Statistics and Probability Letters 78 (2008) 2963–2970

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

Feasible parameter regions for alternative discrete state space modelsI

Paul D. Feigin a, Phillip Gould b,c, Gael M. Martin b,∗, Ralph D. Snyder ba Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Israelb Department of Econometrics and Business Statistics, Monash University, AustraliacMonash University Accident Research Centre, Australia

a r t i c l e i n f o

Article history:Received 2 October 2007Received in revised form 9 May 2008Accepted 11 May 2008Available online 24 May 2008

JEL classification:C22C46C52

a b s t r a c t

This paper provides a comparison of a parameter-driven and an observation-drivendiscrete state space model. The two models are shown to have non-overlapping feasibleregions for dispersion and first-order autocorrelation, with the region for the parameter-driven model being much larger than that of the observation-driven model, as well asproviding amuch better representation of the empiricalmoments of observed count series.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Models for time series of counts can be categorized as either ‘observation-driven’ or ‘parameter-driven’; see Cox (1981).In the former case, serial correlation is modelled directly via lagged values of the count variable, with strategies adoptedto ensure that the integer nature of the data is preserved (e.g. the binomial thinning operation used in the integer-valuedautoregressive (INAR) models of Al-Osh and Alzaid (1987) and McKenzie (1988)). In the case of parameter-driven models,correlation is introduced indirectly by specifying the parameter(s) of the conditional distribution to be a function of acorrelated latent stochastic process. The random parameter model is equivalent, in turn, to a dual source of error (DSOE)discrete state spacemodel, inwhich bothmeasurement and state equation contain a source of randomness; see, for example,West et al. (1985), Zeger (1988), Harvey and Fernandes (1989), West and Harrison (1997), Davis et al. (2000), Durbin andKoopman (2001) and McCabe et al. (2006).1An intermediate class of models includes the generalized linear autoregressive moving average (GLARMA) models of

Shephard (1995) and Davis et al. (1999, 2003), the autoregressive conditional Poisson (ACP) model of Heinen (2003) and theautoregressive conditional ordered probit (ACOP) model of Jung et al. (2006). In these models, autocorrelation is modelledindirectly by allowing (functions of) the parameter(s) of the conditional distribution to be both serially correlated anddependent on lagged counts. Suchmodels are thus, in style, parameter-driven. However, in contrast with a DSOEmodel, thelatent parameter(s) in these models, conditional on lagged values of the counts and initial values for the latent parameters,are deterministic. As a consequence, such models can be referred to as single source of error (SSOE) models and wouldtypically be classified as observation-driven.

I This research has been partially supported by Australian Research Council Discovery Grant No. DP0450257 and Israel Science Foundation Grant No.1046/04. The authors would like to thank a referee for some very constructive and insightful comments on an earlier draft of the paper.∗ Corresponding address: Department of Econometrics and Business Statistics, P.O. Box, 11E, Monash University, Victoria, 3800, Australia.E-mail address: gael.martin@buseco.monash.edu.au (G.M. Martin).1 The term ‘dual’, rather than ‘multiple’ is used here in order to emphasize the fact that randomness characterizes both the measurement and state

equations. These equations could, of course, be defined for vectors, in which case the dual sources of error encompass multiple scalar error terms.

0167-7152/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2008.05.021

2964 P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970

The contrast between the SSOE and DSOE models for count data is analogous to the contrast between a generalizedautoregressive conditionally heteroscedastic (GARCH) model and a stochastic volatility (SV) model for financial returns;see Kim et al. (1998). It also mimics the contrast between the autoregressive conditional duration (ACD) model for tradedurations (Engle and Russell, 1998) and the alternative stochastic conditional duration (SCD) model for the same data type(e.g. Bauwens andVeredas (2004), Strickland et al. (2006)). As in these other non-Gaussian settings, the relativemerits of theSSOE and DSOE approaches to modelling counts remains an open empirical question. Whilst the latter approach may yieldmore flexibility than the former approach, via the introduction of the additional source of random error, it does so at the costof computational ease, with estimation of the dual source models often entailing the use of some form of computationallyintensive simulation methodology (e.g. Chan and Ledolter (1995), Durbin and Koopman (2001), Frühwirth-Schnatter andWagner (2006) and Jung et al. (2006)).2

The aim of this paper is to quantify the relative flexibility of the two alternative discrete state-space models via oneparticular criterion: the region of feasible combinations of a measure of over-dispersion (D = marginal variance

E[conditional variance] ) andfirst-order autocorrelation (C). Using, as the archetypical example, a conditional Poisson distribution (for which D =marginal variancemarginal mean ), we show that, as well as there being no overlap in the feasible regions for the twomodels, the DSOE region ismuch larger than that for the SSOE model. Of 13 empirical count time series analysed only one has empirical moments thatfall within the feasible region for the SSOE model. Ten series tally with the region of the DSOE model, while for two seriesthe sample moments fall within neither theoretical region.

2. Two models for count time series

The count time series variable, yt , at time t , when conditioned on the value of its (possibly) stochastic and time-varying(mean and variance) parameter λt , is assumed to have a Poisson distribution, P(·).3 The SSOE model is defined as

yt |λt ∼ P (λt) (1)λt = λ+ φλt−1 + α(yt−1 − λt−1), (2)

for t = 2, . . . , T , where the restrictions λ > 0, φ ≥ α ≥ 0 and φ ≤ 1 are imposed, and only first-order lags are assumed forthe sake of illustration. Defining It−1 = {λ1, y1, y2, . . . , yt−1}, from (2) it follows that, conditional on It−1, the mean (andvariance) of yt is λt . The SSOE model in (1) and (2), for corresponding parameter values, is equivalent in distribution to theACP (1, 1) model specified in Heinen (2003) and also investigated in Jung et al. (2006).The DSOE model is defined as (1), but with

λt = h(xt) (3)

xt = a+ κxt−1 + ηt; ηt ∼ iidN(0, σ 2η ), (4)

for t = 1, 2, . . . , T , where h(xt) is any function that maps xt into the positive space of λt and the stationarity restriction|κ| < 1 is imposed. For simplicity we assume that h(.) is the exponential function. Conditionally on {λt; t = 1, 2, . . . , T },{yt} is assumed to be an independent sequence of Poisson counts, with corresponding mean and variance sequence {λt}.The model in (1), (3) and (4), extended to cater for covariates, is analysed in Chan and Ledolter (1995) and Jung et al. (2006),amongst other studies.Both models presented here for {yt , λt} are examples of bivariate Markov chains. Under the conditions assumed, they

are ergodic and have unique stationary distributions (see Appendix A). The moment calculations are conducted under theassumption that the Markov chains are in their stationary regimes. For applications, this assumption amounts to assumingthat the process has been running from the past, distant enough such that at time t = 0 it has already reached its limitingstationary regime.In Sections 2.1 and 2.2we summarize themain properties of the twomodels, withmore details provided in Appendices A

and B.

2.1. Properties of the SSOE model

We denote the (stationary distribution) mean of {λt} by µ = λ(1−φ) .

Theorem 1. Given (2), with 0 ≤ α ≤ φ < 1, {λt} has a stationary distribution with moments,

2 In the work ofWest et al. (1985) and Harvey and Fernandes (1989), natural conjugate distributions are chosen in order to enable non-simulation basedinferential treatment of the discrete DSOE models adopted therein.3 Clearly, other parametric discrete distributions could be chosen, with the choice being based on the empirical features of the data. The model would

then allow for the parameter(s) of this distribution to be (possibly) stochastic and time-varying. For simplicity we use the Poisson distribution throughoutthe paper, as well as focussing on models without covariates.

P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970 2965

E(λt) = µ (5)

var(λt) = µα2

(1− φ2). (6)

Theorem 2. Given (1) and (2), with 0 ≤ α ≤ φ < 1, the count variable {yt} has a stationary distribution with marginalmoments,

M ≡ E(yt) = µ (7)

V ≡ var(yt) = µ+ var (λt) =µ[1− φ2 + α2]

1− φ2> µ (8)

cov(yt , yt−1) =µα[1− φ2 + φα]

1− φ2. (9)

Using (7)–(9), the first-order autocorrelation and dispersion are given respectively by

C ≡ cor(yt , yt−1) =α[1− φ2 + φα][1− φ2 + α2]

(10)

D ≡var(yt)

E[var(yt |λt)]=

VE[λt ]

=VM= 1+

α2

1− φ2. (11)

Heinen (2003) provides proofs of the above moment results and refers to the durations analysis of Engle and Russell(1998) for a demonstration of how the proof of stationarity would proceed. For completeness, in Appendix A, Section A.1we provide information about the stationary distribution of the count model via a characterization of its Laplace transform.Given Theorems 1 and 2, the following properties can also be derived.

SSOE: Property 1. Given φ < 1 and λ > 0 and the resulting non-degenerate stationary distribution, the model does notsuffer the fixed point problem (yt → 0 a.s. as t →∞when λ = 0) highlighted in Grunwald et al. (1997).

SSOE: Property 2. Given φ < 1, and conditional on λ1, if α = 0, the λt process is deterministic with a limiting value of µ,such that the stationary distribution of the process has fixed mean λt = µ and yt ∼ P (µ) .

SSOE: Property 3. For fixed 0 ≤ α < 1, C increases from α > 0 to 1 as φ increases from α to 1. Negative correlation is thusnot possible with the SSOE model.

SSOE: Property 4. Given {(α, φ) : 0 ≤ α ≤ φ < 1}, D ≥ 1. If α = 0 then D = 1 (and yt is equidispersed as a consequence),for any φ < 1.

The following Corollary also follows from Theorem 2:

Corollary 1. Given {(α, φ) : 0 ≤ α ≤ φ < 1}, the feasible region for the dispersion D, given 0 ≤ C < 1 is

11− C2

≤ D <1

1− C. (12)

Proof. Provided in Appendix B.1. �

As is clear from Fig. 1, the feasible region of (D, C) values implied by (12) is very limited, with there being only a narrowrange of quite modest values of D possible for a given value of C , unless C is quite large (C > 0.9, say).

2.2. Properties of the DSOE model

We denote the (stationary distribution) mean and variance of xt by µX = a1−κ and σ

2X =

σ 2η(1−κ2)

respectively.

Theorem 3. Given (3) and (4), with |κ| < 1, {λt} has a stationary distribution with moments,

E(λt) = e{µX+0.5σ 2X

}(13)

var(λt) = e2{µX+σ

2X

}− e2µX+σ

2X . (14)

Proof. Straightforward using the properties of the log-normal distribution. See also Appendix A.2. �

2966 P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970

Fig. 1. Feasible values of dispersion (D) and first-order autocorrelation (C) for the SSOE and DSOE models.

Theorem 4. Given (1), (3) and (4), with |κ| < 1, the count variable {yt} has a stationary distribution with marginal moments

M ≡ E(yt) = E (E [Yt |λt ]) = E(λt) = e{µX+0.5σ 2X

}(15)

V ≡ var(yt) = E(var [Yt |λt ])+ var(E [Yt |λt ]) = M +M2(e{σ 2X

}− 1) (16)

cov(yt , yt−1) = M2[e{κσ 2X

}− 1

]. (17)

Given (15)–(17), first-order correlation and dispersion are given respectively by

C ≡ cor(yt , yt−1) =e{κσ 2X

}− 1

e{σ 2X

}− 1+ 1/M

(18)

D ≡var(yt)

E[var(yt |λt)]=

VE[λt ]

=VM= 1+M(e

{σ 2X

}− 1). (19)

Proof. Straightforward using iterated expectations. �

Given Theorems 3 and 4, the following properties can also be derived:

DSOE: Property 1. Given κ < 1, and conditional on x0, if ση = 0, the xt process is deterministic with a limiting value ofµX ,such that the stationary distribution of the process has fixed mean λt = eµX and yt ∼ P (eµX ) .

DSOE: Property 2. C = 0 ⇐⇒ either κ = 0 or ση = 0. C is an increasing function of κ and ση and is negative for κ < 0.DSOE: Property 3. Given {(a, κ, ση) : a ∈ R, |κ| < 1, ση > 0},D ≥ 1. If ση = 0, D = 1, for any |κ| < 1. As |κ| → 1,

D→∞, for any ση > 0.

The following Corollary also follows from Theorem 4:

Corollary 2. Given {(a, κ, ση) : a ∈ R, |κ| < 1, ση > 0}, the feasible region for the dispersion D, for 0 ≤ C < 1, is

D >1

1− C. (20)

For −1 < C < 0, D > 1.

Proof. Provided in Appendix B.2. �

In comparisonwith the feasible region in (12) for the SSOEmodel, for any given value of C the possible values ofD, beyondthe lower bound of 1

1−C , are unrestricted in (20). Furthermore, in the positive correlation region, inwhich bothmodels apply,there is no overlap in the feasible regions for (D, C) for the two alternatives. Only theDSOEmodel is applicable to the negativecorrelation case. Note that by the very nature of the models and definitions, under dispersion is never catered for.

P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970 2967

Table 1Summary statistics for empirical count time series

Time Freq. Sample Sample Sample Sample Sample SSOE DSOEseries size mean variance dispersion ACF(1) lb lbyt a (T ) (M) (V ) (D = V/M) (C) 1

1−C211−C

CUTS(S) MTb 120 6.13 11.80 1.92 0.56 1.46 2.27MEDICAL(D) MT 84 2.08 2.66 1.27 0.08 1.01 1.09FAT_4060(D) MT 110 6.97 10.32 1.48 0.14 1.02 1.17POLICE(D) MT 72 7.22 14.25 1.97 0.15 1.02 1.25CHOKE(D) MT 84 3.24 7.12 2.20 0.18 1.03 1.22UNDETERM(D) MT 84 1.73 2.73 1.58 0.22 1.05 1.28ASTHMA(D) Dc 1461 1.94 2.70 1.39 0.25 1.07 1.33TRADES(D) IDd 360 4.52 15.07 3.33 0.33 1.13 1.50UNINTENT(D) MT 84 2.96 10.25 3.46 0.39 1.18 1.65DEFAULTS(D) Ae 85 1.74 9.62 5.53 0.52 1.38 2.09LATE(D) MT 84 1.62 1.97 1.22 −0.14 1.02 0.88FAT_60(N) MT 78 2.06 2.01 0.97 0.15 1.02 1.18BURNS(N) MT 120 0.18 0.16 0.93 0.22 1.05 1.29a Model that fits the sample (D, C) values for yt is denoted by the superscript: S = SSOE model; D = DSOE model; N = neither model.b MT = monthly frequency.c D = daily frequency.d ID = intraday frequency.e A = annual frequency.

2.3. Empirical count time series

In Table 1 the descriptive properties of 13 empirical count time series are summarised. The data sets fall into six differentcategories which span a wide range of empirical settings. In particular, all data series are low count series, thereby requiringexplicit modelling via a discrete conditional distribution. Although no formal stationarity tests are conducted, the estimatedvalue of C suggests that the stationarity assumption is valid for these data sets.

1. Australian monthly road accident deaths and injuries: (i) fatalities in 60 km/h zones in Victoria, Australia from January1996 to June 2002 (FAT_60); (ii) fatalities in 40–60 km/h zones in Victoria from January 1996 to February 2005(FAT_4060); (iii) non-fatal police-car-related injuries in New SouthWales from January 1987 to December 2004 (POLICE).

2. Australian monthly deaths by other (non-road-related) causes: (i) medical-related deaths (MEDICAL); (ii) choking-related deaths (CHOKE); (iii) death inflicted by another but with undetermined intent (UNDETERM); (iv) death inflictedby another but unintentionally (UNINTENT); (v) deaths by late effects (LATE).

3. Monthly wage loss benefit claims for injuries sustained in the British Columbia (Canada) logging industry: (i) claims forburn injuries from January 1984 to December 1994 (BURNS); (ii) claims for cut and/or laceration injuries for the sameperiod (CUTS).

4. Annual corporate failures (DEFAULTS) of investment grade firms from the U.S., over the period 1920–2000.5. One-minute trade counts (TRADES) for the Australian firm Broken Hill Proprietary (BHP) Limited for 1 August 2001,10.00 am to 4.00 pm.

6. Daily admissions for asthma treatment (ASTHMA) to a Sydney hospital from 1 January 1990 to 31 December 1993.

The claims data sets in 3. have been extensively analysed in the literature using INAR-type specifications, most notablyin Freeland and McCabe (2004a,b), McCabe and Martin (2005) and Zhu and Joe (2006). The admissions data in 6. have beenanalysed previously by Davis et al. (1999) using a GLARMA model and Davis et al. (2000) using a generalized linear model(with allowance made for an autocorrelated latent process). Jung et al. (2006) also use the admissions data in an empiricalcomparison of a range of count time series models, including SSOE and DSOE models.4

The term in the superscript attached to the series name in the first column of the table indicates which model fits thesample (D, C) characteristics of the observed data (S = SSOE model; D = DSOE model; N = neither model). As is clear,based on the estimated parameter functions, only one series (CUTS) falls into the narrow feasible set for the SSOE model( 11−C2≤ D < 1

1−C ), with nine of the serieswith positive correlation satisfying theDSOE criterion (D >11−C ). Twoof the series

are underdispersed; hence neither model is suitable. The dispersion value of the single series with negative autocorrelation(LATE) exceeds the lower bound of the DSOE model (D > 1), the only model of the two which is valid for the negativecorrelation case.

4 All data in 1. and 2. have been provided by the Monash University Accident Research Centre. The stock market data in 4. is used in the trade durationanalysis of Strickland et al. (2006).

2968 P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970

3. Conclusions

In summary then, the dual and single source of error discrete state space models have markedly different dispersionand autocorrelation properties, with the DSOE model seeming to provide a better match to the corresponding empiricalproperties of observed count data, at least based on point estimates of the relevant moment functions. Obvious extensionsto the analysis include the generalization of the conditional Poisson distribution to more flexible distributions such as thenegative binomial and the double Poisson, that allow for separate dynamic specifications for the mean and variance (see,for e.g. Heinen (2003), McCabe et al. (2006)).5In particular, a more flexible parameterisation of the conditional distributionmay lead to a larger feasible dispersion/autocorrelation region for the single source model than is associated with the singleparameter Poisson case. Also, although the focus here is on count data, these results suggest that similar results may obtainfor anydata defined on a restricteddomain. In particular, an obvious focus for future research is the production of comparableresults for the alternative state space representations of positive durations data.

Appendix A. The stationary distribution of {λt} and {yt}

A.1. The SSOE model

We seek to characterize the stationary distributions of yt and λt when φ < 1, and derive an expression for their Laplacetransforms. Note that the pair {yt , λt} form a bivariate Markov chain. Denote the Laplace transform of yt by Lt(·) and thatof λt by Mt(·). Given the conditional Poisson distribution for yt in (1), and defining It−1 as in the text, it follows thatLt(u) = E [E(exp(−uyt)|It−1)] = E exp

[−λt(1− e−u)

]= Mt(1 − e−u). If the limit M(·) of Mt(·) exists, then the Laplace

transform, L(·), of the stationary distribution of yt will also exist and satisfy

L(u) = M(1− e−u). (21)

From (2) we deduce that

Mt(v) = E [E(exp (−vλt) |It−2)] = exp {−vλ}Mt−1 (g(v; δ, α)) , (22)

where g(v) ≡ g(v; δ, α) = vδ + 1 − e−vα and 1 > δ ≡ φ − α ≥ 0. Define the kth iterate of g(v) by g(k)(v) =g(g(k−1)(v)

); g(0)(v) ≡ v. Applying (22) iteratively we deduce:

Theorem 5. Given 0 ≤ φ < 1,Mt (v) converges to

M (v) = exp

{−λ

∞∑k=0

g(k) (v)

}(23)

as t →∞, where∑∞

k=0 g(k) (v) <∞.

By (21) and (23), the Laplace transform of the stationary distribution of yt satisfies L(u) = exp{−λ

∑∞

k=0 g(k)(1− e−u)

}for u ≥ 0.

A.2. The DSOE model

The existence of a unique limiting stationary distribution for yt follows from the fact that {yt , λt} is a bivariate Markovchain and that {xt = log(λt)} is a Gaussian autoregressive process with known stationary distribution under the conditionsgiven (namely, |κ| < 1). Conditionally on {λt}, the yt are independent Poisson variables with E(ys|{λt}) = λs.

Appendix B. Feasible (D,C) regions

B.1. The SSOE model: Proof of Corollary 1

FromM ≡ E(yt) = λ1−φ , V ≡ var(yt) =

M[1−φ2+α2]1−φ2

and

C ≡ cor(yt , yt−1) =α[1− φ2 + φα][1− φ2 + α2]

(24)

D ≡VM= 1+

α2

1− φ2, (25)

5 For general non-Poisson based models, with conditional distribution parameter (or parameter vector) θt , the more general definition of dispersionwould be used. Using the expansion, var(yt ) = E[var(yt |θt )] + var[E(yt |θt )], we see that the general version of D also only measures overdispersion (andequidispersion), since D = var(yt )

E[var(yt |θt )]= 1+ var[E(yt |θt )]

E[var(yt |θt )]≥ 1.

P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970 2969

we can deduce the following inversions from (M, V , C) to (λ, φ, α):

φ = C +

√D(1− C2)− 1D(D− 1)

(26)

α = CD− φ(D− 1) (27)λ = M(1− φ). (28)

The solution (26) for φ in terms of C and D follows by noting that (27) follows from (24) and (25), with substitution of (27)into (25) yieldingD = 1+ (CD−φ(D−1))2

1−φ2. Solving this quadratic equation in φ, we obtain (after showing from (24) that C ≤ φ)

the solution in (26), as well as the condition that

D ≥1

(1− C2). (29)

Solving (24) for α we obtain (φ − C)α2 + (1− φ2)α − (1− φ2)C = 0, with (positive) solution α = −1+√1+4C(φ−C)/(1−φ2)2(φ−C)/(1−φ2)

.Thus D from (25) can be written as

D = 1+1+ 2C(φ − C)/(1− φ2)−

√1+ 4C(φ − C)/(1− φ2)

2(φ − C)2/(1− φ2).

Now α ≤ C ≤ φ < 1 follows from (24) — since φ ≥ α so that C/α ≥ 1 — and from (26). Thus, the range of values of Das a function of given C can be determined by allowing φ to vary from C through to 1. As φ ↑ 1, D→ 1 + C 1−C

(1−C)2=

11−C .

Defining U = (φ − C) and k = 4C/(1 − φ2), as φ ↓ C it follows that U = (φ − C) ↓ 0 and k ↓ 4C/(1 − C2). Given that

D = 1 + 2+kU−2√1+kU

kU2/C∼ 1 +

2+kU−2(1+ 12 kU−18 k2U2)

kU2/C∼ 1 + C2

(1−C2), it follows that as φ ↓ C , D→ 1

1−C2. From (29) it follows

that D approaches 11−C2

from above. Moreover, from (24) one can also show that C−αφ−C =

α2

1−φ2, so that for each φ there exists

α such that D = 1+ C−αφ−C ≤ 1+

Cφ−C . As the latter holds for any 1 > φ ≥ C , we can conclude that D approaches 1

1−C frombelow, i.e. that

D <1

1− C. (30)

From (29) and (30) it follows that (12) defines the feasible range of D values for any 0 ≤ C < 1, with both bounds beingtight. Note that when C = 0 then D = 1 (and φ = α = 0).

B.2. The DSOE model: Proof of Corollary 2

FromM ≡ E(yt) = e{µX+0.5σ 2X

}, V ≡ var(yt) = M +M2(e

{σ 2X

}− 1) and

C ≡ cor(yt , yt−1) =e{κσ 2X

}− 1

e{σ 2X

}− 1+ 1/M

(31)

D ≡VM= 1+M(e

{σ 2X

}− 1), (32)

we can deduce the following inversions from (M, V , C) to (a, κ, σ 2η ):

κ =log

[ CDM + 1

]log

[D−1M + 1

] (33)

a = (1− κ)(log(M)−

12log

[D− 1M+ 1

])(34)

σ 2η = (1− κ2) log

[D− 1M+ 1

]. (35)

Note that if C = 0 we obtain κ = 0 from (33), but we need not have D = 1 as in the case of the SSOE model. From Eqs. (31)and (32), and using the fact that exp{κσ 2X } < exp{σ

2X } for 0 ≤ κ < 1, we can conclude that for κ ≥ 0(⇒ C ≥ 0), C <

D−1D

which, in turn, implies that D > 11−C . It also follows from (33) that for fixed C , D can be arbitrarily close to

11−C by choosing

κ sufficiently close to 1. Thus this lower bound for D is tight.

2970 P.D. Feigin et al. / Statistics and Probability Letters 78 (2008) 2963–2970

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