random variables having the same normaldistribution but this time with a random number of terms distributed according to thegeometric distribution with parameter p = 1=1000 and independently of the terms themselves.The data were centered on their mean and scaled by their variation. We clearly see asymmetryand peakedness in the case of random summation scheme.-3 -2 -1 0 1 2 3

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Figure 1: Histograms of non-random sums (left) and sums with geometrically distributednumber of terms (right).Apart from its interesting theoretical properties, the random summation scheme appearsnaturally in various �elds, particularly in insurance mathematics. In risk theory, we areinterested in the distribution of aggregate claims generated by the portfolio of insurancepolices. If the individual claims are denoted by Xi's (usually assumed to be i.i.d.) and therandom variable �p denotes the number of claims in a given time period, than the aggregate2

claim Sp is given by Sp = X1 + � � �+X�p : (1)We consider a class of distributions that approximate geometric compounds, that iscompound distributions (1) with the geometric number of terms:P (�p = k) = p(1� p)k�1; k = 1; 2; : : : ; (2)when the parameter p converges to zero (so that the average number of terms in (2) convergesto in�nity). Geometric compounds frequently appear in applied problems from various �elds,including actuarial science, as discussed in Kalashnikov [5]. As shown in Mittnik and Rachev[16], the geometric compounds (1), appropriately normalized, converge (in distribution) toa geometric stable random variable, which is a location - scale mixture of stable randomvariables.In this paper we focus on an important special case, where the random variables Xi'shave a �nite second moment (variance). Then, the limiting distribution of normalized Sp isa random variable with the following characteristic function: (t) = [1 + �2t2 � i�t]�1 (3)(see for example Mittnik and Rachev [16]). By specifying � = 0, we obtain Laplace distribu-tions which are the only symmetric distributions within this class. Thus, it seems pertinentto name the distribution with ch.f. (3) an asymmetric Laplace distribution (AL).We introduce AL laws in Section 2, where we present their basic properties and pro-cedures for simulation and estimation. We show that the probability distribution of everyAL random variable is the same as that of the di�erence of two independent exponentiallydistributed random variables. This crucial observation leads to explicit formulas for densitiesand distribution functions of AL distributions, facilitating their practical implementation.We also de�ne a time dependent random process through an AL distribution, which playsan analogous role to Brownian motion. In the symmetric case, this process was applied tomodel �nancial data in Madan and Seneta [14], where it was termed the Variance Gammaprocess.We think that AL laws should provide an alternative to normal distributions as distri-butional models in a variety of settings. This class is particularly well suited for modelingphenomena where the variable of interest results from of a large random number of inde-pendent innovations, while the empirical distribution appears to be asymmetric, \peaky",and has tails heavier than those allowed by normal distribution. One area of applicationwhere modeling with AL laws should be explored is mathematical �nance, where the empir-ical data often have the above features. The idea that the price change during a period oftime is produced by a random number of \individual e�ects" �rst appeared in Mandelbrotand Taylor [15] and Clark [3], and was further explored in Mittnik and Rachev [16, 17] andKozubowski and Rachev [8]. In Section 3, we apply the AL model to the interest rates datastudied by Klein [6], showing the consistency with our model. In the Appendix, we collect3

main properties of AL laws and comment on their various further extensions. The results arebrief and presented without proofs, as the more detailed treatment of AL laws will appearelsewhere.Notation.� Z� { exponential random variable with the densityf�(x) = 1� exp(�x=�); x > 0; (4)� Z = Z1 { standard exponential random variable,� For a vector (or matrix) t, t0 denotes the transpose of t,� s0t =Pdi=1 tisi { the inner product of s = (s1; : : : ; sd)0 and t = (t1; : : : ; td)0,� jjtjj = (t0t)1=2 = (Pdi=1 t2i )1=2 { the Euclidean norm in Rd,� \ d! " { convergence of distributions,� \ d= " { equality of distributions,� For � > 0, �(�) = R10 x��1e�xdx (the gamma function),� sign(x) equals 1 for x > 0, �1 for x < 0, and 0 for x = 0,� � 2 R, � � 0 { location and scale parameters of AL distribution,� � = 2�=(�+p4�2 + �2) { scale invariant parameter.2 Asymmetric Laplace distributionsIn this Section we de�ne univariate asymmetric Laplace distributions and derive their basicproperties. We omit most proofs and refer an interested reader to Kozubowski and Podg�orski[11] for a more detailed treatment.De�nition 2.1 A random variable is said to have an asymmetric Laplace (AL) distributionif there are parameters � 2 R and � � 0 such that its characteristic function has the form(3). We denote such r.v. and its distribution as Y�;� and AL(�; �), respectively, and writeY�;� � AL(�; �).Note the following relations among the parameters:1� � � = �� ; 1� + � = s4 + ����2; 1�2 + �2 = 2 + ����2 :4

2.1 Special casesWhile the distribution makes sense for every � 2 R and � � 0, we have several special cases.1. If � = � = 0, then (t) = 1 for every t 2 R, and the distribution is degenerate at 0.2. For � = 0 and � > 0, we have an exponential distribution with mean �, denotedthroughout as Z�. Similarly, if � = 0 and � < 0, we have �Z��.3. If � = 0 and � 6= 0, we have the Laplace distribution with location zero and scale �,whose density is f(x) = 12�e�jx=�j; x 2 R: (5)2.2 Mixture representationsIn this section we present various representations of AL distributions. The representationslead to explicit formulas for AL densities and distribution functions, and facilitate computersimulations of AL random variates.Mixture of normal distributions. Let N and Z be independent and standard normaland exponential distributions, respectively. Then, the following relation takes place:Y�;� d= �Z +p2�2Z �N: (6)Thus, conditionally on Z = z, the r.v. Y�;� � AL(�; �) is normal with mean �z and variance2�2z.An exponential mixture. Let I� be a discrete r.v. taking values �� and 1=� with prob-abilities p = �2=(1 + �2) and q = 1=(1 + �2), respectively. Let Z be standard exponentialindependent of I�. Then Y�;� d= � � I� �Z: (7)In the symmetric case (� = 0 and � = 1), the random variable I� takes values �1 withprobabilities 1/2 each, and we obtain the well-known representation of symmetric Laplacedistribution.Mixture of exponentials. The ch.f. of Y�;� can be factored as (t) = 1(1 + it��)(1� it�=�) ; (8)which shows that every AL random variable has the same distribution as the di�erence oftwo independent exponential random variables:Y�;� d= � � Z1=� � � �Z�: (9)5

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Figure 2: Asymmetric Laplace densities, � = 1 and � =0 , 0.8, 1.5, 2, 3, 4, 6, 8, 10 whichcorrespond to � � 1.0, 0.68, 0.50, 0.41, 0.30, 0.24, 0.16, 0.12, 0.1.2.3 DensitiesLet p�;� and F�;� denote probability density function (p.d.f.) and cumulative distributionfunction (c.d.f.) of an AL(�; �) distribution, respectively. The representation (9) producesthe following explicit formulas:p�;�(x) = 1� �1 + �2 ( exp ����x� ; if x � 0exp � 1��x� ; if x < 0; (10)and F�;�(x) = ( 1� 11+�2 exp ����x� ; if x > 0�21+�2 exp � 1��x� ; if x � 0:Figure 2 shows AL densities for various values of the parameters. It is clear that the distri-bution is unimodal with the mode equal to zero. We see the characteristic peakedness of thedensity at zero. Some basic properties of AL densities are collected in the Appendix.2.4 Moments and related parametersSince the density of an AL law is a simple exponential function, the values of moments andother related parameters of AL laws follow. We summarize them in Table 1.6

Parameter De�nition ValueAbsolute moment EjY ja ����a �(a+ 1)1 + �2(a+1)1 + �2nth moments EY n n!����n 1 + (�1)n�2(n+1)1 + �2Mean EY �Variance E(Y �EY )2 �2 + 2�2Mean deviation EjY �EY j 2�e(�2�1)�(1 + �2)Coe�cient of Variation pV ar(X)jEX j s2�2�2 + 1 = p1=�2 + �21=�� �Coe�cient of Skewness 1 = E(X �EX)3(E(X � EX)2)3=2 2 1=�3 � �3(1=�2 + �2)3=2Kurtosis (adjusted) 2 = E(X �EX)4(V ar(X))2 � 3 6� 12(1=�2 + �2)2Table 1: Moments and related parameters of Y � AL(�; �).Remark 1. The mean deviation equals � for � = 0. Further, we havemean deviationstandard deviation = 2e�2�1(1 + �2)p1 + �4 : (11)For the symmetric Laplace distribution (� = 0, � = 1), the above ratio is equal to 1=p2.Remark 2. For a distribution with �nite third moment and standard deviation greater thanzero, the coe�cient of skewness is a measure of symmetry (for symmetric distributions itsvalue is zero), and is independent of scale.Remark 3. For a distribution with a �nite 4th moment, kurtosis (adjusted, so that 2 = 0for normal distribution) measures peakedness, and is independent of scale. If 2 > 0, thedistribution is said to be leptokurtic, and if 2 < 0, the distribution is said to platykurtic.7

We see that an AL(�; �) distribution is leptokurtic and 2 varies from 3 (the least value forthe symmetric Laplace distribution, where � = 1) to 6 (the greatest value for exponentialdistribution, where � = 0).2.5 The median and skewnessThe calculation of the median and other percentiles is straightforward. We have the followingequation for the median m of an AL(�; �) distributionm = � log(2=(1+ �2))1� �2 : (12)Note if � = 0, equation (12) yields m = 0, which is the median of symmetric Laplacedistribution. Similarly, for � = 0, we get m = � log 2, which is the median of an exponentialdistribution with mean � (to which AL law simpli�es in this case).Further, the following inequalities hold for the three common measures of the center:If � > 0, then Mode < Median < Mean.If � < 0, then Mode �Median �Mean:All three measures are equal to zero if � = 0 (and the distribution is symmetric).Finally, we note that another measure of skewness of a distribution with c.d.f. F , pro-vided by the limit limx!1 1� F (x)� F (�x)1� F (x) + F (�x) ;in case of an AL(�; �) distribution is equal to sign(�).2.6 SimulationSince the distribution function of an AL distribution, as well as its inverse, can be writtenin closed form, the inversion method of simulation is straightforward to implement. Alter-natively, mixture representations (6), (7), and (9) can be used for simulation. Below is agenerator of a random variate from an AL distribution, based on the representation (7) for� > 0 and on the representation (6) for � = 0.An AL(�; �) generator.� Generate a standard exponential variate Z.� IF � = 0THEN Y � �Z.ELSE f 8

Set � 2��+p�2+4�2 .Generate uniform [0; 1] variate U , independent of Z.IF U < �2=(1 + �2)THEN Set I ��.ELSE Set I 1=�.Set Y � � I � Z. g� RETURN Y .Numerical subroutines (written in SPlus c ) for simulating AL distributions as well as forcomputing densities, quantiles, c.d.f.'s, and estimators are available from the authors uponrequest.2.7 EstimationHere we derive moment and maximum likelihood estimators for AL parameters � and �. Weassume that Y1; : : : ; Yn is an i.i.d. random sample from an AL(�; �) distribution given bych.f. (3), and write our parameters in vector notation as � = [�; �]0.Method of moments. Letm1 = E(Y�;�) = � and m2 = E(Y�;�)2 = 2�2 + 2�2 (13)be the �rst two moments of an AL(�; �) distribution (see Table 1). When we solve equations(13) for � and � and substitute the sample moments,bm1n = 1n nXi=1 Yi and bm2n = 1n nXi=1 Y 2i ; (14)for m1 and m2, we obtain the method of moments estimators:b�n = " b�nb�n # = " bm1nqbm2n=2� bm21n # : (15)Standard arguments of the large sample theory show that the method of moments estimatorsof � and � are consistent and asymptotically normal. Namely, if � > 0, then b�n given by (15)is strongly consistent estimator of � and pn(b�n � �) is asymptotically normal with (vector)mean zero and covariance matrix�MME = �2 " 2 + �2=�2 12�=�12�=� �2=�2 + 14�4=�4 + 54 # : (16)9

Maximum likelihood. Maximum likelihood estimators (MLE's) in this case are e�cient(their asymptotic covariance matrix is the inverse of the Fisher information matrix). Thefollowing standard notation allows for a compact formulas for estimators. For real y, lety+ = max(y; 0) and y� = max(�y; 0) be the positive and negative parts of y, respectively.Applying the above notation to the random sample Y1; : : : ; Yn, we write Y + = Pni=1 Y +i =n,and Y � =Pni=1 Y �i =n. Now, we can express the MLE's for �, �, and � as follows:b�n = 4qY �=Y +; b�n = Y ; b�n = 4qY � 4qY + �qY + +qY �� : (17)The MLE b�n = [b�n; b�n]0 is consistent and asymptotically normal. The asymptotic distributionof pn(b�n � �) is bivariate normal with (vector) mean zero and covariance matrix�MLE = �28 " 8(1=�2 + �2) 4(1=�� �)4(1=�� �) 1=�2 + �2 + 6 # : (18)2.8 Generalized Laplace laws { Laplace motionWe can de�ne a L�evy process on [0;1) with independent increments, Laplace motion fY (t); t �0g, so that Y (0) = 0, Y (1) is given by (3), and for 0 < � the distribution of Y (�) is given bythe ch.f. (t) = [1 + �2 jtj2 � i�t]�� : (19)One may call it a generalized asymmetric Laplace distribution. Denote the corresponding r.v.by Y�;�;� . It is clear that Y�;�;� d= �Y1;�;� , where � = �=�, so we study the latter in the sequel.Note that factorization (8) shows that the ch.f. of Y1;�;� can be written as the product of twogamma ch.f.'s: 1;�;�(t) = � 11 + it��� � 11� it=��� : (20)Recall that � 11�it=��� is the ch.f. of the gamma r.v. ��;1=�, whose density is given byg�;1=�(x) = ���(�)x��1e��x; x � 0: (21)(For � = 1 it reduces to the exponential distribution with mean 1=�). Thus, we have therelation Y1;�;� d= ��;1=� � ��;� ; (22)that generalizes (9). Since these distributions will be considered in another paper, we limitour discussion here to their densities. We can get the following expression for the density ofY1;�;� via the standard transformation theorem of random variables:p1;�;�(�x) = [�(�)]�2e���1x Z 10 y��1(x+ y)��1e��ydy; x > 0; (23)10

where � = p1 + �2. Note that in case � = 1 the above simpli�es to (10), as it should. Wecan write the density (23) asp1;�;�(�x) = 1�(�)p� �x����1=2 e��x=2K��1=2��x2 � ; x > 0; (24)where Ku is the modi�ed Bessel function of the third kind:Ku(z) = (z=2)u�(1=2)�(u+ 1=2) Z 11 e�zt(t2 � 1)u�1=2dt: (25)If � = k is an integer, then the density (23) is a mixture of k densities on (�1;1) and hasan explicit form. For j = 0; : : : ; k� 1, the jth density has the formfk;j(x) = pk;jgk�j;1=�(x)I[0;1)(x) + qk;jgk�j;�(�x)I(0;1)(�x); (26)where g�;� stands for a gamma density as before, andpk;j = pkqjpkqj + pjqk ; qk;j = 1� pk;j = pjqkpkqj + pjqk ; (27)with p = �2=(1 + �2) and q = 1=(1 + �2). Note that p1;0 = p and q1;0 = q. For k = 1 andj = 0 the density (26) coincides with (10). Under the above notation, the density of Y1;�;ktakes the form p1;�;k(x) = k�1Xj=0 (k + j � 1)!j!(k� 1)! (pkqj + pjqk)fk;j(x): (28)3 ApplicationsIn this section we present two applications of AL distributions. The �rst one is in modelinginterest rates on 30-year Treasury bonds. Klein [6] studied yield rates on average daily30-year Treasury bonds from 1977 to 1990, �nding that the empirical distribution is too\peaky" and \fat-tailed" to have been from a normal distribution. He rejected the traditionallognormal hypothesis and proposed the stable Paretian hypothesis, which would \account forthe observed peaked middle and fat tails". The paper was followed by several discussions,where some researchers objected to the stable hypothesis and o�ered alternative models,including a �rst-order moving average model of Huber. In our approach, we assume that thesuccessive logarithmic changes in interest rates are i.i.d. observations from an AL distribution.Our model is simple, allows for peakedness, fat-tails, skewness, and high kurtosis observed inthe data. We were inspired by the ideas of Mittnik and Rachev [17], regarding the interestrate change as the random sum of a large number of small changes:interest rate change = �pXi=1 (small changes),11

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Figure 3: Top-left: Empirical c.d.f. vs. normal c.d.f. Top-right: Empirical c.d.f. vs. ALc.d.f. Bottom-left: Histogram of interest rates on 30-year Treasury bonds. Bottom-right:Non-parametric estimator of the density (thin solid line) vs. the theoretical ones (normal {dashed line, AL { thick solid line).where the number of terms, �p, that has a geometric distribution. Thus, provided the smallchanges have �nite variance, the AL law (3) can approximate the distribution of the interestrate change. We think of �p as the moment when the probabilistic structure governing theinterest rates breaks down. Such event could be a new information, political, economical orother event that a�ect the fundamentals of the �nancial market.Our goal is to present the idea of modeling interest rates using AL laws. The data setconsists of interest rates on 30-year Treasury bonds on the last working day of the month andis published in Huber's discussion of Klein's paper [6], p. 156. The data covers the periodof February 1997 through December 1993. We convert the data to the logarithmic changesaccording to the formula: Yt = log(it=it�1), where it is the is the interest rate on 30-yearTreasury bonds on the last working day of the month t. There were the total of 202 valuesof the logarithmic changes Yi.First, we have plotted the histogram of the data set (Figure 3 (bottom-left). We cansee the typical shape of a AL density: the distribution has high peak near zero, and appearsto have tails thicker than that of the normal distribution. Comparisons of the c.d.f. ofthe normal distribution and the empirical c.d.f. seen on Figure 3 (top-left) and the densityfunctions Figure 3 (bottom-right) con�rm these �ndings. We see a disparity around the center12

Parameter Theoretical value Empirical valueMean �0:001018163 �0:001018163Variance 0:001733809 001372467Mean deviation 0:02944785 0:02945773Mean dev./ Std dev. 0:7072175 0:7582487Coe�cient of Skewness �0:07334177 �0:2274964Kurtosis (adjusted) 3:003586 3:599207Table 2: Theoretical versus empirical moments and related parameters of Y �L(b�; b�)of the distribution due to a high peak in the observed data. In order to �t an AL model,we need to estimate the parameters � and �. We used the maximum likelihood estimators�nding b� = �0:001018163 and b� = 0:029434439. Further, we calculated the parameter � aswell as the theoretical values of various parameters presented in Table 2. We also calculatedthe empirical counterparts of the parameters, where we used the following statistics:� Mean: 1n P Yi.� Variance: 1n P(Yi � Y )2.� Mean deviation: 1nP jYi � Y j.� Coe�cient of skewness: b 1 = 1n P(Yi � Y )3=( 1n P(Yi � Y )2)3=2.� Kurtosis (adjusted): b 2 = 1n P(Yi � Y )4=( 1nP(Yi � Y )2)2.We present the empirical and theoretical values in Table 2. Except for a slight dis-crepancy for the skewness, the match between empirical and theoretical values is striking.In addition, we show, in Figure 3 (top-right), the theoretical AL c.d.f. compared with theempirical c.d.f. and, in Figure 3 (bottom-right), the density kernel estimator based on thedata against the theoretical densities of normal and AL distributions with the estimated pa-rameters. We see that at the mode agreement is better for the AL distribution than for thenormal one.The second example illustrates how AL laws can account for asymmetry in the data.The data consists of currency exchange rates: the German Deutschemark versus the USDollar (DMUS). The observations are daily exchange rates from 1/1/80 to 12/7/90 (2853data points). As usual, we consider the change in the log(rate) from day t to day t + 1.First, we plotted the histogram (Figure 4 (left)) We see the typical shape of a AL density.The distribution has a high peak near zero, and appears to have non-symmetric tails thickerthan that of the normal distribution. A normal QQ plot (see Figure 4 (middle)) con�rms13

these �ndings. We used maximum likelihood for estimating the AL parameters obtainingb� = 0:0007558 and b� = 0:00521968. The quantile plot of the data set and the theoreticalAL distribution is presented in Figure 4 (right). It shows only very slight departures fromthe straight line. We conclude that AL distributions model this data set more correctly thannormal distributions.-0.03 -0.02 -0.01 0.0 0.01 0.02 0.03

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Figure 4: Analysis of the currency data.4 Appendix4.1 PropertiesWe collect here the main properties of AL laws. We omit most proofs and refer an interestedreader to Kozubowski and Podg�orski [11] for a more detailed treatment.4.1.1 StabilityParetian stable distributions, that include normal laws, have two fundamental properties.First, they are limiting laws for appropriately normalized sums of i.i.d. random variables.Thus, they work well as approximations to sums of i.i.d. random variables. Second, theyare stable: the sum of i.i.d. normal (Paretian stable) r.v.'s has a normal (stable Paretian)distribution. These properties are shared by AL laws, if deterministic summation is replacedby geometric summation. By de�nition, AL laws are limits of geometric compounds of i.i.d.components. The stability properties of exponential and Laplace distributions are well known(see for example Arnold [2] for exponential and Lin [12] for Laplace):Y d= ap �pXi=1 Yi; (29)where �p is geometric (2), Yi's are i.i.d. copies of Y , and �p and (Yi) are independent (theconstants a(p) are equal to pp for Laplace and p for exponential distributions). Although14

general AL r.v.'s do not satisfy (29), they have the following characterization (see Kozubowski[7]): A r.v. Y with �nite variance is AL if and only ifap �pXi=1(Yi + bp) d! Y; (30)where �p is geometric (2), Yi's are i.i.d. copies of Y , and �p and (Yi) are independent.Moreover, if � > 0, the normalizing constants in (30) can be taken asap = Cp1=2; bp = (p1=2 � C)�=C; C = q2=(2 + (�=�)2):In addition, all AL laws are geometric in�nitely divisible, that isY d= �pXi=1 Y (i)p ; (31)where �p is geometric (2), Y � AL(�; �), Y (i)p 's are i.i.d. with AL(�pp; �p) distribution foreach p, and �p, (Y (i)p ) are independent.Remark. AL laws are in�nitely divisible in the classical sense as well. Please see Kozubowskiand Podg�orski [11] for the exact form of their L�evy measure.4.1.2 Self-decomposabilityRelations (7) and (9) are special cases with c = 0 of the more general representations:Y�;� d= c � Y�;� + (�1=�� �2�)Z d= c � Y�;� + �1�Z1=� � �2�Z�; 0 � c � 1; (32)where (�1; �2) has the following joint distribution:P (�1 = 0; �2 = 0) = c2;P (�1 = 1; �2 = 0) = (1� c)�c+ 1� c1 + �2� ;P (�1 = 0; �2 = 1) = (1� c) c+ (1� c)�21 + �2 ! ;and the r.v.'s Y�;�, (�1; �2), and Z (correspondingly, Y�;�, (�1; �2), Z1=�, and Z�) are mutuallyindependent. Written in terms of ch.f.'s, the relation (32) takes the form (t) = (ct) c(t); 0 � c � 1; (33)where and c are ch.f.'s of Y�;� and (�1=�� �2�)Z, respectively. Recall that a ch.f. thatsatisfy (33) is said to be self-decomposable, and the corresponding distribution (and r.v.) issaid to be in class L. Thus, all AL laws are in class L, which implies that they are unimodal,as self-decomposability implies unimodality (Yamazato [21]). It is clear from the explicitformula for the density, that modes of all AL laws are actually equal to zero.Remark. We note that self-decomposability and unimodality of AL laws was established inRamachandran [19]. 15

4.1.3 Properties of the densitiesWe discuss here basic properties of AL densities.Values at zero. We have the following relations:p�;�(0) = 1� �1 + �2 ; F�;�(0) = P (Y�;� � 0) = �21 + �2 ;lim�!�1 p�;�(0) = 12� ; lim�!1 p�;�(0) = 1j�j ; lim�;�!0 p�;�(0) =1:Symmetry with respect to �. The densities p�;� are symmetric with respect to theparameter �: p�;�(x) = p�;��(�x);which follows from (10) and the relation� = 2�p4�2 + �2 + � = p4�2 + �2 � �2� :Asymmetry of the density. The density p�;� of a AL distribution with � > 0 satis�es therelation p�;�(x) > p�;�(�x); x > 0:By symmetry, an analogous results hold for � < 0. In fact, we havep�;�(x)p�;�(�x) = exp(x(1=�� �)=�) = exp(x�=�2);so that p�;�(�x)=p�;�(x)! 0 as x!1.Derivatives. Except for x = 0, AL densities have derivatives of any order n > 0:f (n)�;� (x) = ( (�1)n ��� �n+1 11+�2 e��x=� ; if x > 01(��)n+1 �21+�2 ex=(��); if x < 0 : (34)Further, we havelimx!0+(�1)np(n)�;�(x) = 11 + �2 ����n+1 ; limx!0� p(n)�;�(x) = �21 + �2 � 1���n+1 :The two limits are equal if either n = 0 (showing the continuity of the density at zero) or� = 1 (and thus � = 0), producing the symmetric Laplace distribution.Complete monotonicity. A function f de�ned on I � R is called completely monotonic(respectively, absolutely monotonic) if it is in�nitely di�erentiable on I and (�1)kf (k)(x) � 0(respectively, f (k)(x) � 0) for any x 2 I and any k = 0; 1; 2; : : :. The complete and absolutemonotonicity of AL densities follow directly from (34). Namely, if p�;� is the density of anAL(�; �) distribution, then the functions p�;�(�x) are completely monotonic on (0;1) andabsolutely monotonic on (�1; 0). 16

4.2 Further extensionsThe class of AL laws can be extended in various ways. First, the distributions may be shifted,allowing for arbitrary modes. Next, one can consider a more general class of distributionsgiven by an AL ch.f. (3) raised to a positive power. These are marginal distributions of theL�evy process fY (t); t � 0g with independent increments, for which Y (1) � AL(�; �) as it wasdescribed at the end of Section 2. Further, one obtains a richer class of limiting distributions,consisting of geometric stable laws, by allowing for in�nite variance of the components inthe geometric compounds (1). More generally, if the random number of components in thesummation (1) is not geometrically distributed, a wider class of �-stable laws is obtainedas the limiting distributions. Finally, if the components in (1) are multi-dimensional, themultivariate AL distributions are obtained.4.2.1 Translated AL lawsIf Y � AL(�; �), then Y + � is a r.v. with a three-parameter densityp�;�;�(x) = 1� �1 + �2 ( exp ���� (x� �)� ; if x � �exp � 1�� (x� �)� ; if x < �; (35)and distribution functionF�;��(x) = ( 1� 11+�2 exp ���� (x� �)� ; if x > ��21+�2 exp � 1�� (x� �)� ; if x � �:Although these three-parameter distributions are no longer limiting laws for geometric com-pounds (1), nor do they have stability property (29), they do provide more exibility in datamodeling by allowing for arbitrary modes.4.2.2 Geometric stable lawsIf the random variables in (1) have in�nite variance, than the geometric compound no longerconverge to an AL law. Instead, the limiting distributions form a broader class of geometricstable (GS) laws. It is a four-parameter family best described in terms of characteristicfunction: (t) = [1 + �� jtj� !�;�(t)� i�t]�1; (36)where !�;�(x) = ( 1� i�sign(x) tan(��=2); if � 6= 1;1 + i� 2� sign(x) log jxj ; if � = 1:The parameter � 2 (0; 2] is the index that determines the tail of the distribution: P (Y >y) � Cy�� (as y !1) for 0 < � < 2. For � = 2 the tail is exponential and the distributionreduces to AL law, as !2;� � 1. The parameter � 2 [�1; 1] is the skewness parameter, while17

� 2 R and � � 0 control the location and scale, respectively. We brie y comment on someof the features of GS laws, and refer an interested reader to Kozubowski and Rachev [10] foran up to date information and extensive references on GS laws and their special cases.Remark 1. Special cases of GS laws include Linnik distribution, for which � = 0 and � = 0(see Linnik [13]), and Mittag-Le�er laws, which are GS with � = 1 and either � = 1 and� = 0 (exponential distribution) or 0 < � < 1 and � = 0. The latter are the only non-negativeGS r.v.'s (see Pillai [18]).Remark 2. GS laws share many, but not all, properties of Paretian stable distributions,which were discussed in the actuarial context in Klein [6]. In fact, stable and GS laws arerelated through their characteristic functions, ' and , as shown in Mittnik and Rachev [16]: (t) = (� log'(t)); (37)where (x) = 1=(1 + x) is the Laplace transform of the standard exponential distribution.Relation (37) produces the representation (36), as well as the mixture representation of a GSrandom variable Y in terms of independent standard stable and exponential r.v.'s , X andZ: Y d= ( �Z + Z1=��X; � 6= 1;�Z + Z�X + �Z�(2=�) log(Z�); � = 1: (38)Note that the above representation reduces to (6) in case � = 2, as then X has the normaldistribution with mean zero and variance 2.Remark 3. The asymmetric Laplace distribution, which is GS with � = 2, plays the samerole among GS laws, as normal distribution does among stable laws. As normal distributionis convenient in application, so is AL law, as its p.d.f., c.d.f. have explicit expressions.Remark 4. Like stable laws, GS laws lack explicit expressions for densities and distributionfunctions, which handicap their practical implementation. Also, they are \fat-tailed", havestability properties (with respect to random summation), and generalize the central limittheorem (as they are the only limiting laws for geometric compounds). However, they aredi�erent from stable (and normal) laws in that their densities are more \peaked", while stillbeing heavy-tailed. Unlike stable densities, GS densities \blow-up" at zero if � < 1. Sincemany �nancial data are \peaked" and \fat-tailed", they are often consistent with the GSmodel (see for example Kozubowski and Rachev [8]).4.2.3 �-stable lawsSuppose that the random number of terms in the summation (1) is any integer-valued ran-dom variable, and, as p converges to zero, �p approaches in�nity (in probability) while p�pconverges in distribution to a r.v. � with Laplace transform . Then, the normalized com-pounds (1) converge in distribution to a �-stable distribution, whose characteristic functionis (37) (see for example Kozubowski and Panorska [9]). The class of �-stable laws contains18

GS and generalized AL laws as special cases: if �p is geometric (2), then p�p converges tothe standard exponential and (37) produces (36); similarly, if �p is negative binomial withparameters � and p, then p�p converges to a gamma distribution and (37) produces (19).The tail behavior of �-stable laws is essentially the same as that of stable and GS laws (seeKozubowski and Panorska [9]).4.2.4 Multivariate extensionThe theory of AL laws can be extended to random vectors. Namely, a multivariate AL lawcan be de�ned as the limit (in distribution) as p ! 0 of appropriately normalized randomsums Sp = X(1) + � � �+X(�p): (39)Here, (X(i)) is a sequence i.i.d. random vectors with �nite second moments and �p has ageometric distribution (2), independent of X(i)'s. It follows that the limiting distributionsfor normalized geometric compounds (39) are laws with the following ch.f.:(t) = �1 + 12t0�t� it0m��1 ; (40)wherem is an arbitrary vector in Rd and � is a d�d non-negative symmetric matrix (see forexample Mittnik and Rachev [16]). We shall call a distribution given by (40) a multivariateasymmetric Laplace law and denote it by AL(�;m). The symmetric case with m = 0 wasdiscussed in the literature before (see Johnson and Kotz [4], Madan and Seneta [14]). If� is positive-de�nite, the distribution is truly d-dimensional and has a probability densityfunctiong(y) = 2ey0��1m(2�)d=2j�j1=2 y0��1y2 +m0��1m!v=2Kv �q(2 +m0��1m)(y0��1y)� ; (41)where v = (2 � d)=2 and Kv is the modi�ed Bessel function (25). In the symmetric case(m = 0) this density was derived in Anderson [1]. In the one-dimensional case, where � =�11, we have v = (2�1)=2 = 1=2 and the Bessel function simpli�es toK1=2(u) = p�=(2u)e�u.Consequently, the density (41) simpli�es to the density (10) of a univariate AL law withparameters � = p�11=2 and � = m. For d > 1 the density (41) blows up at zero. Further,if the dimensionality d is odd, d = 2r + 3, the density has a closed form:g(y) = Crey0��1m�Cjjyjj �1(2�jjyjj��1)r+1j�j1=2 rXk=0 (r+ k)!(r� k)!k!(2Cjjyjj��1)�k; y 6= 0; (42)where v = (2� d)=2, C = p2 +m0��1m, and jjyjj��1 = py0��1y is a norm in Rd. In thethree dimensional space, we have r = 0 and the density is particularly simple:g(y) = ey0��1m�Cjjyjj �12�jjyjj��1j�j1=2 ; y 6= 0: (43)19

Multivariate AL laws share many, but not all, properties of univariate AL laws. Since amore extensive study of the multivariate case will appear elsewhere, we only give few generalremarks.Stability. Multivariate AL laws are geometrically in�nitely divisible as well as in�nitelydivisible in the classical sense, and satisfy the stability property (29) whenever either m or� equals zero. However, relation (30) does not generally hold for d > 1.Mixture representation. Mixture representation (6) extends to the multivariate case asfollows. Let Y � AL(�;m) and let X � N(0;�) (multivariate normal with mean zero andvariance-covariance �). Let Z be an exponentially distributed r.v. with mean 1, independentof X: Then the following representation holdsY d=mZ + Z1=2X: (44)The mean and variance-covariance matrix. Representation (44) leads to the followingformulas for the mean and variance-covariance of Y � AL(�;m):EY =m; E(Y� EY)(Y�EY)0 = �+mm0: (45)Linear transformations. Any linear transformation of an AL r.v. leads to another ALrandom variable. Let Y = (Y1; : : : ; Yd)0 � AL(�;m) and let A be an l � d real matrix.Then, the random vector YA = AY is AL(�A;mA), where mA = Am and �A = A�A0.In particular, multivariate and univariate marginals of an AL random vector are AL, as areall linear combinations of its components.5 SummaryThe class of AL laws plays an analogous role among geometric stable laws to that played bythe class of normal distributions among stable laws. AL laws arise as limiting distributions forgeometric compounds, as normal laws do for deterministic sums, of i.i.d. random variableswith �nite second moments. AL laws have a stability property with respect to geometricsummation, as normal laws do with respect to classical summation. Both, AL and normallaws are convenient in applications, as their densities have explicit forms and estimationprocedures are easily implemented. However, there are important di�erences between thetwo families of distributions: AL laws are more \peaky" and have tails heavier than normallaws and allow for asymmetry, whereas all normal distributions are symmetric. We hope thatour survey of results and methods for AL laws will lead to more frequent applications of ALlaws in actuarial science and other areas of applied research.20

References[1] D.N. Anderson, A multivariate Linnik distribution, Statist. Probab. Lett. 14, 333-336(1992).[2] B.C. Arnold, Some characterizations of the exponential distribution by geometric com-pounding, SIAM J. Appl. Math. 24(2), 242-244 (1973).[3] P.K. Clark, Subordinated stochastic process model with �nite variance, Econometrica41, 135-155 (1973).[4] N.L. Johnson and S. Kotz, Distributions in Statistics: Continuous Multivariate Distri-butions. (John Wiley & Sons, New York, 1972).[5] V. Kalashnikov, Geometric Sums: Bounds for Rare Events with Applications (KluwerAcad. Publ., Dordrecht, 1997).[6] G.E. Klein, The sensivity of cash- ow analysis to the choice of statistical model forinterest rate changes (with discussions), TSA XLV, 79-186 (1993).[7] T.J. Kozubowski, The inner characterization of geometric stable laws, Statist. Decisions12, (1994) 307-321.[8] T.J. Kozubowski and S.T. Rachev, The theory of geometric stable distributions and itsuse in modeling �nancial data, European J. Oper. Res. 74, 310-324 (1994).[9] T.J. Kozubowski and A.K. Panorska, On moments and tail behavior of �-stable randomvariables, Statist. Probab. Lett. 29, 307-315 (1996).[10] T.J. Kozubowski and S.T. Rachev, Univariate geometric stable distributions, J. Comput.Anal. Appl. 1(2) (to appear).[11] T.J. Kozubowski and K. Podg�orski, Asymmetric Laplace distributions, preprint (1998).[12] G.D. Lin, Characterizations of the Laplace and related distributions via geometric com-pounding, Sankhya Ser. A 56, 1-9 (1994).[13] Ju.V. Linnik, Linear forms and statistical criteria, I, II, Selected Translations in Math.Statist. Probab. 3 1-90 (1963). (The original paper appeared in: Ukrainskii Mat.Zhournal 5 (1953) 207-290.)[14] D.B. Madan and E. Seneta, The variance gamma (V.G.) model for share markets returns,J. Business 63, 511-524 (1990).[15] B. Mandelbrot and H.M. Taylor, On the distribution of stock price di�erences, Oper.Res. 15, 1057-1062 (1967).[16] S. Mittnik, and S.T. Rachev, Alternative multivariate stable distributions and theirapplications to �nancial modelling, in S. Cambanis et al. eds., Stable Processes andRelated Topics (Birkhauser, Boston, 1991) 107-119.[17] S. Mittnik and S.T. Rachev, Modeling asset returns with alternative stable distributions,Econometric Rev. 12(3), 261-330 (1993).21

[18] R.N. Pillai, On Mittag-Le�er functions and related distributions, Ann. Inst. Statist.Math. 42(1), 157-161 (1990).[19] B. Ramachandran, On geometric stable laws, a related property of stable processes, andstable densities, Ann. Inst. Statist. Math. 49(2), 299-313 (1997).[20] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes (Chapman &Hall, New York, 1994).[21] M. Yamazato, Unimodality of in�nitely divisible distribution functions of class L, Ann.Probab. 6, 523-531 (1978).

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