A Gini Autocovariance Function for TimeSeries Modeling

Marcel Carcea1 and Robert Serfling2

Western New England Universityand

University of Texas at Dallas

February 11, 2015

1Department of Mathematics, Western New England University, Springfield, MA,01119, USA. Email: marcel.carcea@wne.edu.

2Department of Mathematical Sciences, University of Texas at Dallas, Richard-son, Texas 75080-3021, USA. Email: serfling@utdallas.edu. Website:www.utdallas.edu/∼serfling.

Abstract

In stationary time series modeling, the autocovariance function (ACV) throughits associated autocorrelation function provides an appealing description of thedependence structure but presupposes finite second moments. Here we providean alternative, the Gini autocovariance function (Gini ACV), which capturessome key features of the usual ACV while requiring only first moments. Forfitting autoregressive, moving average, and ARMA models under just firstorder assumptions, we derive equations based on the Gini ACV instead ofthe usual ACV. As another application, we treat a nonlinear autoregressive(Pareto) model allowing heavy tails and obtain via the Gini ACV an explicitcorrelational analysis in terms of model parameters, whereas the usual ACVeven when defined is not available in explicit form. Finally, we formulate asample Gini ACV that is straightforward to evaluate.

AMS 2000 Subject Classification: Primary 62M10 Secondary 62N02

Key words and phrases: Autocovariance function; Linear time series; Nonlinearautoregressive; Pareto; Heavy tails; Gini covariance.

1 Introduction

In stationary time series modeling, the autocovariance function (ACV) throughits associated autocorrelation function (ACF) provides a straightforward andappealing description of the dependence structure, when second moments arefinite. Without such an assumption, the ACV is no longer well-defined as apopulation entity, although sample versions can still be used fruitfully. Herewe provide an alternative, the Gini autocovariance function (Gini ACV), whichcaptures some key features of the usual ACV while imposing finiteness only offirst moments. While thus serving heavy tailed time series modeling, the GiniACV complements the usual ACV when standard second order assumptionsholds and for some models is even more tractable. For Gaussian models, theGini autocorrelation function (Gini ACF) and the usual ACF coincide. Forfitting autoregressive, moving average, and ARMA models under just firstorder assumptions, we derive equations based on the Gini ACV instead ofthe usual ACV. For an important nonlinear autoregressive (Pareto) modeldefined under a range of moment assumptions including the heavy tailed case,we obtain via the Gini ACV an explicit correlational analysis in terms ofmodel parameters, whereas the usual ACV even when defined is not explicitlyavailable. Finally, we formulate a straightforward sample Gini ACV. (Ourtreatment extends in part a preliminary paper, Serfling, 2010, also cited inShelef and Schechtman, 2011, who formulate a Gini partial ACF and developinference procedures exploiting the two separate Gini autocovariances of eachlag.) Our setting for the present paper is that of a strictly stationary timeseries with continuous marginal distribution F .

Just as the usual ACV is based on the usual covariance, the Gini ACVadapts to the time series setting the “Gini covariance” of Schechtman andYitzhaki (1987), a measure well-defined under just first moment assumptions.Specifically, for X and Y jointly distributed with finite first moments, thereare two associated Gini covariances, β(X, Y ) = 4Cov(X,FY (Y )) and β(Y,X)= 4Cov(Y, FX(X)), each involving the usual covariance of one of the variableswith the rank of the other. As discussed in Yitzhaki and Olkin (1991), thesecompromise between Pearson covariance Cov(X, Y ) and the Spearman versionCov(FX(X), FY (Y )). Special applications of Gini covariance are developed inOlkin and Yitzhaki (1992), Xu, Hung, Niranjan, and Shen (2010), and Yitzhakiand Schechtman (2013).

As other relevant background, we mention the connection with L-momentsand L-comoments. Hosking (1990) extended the Gini mean difference into acomplete series of descriptive measures of all orders, called L-moments, whichmeasure spread, skewness, kurtosis, etc., just as do the central moments, butunder merely a first moment assumption. The sample L-moments have lesssensitivity to extreme data values than the sample central moments. Serfling

1

and Xiao (2007) extended L-moments to the multivariate case, defining L-comoments and L-comoment matrices, measuring L-covariance, L-coskewness,L-cokurtosis, etc., pairwise across the variables, again under first moment as-sumptions. The second order L-comoments are, in fact, the Gini covariances.

In Section 2 we formulate the Gini ACV and Gini ACF as γ(G)(k) =β(X1+k, X1) and ρ(G)(k) = γ(G)(k)/γ(G)(0), k = 0,±1,±2, . . ., respectively.Here we also provide relevant technical background on Gini mean differenceand Gini correlation.

Section 3 treats the Gini ACV for linear time series models. CorrespondingGini cross-correlations are introduced and their key properties developed. Forfitting AR, MA, and ARMA models, systems of equations based on the GiniACV are derived. For example, for the zero mean AR(1) process, i.e., Xt+1 =φXt+ξt, we obtain the equation φ = γ(G)(+1)/γ(G)(0), paralleling the equationbased on the usual ACV, but using the first order Gini quantities.

Section 4 applies the Gini ACV to a nonlinear autoregressive time seriesmodel of Pareto type introduced by Yeh, Arnold, and Robertson (1988) andtreated recently by Ferreira (2012). For this model, no closed form expressionsexist for the usual ACF when it is defined under second moment assumptions.However, closed form expressions for the lag ±1 Gini autocorrelations in termsof model parameters are obtained, even under first order assumptions. Thesefacilitate an explicit correlational type representation of dependence structurefor this model and show how that structure changes as a function of modelparameters.

Section 5 formulates a straightforward sample Gini ACV. This supportsuseful nonparametric exploratory analysis of a time series without assuming aparticular type of model or assuming second order conditions.

We mention several goals for further work. It is desirable to develop robustversions of the sample Gini ACV, especially for use with heavy tailed dataarising either as outliers or as innovations. Efficient numerical algorithmsare desired for the nonlinear “Gini” systems of equations for fitting MA(q)and ARMA(p, q) models. A standard approach with the usual Yule-Walkerestimates is to recursively fit AR models using the Durbin-Levinson algorithm,and extension to the Gini-Yule-Walker estimates is desired. It is of interest topursue spectral analysis based on the Gini ACV.

Throughout the paper, well-known facts are invoked as needed withoutciting particular sources. In such cases, suitable sources are Box and Jenkins(1976) and Brockwell and Davis (1991).

Finally, for added perspective, we note that several tools already exist fortreating time series under first order (or lower) moment assumptions. Forexample, a variant of the usual ACV having a sample version that estimatesit consistently as the sample length increases has a long history (Davis andResnick, 1985a, 1985b, 1986, Brockwell and Davis, 1991, 3.3, and Resnick,

2

1997), although this function lacks a straightforward interpretation and hasother limitations. For other approaches, see Peng and Yao (2003), Yang andZhang (2008), and Chen, Li, and Wu (2012). In Section 3.3.4 below, wealso discuss LAD approaches for fitting AR models under minimal momentassumptions. The Gini ACV does not replace any of the diverse existingapproaches but rather complements them with an attractive new tool.

2 The Gini autocovariance function

The formulation of our Gini autocovariance function (Gini ACV) draws uponthe familiar Gini mean difference and the less familiar Gini covariance. Wefirst introduce these with some relevant details and then proceed to the GiniACV.

2.1 The Gini mean difference

For a random variable X with distribution F , an alternative to the standarddeviation as a spread measure was introduced by Gini (1912):

α(X) = E|X1 −X2| = E(X2:2 −X1:2), (1)

with X1:2 ≤ X2:2 the ordered values of independent conceptual observationsX1 and X2 having distribution F . Now known as the Gini mean difference(GMD), α(X) is finite if F has finite mean. An important representation,

α(X) = 2Cov(X, 2F (X)− 1) = 4 Cov(X,F (X)), (2)

facilitates an illuminating interpretation: α(X) is 4 times the covariance of Xand its “rank” in the distribution F , or, more precisely, twice the covarianceof X and the classical centered rank function 2F (X)− 1.

The GMD may also be expressed as an L-functional (weighted integral of

quantiles), α(X) = 2∫ 1

0F−1(u) (2u− 1) du, with F−1(u) = inf{x : F (x) ≥ u},

0 < u < 1, the usual quantile function of F . This representation, as well as(1), yields still another useful expression,

α(X) = 2

∫x (2F (x)− 1) dF (x), (3)

which defines an estimator of α(X) by substitution of a sample version of F .For elaboration of the above details, see Hosking (1990) and Serfling and

Xiao (2007).

3

2.2 The Gini covariance

For a bivariate random vector (X, Y ) with joint distribution FX,Y and marginaldistributions FX and FY having finite means, the so-called “Gini covariance”introduced by Schechtman and Yitzhaki (1987) has two components,

β(X, Y ) = 2Cov(X, 2FY (Y )− 1) = 4 Cov(X,FY (Y )) (4)

β(Y,X) = 2Cov(Y, 2FX(X)− 1) = 4 Cov(Y, FX(X)), (5)

the Gini covariances of X with respect to Y and of Y with respect to X,respectively. Note that β(X, Y ) is proportional to the covariance of X andthe FY -rank of Y , and β(Y,X) to the covariance of Y and the FX-rank of X.Thus β(Y,X) and β(X, Y ) need not be equal, and, when not equal, providecomplementary pieces of information about the dependence of X and Y . Thedefinitions (4) and (5) parallel the representation (2) for α(X) and reduce toit when X = Y . Also, paralleling (1), we have

β(X, Y ) = E(X[2:2] −X[1:2]), (6)

where (X1, Y1) and (X2, Y2) are independent observations on F , and, for i =1, 2, X[i:2] denotes the X-value or “concomitant” matched with Yi:2, with Y1:2 ≤Y2:2 the ordered values of Y1 and Y2 (see David and Nagaraja, 2003). Further,paralleling (3), we have

β(X, Y ) = 2

∫ ∫x (2FY (y)− 1) dF (x, y), (7)

facilitating estimation by substitution of appropriate sample distribution func-tions.

Although β(X, Y ) and β(Y,X) are equal for exchangeable (X, Y ), this doesnot hold in general. Such potential asymmetry may at first seem surprisingand even unwanted. However, classical higher order extensions of covariance,i.e., the “comoments” introduced by Rubinstein (1973), (including coskewness,cokurtosis, etc.) in the finance setting, are all quite naturally asymmetric.Symmetry of the classical covariance is thus an exception to the general rule.Such asymmetry is also characteristic (even in the second order case) for therecently introduced “L-comoments” of Serfling and Xiao (2007), which parallelthe classical central comoments while requiring moment assumptions only offirst order instead of increasingly higher order as the order of the comomentincreases. The second order L-comoment happens to be the Gini covariance,whose two components thus provide separate pieces of information on depen-dence. From these one also may craft various symmetric measures if so desired(see Yitzhaki and Olkin, 1991, for further discussion).

4

A scale-free Gini correlation (of X with respect to Y ) is given by

ρ(G)(X, Y ) =β(X, Y )

α(X)=

Cov(X,FY (Y ))

Cov(X,FX(X)), (8)

and the companion ρ(G)(Y,X) is defined analogously. These compare with theusual Pearson correlation ρ(X, Y ) = Cov(X, Y )/σXσY defined under secondorder moment assumptions. In particular, ρ(G)(X, Y ) and ρ(X, Y ) both takevalues in the interval [−1,+1]. Comparatively, ρ(X, Y ) measures the degreeof linearity in the relationship between X and Y and takes values ±1 if andonly if X is a linear function of Y , whereas ρ(G)(X, Y ) measures the degree ofmonotonicity and takes values ±1 if and only if X is a monotone function of Y .They coincide under some conditions which are fulfilled for bivariate normaldistributions and for certain bivariate Pareto distributions, for example.

For elaboration of the above details, see Schechtman and Yitzhaki (1987)and Serfling and Xiao (2007).

2.3 A Gini autocovariance function

Consider a strictly stationary stochastic process {Xt}. When the variance isfinite, a standard tool is the autocovariance function, consisting of the lag kcovariances

γ(k) = Cov(X1+k, X1), k = 0,±1,±2, . . . .

Of course, γ(−k) and γ(+k) are equal. Here, however, we assume only firstorder moments and introduce the Gini autocovariance function (Gini ACV).For each time t and each lag k ≥ 1, there are two Gini covariances of lagk, β(Xt+k, Xt) and β(Xt, Xt+k). By the stationarity assumption, β(Xt+k, Xt)= β(X1+k, X1) and β(Xt, Xt+k) = β(X1, X1+k), for each t = 0,±1,±2, . . .,and also β(X1, X1+k) = β(X1−k, X1), each k ≥ 1. Consequently, the Ginicovariance structure of the time series {Xt} may be characterized succinctlyby the Gini autocovariance function (Gini ACV)

γ(G)(k) = β(X1+k, X1), k = 0,±1,±2, . . . . (9)

For k = 0, we have γ(G)(0) = α(X) = β(X,X), the GMD of F . For lagk 6= 0, γ(G)(|k|) and γ(G)(−|k|) provide two measures of dependence whichare not necessarily equal, namely, with factors of 4, the covariance betweenan observation and the rank of the lag k previous and future observations,respectively. On this basis, directly facilitating practical interpretations, weexhibit the Gini ACV (9) in terms of two component functions, each indexedby k ≥ 0:

γ(A)(k) = β(X1+k, X1), k = 0, 1, 2, . . . ,

5

andγ(B)(k) = β(X1, X1+k), k = 0, 1, 2, . . . ,

although for theoretical treatments the function γ(G)(k) as a whole is moreconvenient. Corresponding Gini autocorrelation functions (Gini ACFs) aregiven by ρ(A)(k) = γ(A)(k)/γ(G)(0) and ρ(B)(k) = γ(B)(k)/γ(G)(0).

3 The Gini ACV for Linear Models

Let us consider the linear process {Xt} generated by a linear filter applied toa series of independent shocks or innovations,

Xt =∞∑i=0

ψiξt−i, (10)

where the ξt are IID with mean 0, ψ0 = 1 and∑∞

i=0 |ψi| < ∞. Under secondorder assumptions, the ξt have finite variance σ2

ξ . For the process {Xt} in ourheavy-tailed setting not requiring finite variance, we have the following keyresult.

Lemma 1 For any sequence of random variables {ξt} such that suptE|ξt| <∞, and for any sequence of constants {ψi} such that

∑∞i=0 |ψi| <∞, the series∑∞

i=0 ψiξt−i converges absolutely with probability 1.

This is just the first statement of Proposition 3.1.1 of Brockwell and Davis(1991). The assumption suptE|ξt| < ∞ is implied by our stationarity andfirst order assumptions.

Some widely used linear models have only finitely many parameters ψi,and under second order assumptions these parameters may be represented interms of the usual ACV and thus may be estimated via the sample ACV. Here,requiring only first order assumptions, we obtain alternative methods basedon the Gini ACV, thus supporting parameter estimation via the sample GiniACV (which we introduce in Section 5). In particular, below we obtain Giniequations for model parameters in three important cases: moving average(MA), autoregressive (AR), and ARMA models. For this purpose, we firstdevelop some results on Gini cross-covariances for linear models in general.

3.1 Gini cross-covariances

A useful quantity under second order assumptions is the xξ cross-covarianceof lag k defined as γxξ(k) = Cov(Xt+k, ξt). It is straightforward that

γxξ(k) = Cov(Xt+k, ξt) = Cov

(∞∑i=0

ψiξt+k−i, ξt

)=

{0, k < 0

ψk σ2ξ , k ≥ 0.

(11)

6

Reversing the roles of ξ and x, a ξx cross-covariance of lag k is also definedbut, noting that γξx(k) = γxξ(−k), yields nothing new.

Under first order assumptions, we take γ(G)xξ (k) = β(Xt+k, ξt) as the Gini

xξ cross-covariance of lag k and γ(G)ξx (k) = β(ξt+k, Xt) as the Gini ξx cross-

covariance of lag k. The first yields a striking parallel to (11):

γ(G)xξ (k) = β(Xt+k, ξt) = β

(∞∑i=0

ψiξt+k−i, ξt

)=

{0, k < 0

ψk α(ξ), k ≥ 0.(12)

The second yields a useful expression for the Gini ACV:

γ(G)(k) = β(X1+k, X1) = β

(∞∑i=0

ψiξ1+k−i, X1

)=

∞∑i=max{0,k}

ψi γ(G)ξx (k−i) (13)

Let us now suppose further that the linear stationary model is invertible,yielding

ξt =∞∑i=0

πiXt−i, (14)

with π0 = 1 and∑|πi| < ∞. The coefficients {πi} and {ψi} are related via

π(z) = ψ−1(z), where π(z) =∑∞

i=0 πi zi and ψ(z) =

∑∞i=0 ψi z

i, provided thatπ(z) and ψ(z) have no common zeros. If follows that the {πi} may be obtainedrecursively from the {ψi} via π0 = ψ0 = 1 and

πi = −(ψi + π1ψi−1 + · · ·+ πi−1ψ1), i ≥ 1, (15)

yielding π1 = −ψ1, π2 = ψ21 − ψ2, π3 = −ψ3

1 + 2ψ1ψ2 − ψ3, etc. In the secondorder case, using (14), we obtain for the usual ξx cross-covariances convenientexpressions in terms of {πi} and the usual ACV:

γξx(k) = γxξ(−k) = Cov(ξt+k, Xt) =

{0, k > 0∑∞

i=0 πi γ(k − i), k ≤ 0.(16)

For the first order case, similar steps yield a striking Gini analogue:

γ(G)ξx (k) = β(ξt+k, Xt) =

{0, k > 0∑∞

i=0 πi γ(G)(k − i), k ≤ 0.

(17)

3.2 Moving Average Processes

We now consider the invertible MA(q) model given by

Xt = ξt + θ1ξt−1 + · · ·+ θqξt−q (18)

7

for some choice of q ≥ 1 and ξt IID with mean 0. Assuming finite variances,the usual ACV has the representation

γ(k) =

{σ2

ξ

∑q−|k|i=0 θiθi+|k|, |k| ≤ q,

0, |k| > q.(19)

Under first order assumptions, the above fails but by (13) we obtain a Ginianalogue:

γ(G)(k) =

{ ∑qi=max{0,k} θi γ

(G)ξx (k − i), |k| ≤ q,

0, |k| > q.(20)

Here we have used the facts that X1+k and X1 are independent for |k| > q andthat ξ1+k and X1 are independent for k > 0 and k < −q.

3.2.1 Solving for θ1, . . . , θq in terms of Gini autocovariances

We start with equations (20), which more explicitly may be written

γ(G)(0)(1)= γ

(G)ξx (0) + θ1γ

(G)ξx (−1) + θ2γ

(G)ξx (−2) + · · ·+ θqγ

(G)ξx (−q)

γ(G)(1)(2)= θ1γ

(G)ξx (0) + θ2γ

(G)ξx (−1) + · · ·+ θqγ

(G)ξx (−q + 1)

γ(G)(2)(3)= θ2γ

(G)ξx (0) + · · ·+ θqγ

(G)ξx (−q + 2)

...

γ(G)(q)(q+1)= θqγ

(G)ξx (0).

From the (q + 1)th equation, we obtain γ(G)ξx (0) = θ−1

q γ(G)(q) and substitutethis into each of the q preceding equations. Also, in these q equations, wesubstitute for the quantities γ

(G)ξx (·) using (15) and further reduce using (17).

This leads to a nonlinear system of q equations for the q quantities θ1, . . . , θq

in terms of the Gini autocovariances γ(G)(0), . . . , γ(G)(q).

Illustration for q = 1. We have γ(G)(0)(1)= γ

(G)ξx (0)+ θ1γ

(G)ξx (−1) and γ(G)(1)

(2)= θ1γ

(G)ξx (0). From (2) we obtain γ

(G)ξx (0) = θ−1

1 γ(G)(1), which substituted into

(1) yields γ(G)(0) = θ−11 γ(G)(1) + θ1γ

(G)ξx (−1). Now γ

(G)ξx (−1) = π0γ

(G)(−1) =

γ(G)(−1), and our equation for θ1 now becomes

γ(G)(−1)θ21 − γ(G)(0)θ1 + γ(G)(1) = 0,

a quadratic equation with solutions

θ1 =γ(G)(0)±

√γ(G)(0)2 − 4γ(G)(−1)γ(G)(1)

2γ(G)(−1).

8

In practice we adopt rules for selecting one of the two solutions, just as is donewith equations in terms of the usual ACV in the strictly stationary case undersecond order assumptions.

Illustration for q = 2. We arrive at the following equations for θ1 and θ2:

γ(G)(0) = θ−12 γ(G)(2) + θ1(γ

(G)(−1)− θ1γ(G)(−2)) + θ2γ

(G)(−2))

γ(G)(1) = θ1θ−12 γ(G)(2) + θ2(γ

(G)(−1)− θ1γ(G)(−2)).

3.3 AR(p) Processes

Consider the AR(p) model given by

Xt = φ1Xt−1 + · · ·+ φpXt−p + ξt (21)

for some choice of p ≥ 1, with ξt IID with mean 0 and with the causalityassumption that the process may be represented in the form (10) with

∑|ψi| <

∞. We also assume invertibility, which yields the recursion

ψj =

{ ∑0<k≤j φkψj−k, 0 ≤ j < p,∑0<k≤p φkψj−k, j ≥ p,

yielding ψ0 = 1, ψ1 = φ1, ψ2 = φ21 + φ2, etc. To obtain “Yule-Walker” linear

systems of p equations for φ1, . . . , φp, in terms of either the ACV or the GiniACV, respectively, we apply a general “covariance approach” of possible widerinterest.

3.3.1 A general covariance approach

Consider a linear structure

η =

p∑j=1

φjαj + ε, (22)

with ε independent of α1, . . . , αp. We seek a linear system

ai =

p∑j=1

bijφj, i = 1, . . . , p, (23)

or in matrix forma = Bφ, (24)

9

with a = (a1, . . . , ap)T , φ = (φ1, . . . , φp)

T , and B = (bij)p×p (here MT is thetranspose of matrix M). For this we introduce the following simple covarianceapproach. Note that for any function Q(α1, . . . , αp) of α1, . . . , αp we have

Cov(η,Q(α1, . . . , αp)) =

p∑j=1

φjCov(αj, Q(α1, . . . , αp)), (25)

provided that these covariances are finite. Any choice of the p functionsQi(α1, . . . , αp), 1 ≤ i ≤ p, in (25) yields a linear system of form (24), with

ai = Cov(η,Qi(α1, . . . , αp), 1 ≤ i ≤ p,

bij = Cov(ξj, Qi(α1, . . . , αp)), 1 ≤ i, j ≤ p.

If, for some function g, we choose these p functions to be of form Qi(α1, . . . , αp)= g(αi), 1 ≤ i ≤ p, we obtain

ai = Cov(η, g(αi)), 1 ≤ i ≤ p, (26)

bij = Cov(αj, g(αi)), 1 ≤ i, j ≤ p. (27)

We apply the foregoing device with (η, α1, . . . , αp) = (Xt, Xt−1, . . . , Xt−p), asper the AR(p) model (21), and obtain both the usual least squares approachunder second order assumptions and a Gini approach under first order as-sumptions.

3.3.2 The standard least squares method

Use g(αi) = αi and take Qi(Xt−1, . . . , Xt−p) = Xt−i, 1 ≤ i ≤ p, yielding, undersecond order moment assumptions, ai = Cov(Xt, Xt−i) = γ(i), 1 ≤ i ≤ p,and bij = Cov(Xt−j, Xt−i), = γ(|i − j|), 1 ≤ i, j ≤ p. In this case, (24) givesthe usual Yule-Walker equations for φ1, . . . , φp. For p = 1, this least squaressolution is simply φ1 = γ(1)/γ(0).

3.3.3 The Gini-Yule-Walker system

With first order assumptions and using g(αi) = 2(2FX(αi) − 1) along withQi(Xt−1, . . . , Xt−p) = 2(2FXt−i

(Xt−i)−1), we obtain ai = β(Xt, Xt−i) = γ(G)(i),1 ≤ i ≤ p, and bij = β(Xt−j, Xt−i) = γ(G)(i−j), 1 ≤ i, j ≤ p. Then (24) gives aGini-Yule-Walker system for φ1, . . . , φp. For p = 1 this Gini solution is simplyφ1 = γ(G)(1)/γ(G)(0). The Gini-Yule-Walker system has the computationalstructure of the least squares Yule-Walker system but under merely first ordermoment assumptions.

10

3.3.4 Comparison of sample versions

As is well known (Maronna, Martin, and Yohai, 2006), the sample least squaresestimates of AR parameters actually work well even in the presence of outliersand/or heavy tailed innovations, although there are some pitfalls in fittingAR models with heavy-tailed innovations (Feigin and Resnick, 1999). Also,the least absolute deviations (LAD) method for estimation of AR parameters(e.g., Bloomfield and Steiger, 1983) imposes minimal moment assumptions,but it does not yield closed form expressions nor simply solve a linear system.Convenient review and further LAD approaches are provided by Ling (2005).

On the other hand, the Gini estimators of AR parameters are explicit interms of an ACV which has a model formulation even under first order momentassumptions. Further, preliminary simulation studies indicate that, in somescenarios of heavy tails and outliers, the Gini estimators perform better thanthe usual least squares estimators.

3.4 ARMA(p, q) Processes

Including AR and MA as special cases, the ARMA(p, q) model has the form

Xt = φ1Xt−1 + · · ·+ φpXt−p + ξt + θ1ξt−1 + · · ·+ θqξt−q (28)

for some choices of p, q ≥ 1 and with θ0 = 1. Assuming both causality andinvertibility, the coefficients {ψi} in (10) satisfy the recursion

ψj =

{θj +

∑0<k≤j φkψj−k, 0 ≤ j < max{p, q + 1},∑

0<k≤p φkψj−k, j ≥ max{p, q + 1},

yielding ψ0 = θ0 = 1, ψ1 = θ1 + φ1, etc.

3.4.1 Solving for φ1, . . . , φp and θ1, . . . , θq in terms of the Gini ACV

We illustrate the technique for ARMA(1, 1), for which ψ0 = 1 and ψj =θ1φ

j−11 + φj

1, j ≥ 1. Then π0 = 1 and πj = (−1)j(θ1 + φ1)φj−11 , j ≥ 1, yielding

γ(G)ξx (0) =

∞∑i=0

πiγ(G)(−i) = (θ1 + φ1)

∞∑i=0

(−1)iφi−11 γ(G)(−i). (29)

Next we derive the equations γ(G)(1)(1)= φ1γ

(G)(0) + θ1γ(G)ξx (0) and γ(G)(2)

(2)=

φ1γ(G)(1). The second of these yields the solution for φ1:

φ1 = γ(G)(2)/γ(G)(1). (30)

11

Using (29), the first equation then becomes

γ(G)(1) = φ1γ(G)(0) + θ1(θ1 + φ1)

∞∑i=0

(−1)iφi−11 γ(G)(−i). (31)

Since now φ1 is given by (30) in terms of γ(G)(2) and γ(G)(1), this representsa quadratic equation for θ1 in terms of the Gini autocovariances. Since theterms in the infinite sum in (31) decrease rapidly in magnitude, only a fewterms are needed.

4 The Gini ACV for a Nonlinear Process

Here we examine the Gini ACV for a nonlinear type of autoregressive processwith possibly infinite variance. Yeh, Arnold, and Robertson (1988) introduce,in different notation, a nonlinear autoregressive Pareto process YARP(III)(1)given by

Xt =

{p−1/αXt−1, with probability p,

min{p−1/αXt−1, εt}, with probability 1− p,

where 0 < p < 1, with {εt} i.i.d. from the Pareto distribution having survivalfunction [

1 +

(x− µ

σ

)α ]−1

, x ≥ 0,

with the parameters µ, σ > 0, and α > 0 corresponding to location, scale, andtail index, respectively. Henceforth we set µ = 0 for convenience and denotethis distribution by G(σ, α). Only moments of order less than α are finite.

It is understood that εt is independent of {Xs, s ≤ t − 1}. If the seriesis initiated at time t = 0 with X0 distributed as G(σ, α), then the series{Xt, t ≥ 0} is strictly stationary with marginal distribution G(σ, α). In anycase, Xt converges in distribution to G(σ, α).

The sample paths of YARP(III)(1) exhibit frequent rapid increases to peaksfollowed by sudden drops to lower levels (Figure 1). As recently discussed inFerreira (2012), these processes are especially appealing for their tractabilityin certain respects and their straightforward asymptotically normal estimatesof p and α.

It is intuitively evident from the definition that a YARP(III)(1) processexhibits weak dependence for p near 0 and increasing dependence as p ↑ 1.In fact, the probability that the series at time t starts afresh with a newinnovation εt decreases with p as follows (proved in the Appendix):

P (Xt = εt) = 1 +p log p

1− p, 0 < p < 1, (32)

12

independently of σ, α, and t. In particular, we have

p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P (Xt = εt) 0.74 0.60 0.48 0.40 0.31 0.23 0.17 0.11 0.05

However, a correlation type analysis of YARP(III)(1) poses challenges. Undersecond order assumptions (α > 2), the usual ACV is defined but problematic.Yeh, Arnold, and Robertson (1988) give an explicit formula for γ(0) but onlyan implicit formula involving an incomplete beta integral for γ(1) and do nottreat γ(k), k ≥ 2. Therefore, alternative approaches have been developed toexplore quantitatively the dependence features of YARP(III)(1). Thus Ferreira(2012) establishes that this process has unit upcrossing index (upcrossingsof high levels do not cluster) and that its lag k tail dependence coefficient(conditional probability that Xt+k is extreme, given that Xt is extreme) is pk,decaying geometrically with k.

On the other hand, via the Gini ACV, we obtain for YARP(III)(1) a moreexplicit correlation type analysis and one that requires only α > 1. That is,we give explicit closed-form expressions for γ(G)(0), γ(G)(−1), and γ(G)(+1) interms of the model parameters σ, p, and α, as follows. First, as shown in Yeh,Arnold, and Robertson (1988), we have

EXn = σ Γ

(1− 1

α

)Γ

(1 +

1

α

)= σ

π

αcsc(πα

). (33)

Since F is continuous, we have EF (X1) = 1/2 and hence our goal becomesevaluation of

γ(G)(k) = 4

[E(X1+kF (X1))−

σ

2Γ

(1− 1

α

)Γ

(1 +

1

α

)], (34)

reducing the problem to evaluation of E(X1+kF (X1)). Deferring details ofproof to the Appendix, here we discuss the solution and its interpretationsrelative to the parameters α and p of YARP(III)(1). In particular, for k = 0we obtain

γ(G)(0) =2σ

αΓ

(1− 1

α

)Γ

(1 +

1

α

)=

2σ

α

π

αcsc(πα

), (35)

which, of course, is the Gini mean difference of X1 and does not depend uponthe parameter p. For k = ±1 we obtain

γ(G)(−1) = 2σp(1− p1/α)

1− p

π

αcsc(πα

)(36)

γ(G)(1) = 2σp(p−1/α − 1)

1− p

π

αcsc(πα

), (37)

13

which depend upon all three parameters σ, α, and p. Thus the lag ±1 Giniautocorrelations of YARP(III)(1) are given by

ρ(G)(−1) =αp(1− p1/α)

1− p(38)

ρ(G)(1) =αp(p−1/α − 1)

1− p

(= p−1/αρ(G)(−1)

). (39)

These explicit functions of p and α provide simple measures showing how thecorrelation type dependence in YARP(III)(1) varies with these parameters.Such information is not available from the usual ACV and ACF, which lackexplicit formulas even when defined (α > 2).

For |k| ≥ 2, explicit formulas for γ(G)(k) have not been obtained, but analgorithm for numerical evaluation of γ(G)(k) for any k and any desired p andα has been developed, although its computational complexity increases withk. The algorithm requires numerical evaluation of 3 integrals for lags ±1 witha computing time of < 1 second, 11 integrals for lags ±2 with a computingtime of about 5 minutes, 49 integrals for lags ±3 with a computing time ofabout 9 hours, 261 integrals for lags ±4, and 1631 integrals for lags ±5. Inprinciple, Gini autocovariances and autocorrelations for YARP(III)(1) for lags|k| ≤ 3 are readily available if desired and provide sufficient information oncorrelation type dependence for many applications.

Views of the Gini autocorrelations ρ(G)(±1) and ρ(G)(±2) of YARP(III)(1)as functions of p, 0 < p < 1, for α = 1.1, 1.5, 1.75, 2.0, and 2.5, are providedin Figure 2 and Figure 3, respectively. We note that ρ(G)(+1) changessignificantly with α but considerably less with p, whereas ρ(G)(−1) changes verysignificantly with p (approximately as the identity function) but considerablyless with α. Thus ρ(G)(−1) and ρ(G)(+1) provide complementary pieces ofinformation which, taken together, nicely describe how the correlation typedependence structure of YARP(III)(1) relates to α and p. Similar commentsapply to the lag ±2 Gini autocorrelations. For a comparative view of Giniautocorrelations for lags ±1, ±2, and ±3, for α = 1.5 and p = 0.2, 0.5, and0.8, see Figure 4. Clearly, the information for lags ±2 and ±3 does not greatlyenhance the information supplied for the lags ±1.

Nonparametric estimation of γ(G)(k) for k = 0,±1, . . . can be carried outusing the sample versions of Section 6. However, model-based estimates of σ,p, and α are more efficient (straightforward, efficient estimators are providedin Ferreira, 2012). Using these in the explicit expressions or in the numericalalgorithms for the Gini autocovariances, one obtains model-based estimates ofγ(G)(k) conveniently up to lags |k| ≤ 3. Empirical investigation indicates thatthese improve upon the nonparametric estimates, as of course they should,being based on the additional information of a parametric model. For higher

14

order lags, the nonparametric estimators are available and can be computedreadily.

5 A Sample Gini ACV

For data {X1, X2, . . . , XT} from a strictly stationary stochastic process {Xt},we provide a sample Gini ACV based on representation of the Gini ACV interms of relevant distribution functions and substitution of sample versions ofthese distribution functions.

5.1 Estimation of the marginal distribution function F

We start with the usual sample distribution function,

FT (x) = T−1

T∑t=1

I{Xt ≤ x}, −∞ < x <∞,

with I(·) the usual “indicator function” defined as I(A) = 1 or 0 accordingas event A holds or not. This estimator is unbiased for estimation of F (x),and, assuming that the stationary process {Xi} is ergodic, converges to F (x)in suitable senses such as in probability or almost surely, as T →∞.

5.2 Estimation of γ(G)(0)

We estimate γ(G)(0) = α(F ) by the sample analogue estimator α(FT ) basedon (3). It is readily checked that this yields

γ(G)(0) = α(FT ) = 2

∫x (2FT (x)− 1) dFT (x) =

2

T 2

T∑t=1

(2t− T )Xt:T , (40)

whereX1:T ≤ X2:T ≤ · · · ≤ XT :T denote the ordered values of the observations.

5.3 Estimation of γ(G)(k), k 6= 0

For k 6= 0, we use (7) to write

γ(G)(k) = 2

∫ ∫x (2FX1(y)− 1) dFX1+k, X1(x, y). (41)

Then, for k ≥ +1, let F ∗ be the sample version of FX1+k, X1 based on the T −klag k bivariate observations

S∗ = {(X1+k, X1), (X2+k, X2), . . . , (XT , XT−k)}.

15

Also, for k ≤ −1, let F ∗∗ be the sample version of FX1+k, X1 based on the T −klag k bivariate observations

S∗∗ = {(X1, X1+|k|), (X2, X2+|k|), . . . , (XT−|k|, XT )},

which are the same pairs as in S∗ but with the components of each pair reversedin order. These sample bivariate distribution functions, along with FT again,yield estimators of γ(G)(k) by substitution into (41):

γ(G)T (k) = 2

∫ ∫x (2FT (y)− 1) dFX1+k, X1(x, y), (42)

where FX1+k, X1 denotes F ∗ for k ≥ +1 and F ∗∗ for k ≤ −1. For k ≥ +1, thisyields

γ(G)T (k) =

2

T − k

T−k∑t=1

(2FT (Xt:T−k)− 1)X[t:T−k],1,2 , k ≥ 1, (43)

where Xt:T−k is the tth ordered second component value, and X[t:T−k],1,2 is thefirst component value concomitant to the tth ordered second component value,relative to the bivariate pairs in S∗. Similarly, for k ≤ −1, we obtain

γ(G)T (k) =

2

T − |k|

T−|k|∑t=1

(2FT (X[t:T−|k|],2,1)− 1)Xt:T−|k| , k ≤ −1,

where Xt:T−|k| is the tth ordered first component value, and X[t:T−|k|],2,1 is thesecond component value that is concomitant to the tth ordered first componentvalue, relative to the bivariate pairs in S∗∗. However, relative to the set S∗ ofbivariate pairs, this latter equation may be expressed

γ(G)T (k) =

2

T − |k|

T−|k|∑t=1

(2FT (X[t:T−|k|],1,2)− 1)Xt:T−|k| , k ≤ −1, (44)

where Xt:T−|k| denotes the tth ordered second component value and X[t:T−|k|],1,2

denotes the first component value that is concomitant to the tth ordered secondcomponent value. Thus equations (43) and (44) conveniently express γ

(G)T (k),

k = ±1,±2, . . ., in a single notation and relative to the single set of bivariatepairs S∗.

As discussed earlier, for practical use with data it is helpful to express theseas separate Gini autocovariance functions. For this purpose, we put

γ(A)T (k) =

2

T − k

T−k∑t=1

(2FT (Xt:T−k)− 1)X[t:T−k],1,2 , k ≥ 1, (45)

γ(B)T (k) =

2

T − k

T−k∑t=1

(2FT (X[t:T−k],1,2)− 1)Xt:T−k , k ≥ 1, (46)

16

where Xt:T−k is the tth ordered second component value, and X[t:T−k],1,2 is thefirst component value concomitant to the tth ordered second component value,relative to the bivariate pairs in S∗.

We view the sample Gini autocovariance function as a descriptive toolrather than an inference procedure. Therefore, asymptotic convergence theoryis primarily of technical interest and as such is beyond the scope of the presentpaper. Moreover, in view of the considerable additional variability in heavytailed data and modeling, asymptotic distributions would not be applicableexcept for enormous sample length. Rather, in lieu of asymptotic theory, abootstrap approach is recommended for practical applications. Also, for arange of fixed sample sizes, and a range of scenarios for heavy tails and foroutliers, simulation studies of the sample Gini ACV are being carried out in aseparate study.

Acknowledgments

The authors thank G. L. Thompson for encouragement and special insights.They also are very grateful to Edna Schechtman and Amit Shelef for helpfuland constructive remarks and a preprint of their paper. The comments andsuggestions of anonymous reviewers are greatly appreciated and have been usedto develop substantial improvement of the paper. Further, support under aSociety of Actuaries grant, National Science Foundation Grants DMS-0805786and DMS-1106691, and National Security Agency Grant H98230-08-1-0106 isgratefully acknowledged.

References

[1] Bloomfield, P. and Steiger, W. S. (1983). Least Absolute Deviations: The-ory, Applications and Algorithms. Birkhauser, Boston.

[2] Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis: forecastingand control, Revised Edition. Prentice Hall.

[3] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Meth-ods, 2nd edition. Springer.

[4] Chen, Z., Li, R., and Wu, Y. (2012). Weighted quantile regression for ARmodel with infinite variance errors. Journal of Nonparametric Statistics24 715–731.

[5] David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd Edition.Wiley, New York.

17

[6] Davis, R. A. and Resnick, S. I. (1985a). Limit theory for moving averagesof random variables with regularly varying tail probabilities. Annals ofProbability 13 179–195.

[7] Davis, R. A. and Resnick, S. I. (1985b). More limit theory for the samplecorrelation function of moving averages. Stochastic Processes and TheirApplications 30 257–259.

[8] Davis, R. A. and Resnick, S. I. (1986). Limit theory for the sample covari-ance and correlation functions of moving averages. Annals of Statistics 14533–558.

[9] Feigin, P. D. and Resnick, S. I. (1999). Pitfalls of fitting autoregressivemodels for heavy-tailed time series. Extremes 1 391-422.

[10] Ferreira M. (2012). On the extremal behavior of the Pareto Process: analternative for ARMAX modeling. Kybernetika 48 31–49.

[11] Gini, C. (1912). Variabilita e mutabilita, contributo allo studio delle dis-tribuzioni e delle relazione statistiche. Studi Economico-Giuridici dellaReale Universita di Cagliari 3 3–159.

[12] Hosking, J. R. M. (1990). L-moments: Analysis and Estimation of Dis-tributions using Linear Combinations of Order Statistics. Journal of theRoyal Statistical Society, Series B 52 105–124.

[13] Ling, S. (2005). Self-weighted least absolute deviation estimation for in-finite variance autoregressive models. Journal of the Royal Statistical So-ciety, Series B 67 381–393.

[14] Maronna, R. A., Martin, R. D., and Yohai, V. J. (2006). Robust Statistics:Theory and Methods. Wiley.

[15] Olkin, I. and Yitzhaki, S. (1992). Gini regression analysis. InternationalStatistical Review 60 185–196.

[16] Peng, L. and Yao, Q. (2003). Least absolute deviations estimation forARCH and GARCH models. Biometrika 90 967–975.

[17] Pires, A. M. and Branco, J. A. (2002). Partial influence functions. Journalof Multivariate Analysis 83 451–468.

[18] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data (with dis-cussion). Annals of Statistics 25 1805–1869.

18

[19] Rubinstein, M. E. (1973). The fundamental theorem of parameter-preference security valuation. Journal of Financial and Quantitative Anal-ysis 8 61–69.

[20] Schechtman, E. and Yitzhaki, S. (1987). A measure of association basedon Gini’s mean difference. Communications in Statistics – Theory andMethods 16 207–231.

[21] Serfling, R. (1980). Approximation Theorems of Mathematical Statistics.John Wiley & Sons, New York.

[22] Serfling, R. (2010). Fitting autoregressive models via Yule-Walker equa-tions allowing heavy tail innovations. Preprint.

[23] Serfling, R. and Xiao, P. (2007). A contribution to multivariate L-moments: L-comoment matrices. Journal of Multivariate Analysis 981765-1781.

[24] Shelef, A. and Schechtman, E. (2011). A Gini-based methodology for iden-tifying and analyzing time series with non-normal innovations. Preprint.

[25] Yang, X. R. and Zhang, L. X. (2008). A note on self-weighted quantileestimation for infinite variance quantile autoregression models. Statisticsand Probability Letters 78 2731–2738.

[26] Yeh, H. C., Arnold, B. C., and Robertson, C. A. (1988). Pareto processes.Journal of Applied Probability 25 291–301.

[27] Yitzhaki, S. and Olkin, I. (1991). Concentration indices and concentrationcurves. In Stochastic Orders and Decision under Risk (K. Mosler andM. Scarsini, eds.), pp. 380–392. IMS Lecture Notes – Monograph Series,Volume 19, Hayward, California.

[28] Yitzhaki, S. and Schechtman, E. (2013). The Gini Methodology. Springer.

Appendix

Proof of (32). Put F (x) = F0(x/σ), where F0(x) = xα/(1 + xα), and putf0(x) = F ′

0(x) = αxα−1/(1 + xα)2. Then it is easily checked that

P (Xt = εt) = (1− p)α

∫ ∞

0

x2α−1

(p+ xα)(1 + xα)2dx,

which using Mathematica yields (32).

19

Proofs of (35), (36), and (37). (i) For k = 0, we obtain E(X1F (X1)) asfollows. We have

E(X1F (X1)) = σE((X1/σ)F0(X1/σ)) = σ

∫ ∞

0

xF0(x)f0(x) dx

= σα

∫ ∞

0

xxα

1 + xα

xα−1

(1 + xα)2dx = σα

∫ ∞

0

x2α

(1 + xα)3dx

= σα1

α

Γ(2α+1α

)Γ(3− 2α+1α

)

Γ(3)= σ

1

2Γ

(2 +

1

α

)Γ

(1− 1

α

).

Here we have used the standard integral formula∫ ∞

0

ua

(m+ ub)cdu =

m(a+1−bc)/b

b

Γ(a+1b

)Γ(c− a+1b

)

Γ(c), (47)

provided that a > −1, b > 0, m > 0, and c > a+1b

. Here we are applying (47)with a = 2α, b = α, c = 3, and m = 1, and these substitutions do satisfy theconstraints provided that α > 1, which we are assuming throughout. (Also,with the substitutions a = b = α, c = 2, and m = 1, which too satisfy theconstraints provided that α > 1, we obtain (33).) Now returning to (34) fork = 0, we readily obtain

γ(G)(0) = 4

[E(X1F (X1))−

σ

2Γ

(1− 1

α

)Γ

(1 +

1

α

)]=

2σ

αΓ

(1− 1

α

)Γ

(1 +

1

α

)=

2σ

α

π

αcsc(πα

).

Of course, γ(G)(0) is the Gini mean difference of X1.(ii) Turning now to the case k = −1, we evaluate E(X0F (X1)), which may

be expressed as σE((X0/σ)F0(X1/σ)) = σE(XF0(Y )), where

Y =

{p−1/αX, with probability p,

min{p−1/αX, Z}, with probability 1− p,

whereX and Z are independent with distribution F0. LettingW be a Bernoulli(p)random variable independent of X and Z, we represent Y as

Y = p−1/αX I(W = 1) + min{p−1/αX, Z} I(W = 0)

and likewise XF0(Y ) as

XF0(Y ) = XF0(p−1/αX) I(W = 1) +XF0(min{p−1/αX, Z}) I(W = 0).

20

By the independence assumption regarding W ,

E(XF0(Y )) = pE(XF0(p−1/αX)) + (1− p)E(XF0(min{p−1/αX, Z})). (48)

Note that

F0(p−1/α x) =

p−1xα

1 + p−1xα=

xα

p+ xα.

For the first expectation on the righthand side of (48), we have

E(XF0(p−1/αX)) =

∫ ∞

0

xF0(p−1/α x)f0(x) dx = αA(p, α), (49)

where

A(p, α) =

∫ ∞

0

x2α

(p+ xα)(1 + xα)2dx.

For the second expectation on the righthand side of (48), we have

E(XF0(min{p−1/αX, Z})) =

∫ ∞

0

[∫ zp1/α

0

xF0(p−1/α x)f0(x) dx

]f0(z) dz

+

∫ ∞

0

F0(z)

[∫ ∞

zp1/α

xf0(x) dx

]f0(z) dz

= α2[B(p, α) + C(p, α)], (50)

where

B(p, α) =

∫ ∞

0

[∫ zp1/α

0

x2α

(p+ xα)(1 + xα)2dx

]zα−1

(1 + zα)2dz

and

C(p, α) =

∫ ∞

0

[∫ ∞

zp1/α

xα

(1 + xα)2dx

]z2α−1

(1 + zα)3dz.

We thus have

E(XF0(Y )) = pαA(p, α) + (1− p)2α2(B(p, α) + C(p, α)). (51)

(iii) For the case k = +1, we evaluate E(F (X0)X1) = σE(F0(X0/σ)(X1/σ))= σE(F0(X)Y ), with X and Y as above. Similarly to the steps for the casek = −1, we obtain

E(F0(X)Y ) = pE(F0(X)p−1/αX)+(1−p)E(F0(X) min{p−1/αX, Z}). (52)

21

For the first expectation on the righthand side of (52), we have from previoussteps

E(F0(X)p−1/αX) = p−1/α

∫ ∞

0

xF0(x)f0(x) dx

=p−1/α

2Γ

(2 +

1

α

)Γ

(1− 1

α

)=

p−1/α

2

(1 +

1

α

)Γ

(1 +

1

α

)Γ

(1− 1

α

)=

p−1/α

2

(1 +

1

α

)π

αcsc(πα

). (53)

For the second expectation on the righthand side of (52), we have

E(F0(X) min{p−1/αX, Z}) =

∫ ∞

0

[∫ zp1/α

0

p−1/α xF0(x)f0(x) dx

]f0(z) dz

+

∫ ∞

0

z

[∫ ∞

zp1/α

F0(x)f0(x) dx

]f0(z) dz

= α2p−1/αD(p, α) + α2E(p, α), (54)

where

D(p, α) =

∫ ∞

0

[∫ zp1/α

0

x2α

(1 + xα)3dx

]zα−1

(1 + zα)2dz

and

E(p, α) =

∫ ∞

0

[∫ ∞

zp1/α

x2α−1

(1 + xα)3dx

]zα

(1 + zα)3dz.

We thus have

E(F0(X)Y ) = (55)

p1−1/α

2

(1 +

1

α

)π

αcsc(πα

)+ (1− p)α2(p−1/αD(p, α) + E(p, α)).

To complete the derivations of γ(G)(k) for k = ±1, we need to evaluate theintegrals A(p, α), B(p, α), C(p, α), D(p, α), and E(p, α). With the help of

22

Mathematica, the following formulas are obtained:

A(p, α) =2α2(1− p)[(1− p)− αp(1− p1/α)]

2α3(1− p)3

π

αcsc(πα

)B(p, α) =

α[p−1/α(−1 + α+ p+ αp)− 1− α+ p− αp]

2α3(1− p)3

π

αcsc(πα

)C(p, α) =

α[(1− p)(1 + p1+1/α)− 2αp(1− p1/α)]

2α3(1− p)3

π

αcsc(πα

)D(p, α) =

p[α(1− p2)− (1− p)2 − 2α2p(1− p1/α)]

2α3(1− p)3

π

αcsc(πα

)E(p, α) =

αp−1/α[2αp2(1− p1/α) + p1/α(1− p)(1− 2p− p1−1/α)]

2α3(1− p)3

π

αcsc(πα

).

Using these in (51) and (55), and combining with (34), we obtain (36) and(37).

23

0 50 100 150 200

010

2030

4050

60

Index

Obs

erva

tions

0 50 100 150 200

05

1015

2025

Index

Obs

erva

tions

0 50 100 150 200

02

46

810

1214

Index

Obs

erva

tions

Figure 1: Sample paths of an YARP(III)(1) process of time length 200, for α= 1.5 and p = 0.2 (upper left), p = 0.5 (upper right), and p = 0.8 (lower).Note that the vertical scales differ.

24

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

Figure 2: Lag +1 (upper curve) and Lag−1 (lower curve) Gini autocorrelationsof YARP(III)(1) processes, for α = 1.1 (upper left), 1.5 (upper right), 1.75(middle left), 2.0 (middle right), and 2.5 (lower), and 0 < p < 1.

25

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

Gin

i Cor

rela

tion

Figure 3: Lag +2 (upper curve) and Lag−2 (lower curve) Gini autocorrelationsof YARP(III)(1) processes, for α = 1.1 (upper left), 1.5 (upper right), 1.75(middle left), 2.0 (middle right), and 2.5 (lower), and 0 < p < 1.

26

p = 0.2

k

Gin

i Aut

ocor

rela

tion

0.0

0.2

0.4

0.6

0.8

1.0

−1 1 −2 2 −3 3

p = 0.5

k

Gin

i Aut

ocor

rela

tion

0.0

0.2

0.4

0.6

0.8

1.0

−1 1 −2 2 −3 3

p = 0.8

k

Gin

i Aut

ocor

rela

tion

0.0

0.2

0.4

0.6

0.8

1.0

−1 1 −2 2 −3 3

Figure 4: Gini autocorrelations of YARP(III)(1) processes, lags k =−1,−2,−3(clear bars) and +1,+2,+3 (shaded bars), for parameters α = 1.5 and p = 0.2(top), 0.5 (middle), and 0.8 (bottom).

27