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A class of correlated weighted Poisson processes
Journal of Statistical Planning and Inference 142 (2012) 366–375
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A class of correlated weighted Poisson processes P. Borges a,, J. Rodrigues b, N. Balakrishnan c a
Departamento de Estatı´stica, Universidade do Espı´rito Santo - UFES. Av. Fernando Ferrari 514, Goiabeiras, CEP 29075-910 Vitøria ES, Brazil ~ Carlos, Brazil Departamento de Estatı´stica, Universidade de Sao c Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 b
a r t i c l e in f o
abstract
Article history: Received 14 February 2011 Received in revised form 30 July 2011 Accepted 4 August 2011 Available online 11 August 2011
In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COMPoisson distributions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Correlated Poisson process on [0,1] Standard uniform variables Exchangeable variables Correlation coefficient Correlated weighted Poisson process
1. Introduction In a recent paper, Balakrishnan and Kozubowski (2008) introduced a class of weighted Poisson processes (WPP) on the interval [0,1]. This family assumes that the sequence of standard uniform variables are independent and the unobservable count variable follows a weighted Poisson distribution with parameter l and weight function wðÞ; see, for example, Castillo and Pe´rezCasany (1998, 2005). The probability mass function (PMF) of the WPP with intensity l, fNw ðtÞ,t 2 ½0,1g, is given by
P½Nw ðtÞ ¼ k ¼
elt ðltÞk wt ðkÞ , k! elð1tÞ El ½wðNÞ
k ¼ 0,1, . . . , l 40, t 2 ½0,1,
P n where wt ðkÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þ kÞ. Following their approach, we generalize in this paper the above WPP by considering a dependence structure between the standard uniform variables. In this manner, we obtain a new class of processes that we refer to as correlated weighted Poisson process (CWPP) with intensity l and correlation r, fNrw ðtÞ,t 2 ½0,1g, whose PMF is 8 w w k ¼ 0, > < rð1tÞ þ rt GN ð0Þ þ ð1rÞGN ð1tÞ, k w lt e ð l tÞ w ðkÞ P½Nr ðtÞ ¼ k ¼ t w > , k ¼ 1,2, . . . : rt P½N ¼ k þ ð1rÞ k! elð1tÞ El ½wðN; fÞ for l 4 0, t, r 2 ½0,1, where Nw is a weighted Poisson variable with parameter l and weight function wðÞ, GNw ðÞ is the P n probability generating function of Nw, and wt ðkÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þkÞ. Particular choices of wðÞ lead to some Corresponding author.
E-mail address: [email protected] (P. Borges). 0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.08.002
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
367
important special cases, including CWPP with geometric as well as negative binomial marginal distributions, which frequently arise in applications; see for example, Kozubowski and Podgo´rski (2009) and the references therein. This new class of processes will enable one to tackle the problem of overdispersion and underdispersion inherent in the analysis of count data (see Hinde and Demetrio, 1998) that may arise due to the presence of some correlation between the events. For example, in models of carcinogenesis (see Yakovlev and Tsodikov, 1996), it is usually assumed that cells in a tissue (modeled by a homogeneous Poisson process with intensity l) can give rise to tumors independently of each other, i.e., they are biologically independent as far as carcinogenesis is concerned. Then, statistical independence is used to model biological independence. However, the biological independence assumption may not be true when the dynamics of the cell population of a normal tissue is considered. It is therefore desirable to construct mathematically tractable models of carcinogenesis that can adequately incorporate biological dependence, and this is what motivated the present research work. The rest of this paper is organized as follows. In Section 2, we introduce the correlated Poisson process before formulating the class of correlated weighted Poisson processes in Section 3. In Section 4, the theoretical results are illustrated through some examples. Finally, some concluding remarks are made in Section 5. 2. The correlated Poisson process Let N have a Poisson distribution with parameter l 4 0, and let U1 ,U2 , . . . ,UN be a sequence of independent standard P uniform variables, independently of N. Then, the random sum NðtÞ ¼ N j ¼ 1 I½0,t ðUj Þ, where IA is an indicator of the set A, is a classical Poisson process on the interval [0,1], fNðtÞ,t 2 ½0,1g, with intensity l. We generalize this Poisson process model by relaxing the assumption of mutual independence between the standard uniform variables. To introduce dependence, we assume that the variables are assigned with equal correlation between them. In this process, we obtain another process on the interval [0,1], fNr ðtÞ,t 2 ½0,1g, with intensity l and correlation r, which we refer to as correlated Poisson process (CPP). Thus, for the correlated Poisson process, we have Nr ðtÞ ¼
N X
I½0,t ðUj Þ
for t 2 ½0,1,
ð1Þ
j¼1
where Uj’s are now equicorrelated standard uniform variables, independently of N; i.e.,
CorrðUi ,Uj Þ ¼ r, iaj, i,j ¼ 1,2, . . . ,N ¼ n:
ð2Þ
Remark 1. This model is strongly connected with the concept of exchangeability which restricts the possible correlation structure for the sequences of random variables (r.v.’s) with finite first two moments (Aldous et al., 1985). If ðU1 ,U2 , . . . ,Un Þ is a finite sequence of exchangeable r.v.’s, then 1 ,1 , iaj: ð3Þ CorrðUi ,Uj Þ ¼ r 2 n1 Conversely, every r satisfying (3) serves as a correlation in some exchangeable sequence. To see this, let ðX1 , . . . ,Xn Þ be a sequence of independent r.v.’s with EðXi Þ ¼ 0, VðXi Þ ¼ 1, and define Ui ¼ Xi þd
n X
Xk ,
i ¼ 1,2, . . . ,n,
k¼1
for a constant d Z 1=n. Thus, ðU1 , . . . ,Un Þ is an exchangeable sequence and
r ¼ EðUi Uj Þ ¼ 1
1 2
nd þ2d þ1
,
iaj:
Varying d, it is possible to obtain all values of r 2 ½1=ðn1Þ,1. For example, r ¼ 1=ðn1Þ when d ¼ 1=n, and r ¼ 0 when d ¼0. For infinite exchangeable sequences, we of course have r 2 ½0,1. Now, let us note that distributionally, the variables Zj ¼ I½0,t ðUj Þ have the same Bernoulli distribution with parameter t ¼ P½Uj o t 2 ½0,1 and corresponding probability generating function (PGF) as GZ ðsÞ ¼ ð1tÞ þ ts, s 2 ½0,1. However, now U1 , . . . ,Un are equicorrelated (CorrðUi ,Uj Þ ¼ r for iaj) with identical marginal distributions. Therefore, Z1 ,Z2 , . . . ,Zn are exchangeable Bernoulli variables with CorrðZi ,Zj Þ ¼ r1 , for some r1 2 ½1=ðn1Þ,1 and iaj, as noted above in Remark 1. The following lemma gives a necessary and sufficient condition for r1 ¼ r. Lemma 1. Let CorrðUi ,Uj Þ ¼ r for iaj, where t 1t 1 , , ,1 : r 2 max 1t t n1
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Then, CorrðZi ,Zj Þ ¼ r iff
P½Zi ¼ 19Zj ¼ 1 ¼ t þ ð1tÞr and P½Zi ¼ 09Zj ¼ 0 ¼ ð1tÞ þ tr:
ð4Þ
Proof. The r.v.’s Zi (i¼1,2,y) are identically distributed as Bernoulli with parameter t, and so EðZÞ ¼ t ¼ PðUi o tÞ and VarðZÞ ¼ tð1tÞ. Then, CorrðZi ,Zj Þ ¼ r for iaj iff CovðZi ,Zj Þ ¼ r VarðZÞ, i.e., when EðZi Zj Þt 2 ¼ tð1tÞr. On the other hand, we simply have
EðZi Zj Þ ¼ P½Zi ¼ 1,Zj ¼ 1: As a result, we have P½Zi ¼ 1,Zj ¼ 1 ¼ t 2 þ tð1tÞr, and by the conditional probability formula, we then obtain P½Zi ¼ 19Zj ¼ 1 ¼ t þ ð1tÞr. The second relation in (4) follows from the fact that Zin ¼ 1Zi ,
i ¼ 1,2, . . .
are also Bernoulli r.v.’s with P½Zin ¼ 0 ¼ 1ð1tÞ ¼ t, P½Zin ¼ 1 ¼ 1t, and consequently
CorrðZin ,Zjn Þ ¼
E½ð1Zin Þð1Zjn Þð1tÞ2 tð1tÞ
¼ r:
The conditional probabilities should be non-negative, and therefore we have the restriction that t 1t 1 , , ,1 : & r 2 max 1t t n1
The indicator r.v.’s Zi are equicorrelated, being exchangeable. Then, the conditional probabilities in (4) show when the existing correlation structure between the original r.v.’s Ui can be preserved by the r.v.’s Zi given the value of the correlation coefficient r. Hence, the process in (1) is a sum of equicorrelated Bernoulli variables (i.e., CorrðZi ,Zj Þ ¼ r for iaj). 2.1. Properties of the CPP Here, we establish some basic properties of the correlated Poisson process in (1). Let us take N to be a Poisson variable with PMF n
pn ¼ P½N ¼ n ¼
el l , n!
l 4 0 and n ¼ 0,1,2, . . . :
ð5Þ
The mean and variance of this model are known to be EðNÞ ¼ VarðNÞ ¼ l. 2.1.1. Marginal distributions P Given N ¼n and for each t 2 ½0,1, the process Nr ðtÞ ¼ nj¼ 1 I½0,t ðUj Þ is the sum of equicorrelated Bernoulli variables with probability of success t and an equal correlation coefficient r. Then, Tallis (1962) showed that the conditional PGF takes on the form
GNr ðtÞ9ðN ¼ nÞ ðzÞ ¼ EðzNr ðtÞ9ðN ¼ nÞ Þ ¼ rð1t þ tzn Þ þ ð1rÞð1t þ tzÞn :
ð6Þ
The corresponding distribution has been referred to in the literature as correlated binomial with parameters n, t and r ˜ o, 1995), denoted by CBðn,t, rÞ, i.e., Nr ðtÞ9ðN ¼ nÞ CBðn,t, rÞ. The PMF, the mean and the variance of Nr ðtÞ9ðN ¼ nÞ (Lucen (Diniz et al., 2010) are given, respectively, by n k P½Nr ðtÞ ¼ k9n,t, r ¼ ð1rÞ ð7Þ t ð1tÞnk IA1 ðkÞ þ rt k=n ð1tÞ1ðk=nÞ IA2 ðkÞ, k where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼0,y,n, and r 2 ½0,1, and
EðNr ðtÞ ¼ k9n,t, rÞ ¼ nt and VarðNr ðtÞ9n,t, rÞ ¼ tð1tÞfn þ rnðn1Þg:
ð8Þ
Note that the CBðn,t, rÞ model is equivalent to the binomial model when r ¼ 0. Now, by unconditioning with respect to N, we can readily derive the marginal PGF, PMF, mean and variance of Nr ðtÞ. For example, we obtain the PGF as
GNr ðtÞ ðzÞ ¼ EðzNr ðtÞ Þ ¼ EðEðzNr ðtÞ 9NÞÞ ¼
1 X
GNr ðtÞ9n ðzÞP½N ¼ n ¼
n¼0 lð1zÞ
¼ rð1tÞ þ t re
ltð1zÞ
þ ð1rÞe
1 X
frð1t þ tzn Þ þ ð1rÞð1t þ tzÞn g
n¼0
:
ln el n! ð9Þ
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
369
Next, we obtain the PMF of Nr ðtÞ in a similar manner as
8 l lt > < rð1tÞ þ rte þð1rÞe , k P½Nr ðtÞ ¼ k9n,t, rP½N ¼ n ¼ rtl el ð1rÞðt lÞk elt gNr ðtÞ ðkÞ ¼ P½Nr ðtÞ ¼ k ¼ > þ , : n¼k k! k! 1 X
k ¼ 0, k ¼ 1,2, . . . :
ð10Þ
Furthermore, we also find
EðNr ðtÞÞ ¼ EðEðNr ðtÞ9NÞÞ ¼ EðNTÞ ¼ lt
ð11Þ
VarðNr ðtÞÞ ¼ EðVarðNr ðtÞ9NÞÞ þ VarðEðNr ðtÞ9NÞÞ ¼ tð1tÞðl þ l2 rÞ þt2 l:
ð12Þ
and
Remark 2. From the expressions of mean and variance of Nr ðtÞ in (11) and (12), respectively, we readily find the Fisher index of Nr ðtÞ to be
VarðNr ðtÞÞ ¼ 1 þ lrð1tÞ for t 2 ½0,1, EðNr ðtÞÞ which reveals that the proposed correlated Poisson process accommodates overdispersion naturally. 2.1.2. Distributions of the increments Consider the increment variable Nr ðtÞNr ðsÞ for 0 r s o t r 1. The computation of the PGF of Nr ðtÞNr ðsÞ is analogous to that of Nr ðtÞ presented above. In this case, we have Pn 1 X ðI ðU ÞI ðU ÞÞ GNr ðtÞNr ðsÞ ðtÞ ¼ EðzNr ðtÞNr ðsÞ Þ ¼ Eðz j ¼ 1 ½0,t j ½0,s j ÞP½N ¼ n: ð13Þ n¼0
Since this time the quantities I½0,t ðUj ÞI½0,s ðUj Þ are equicorrelated Bernoulli variables with probability t s and correlation coefficient r 2 ½0,1, then, given N ¼n, the increment variable Nr ðtÞNr ðsÞ ¼
n X
ðI½0,t ðUj ÞI½0,s ðUj ÞÞ CBðn,ts, rÞ:
j¼1
Hence,
GNr ðtÞNr ðsÞ ðtÞ ¼ rf1ðtsÞg þðtsÞrelð1zÞ þ ð1rÞelðtsÞð1zÞ :
ð14Þ
Interestingly, we observe that the distribution of the increment is stationary, depending only on the difference t s. 2.1.3. Joint distributions We now consider the joint distribution of variables Nr ðt1 Þ, . . . ,Nr ðtk Þ for 0 r t1 o o tk r 1. For simplicity, we present here the expression for the case k¼2. We consider in this case the vector ðNr ðsÞ,Nr ðtÞÞ for 0 r s ot r 1, and derive the PMF of this vector. For any 0 rk rn, we have
P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼ P½Nr ðsÞ ¼ k9Nr ðtÞ ¼ nP½Nr ðtÞ ¼ n: Upon noting that, conditioned on Nr ðtÞ ¼ n, the variable Nr ðsÞ CBðn,s=t, rÞ, we readily find k s k=n n s s nk s 1ðk=nÞ P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼ ð1rÞ 1 IA1 ðkÞ þ r 1 IA2 ðkÞ P½Nr ðtÞ ¼ n, t t t t k
ð15Þ
where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼ 0,y,n, r 2 ½0,1, and P½Nr ðtÞ ¼ n is as given earlier in (10). Next, we derive the characteristic function (ChF) of the vector ðNr ðsÞ,Nr ðtÞÞ, for 0 rs o t r1, as follows:
EðeiuNr ðsÞ þ ivNr ðtÞ Þ ¼
1 X n X n¼0k¼0
eiuk þ ivn P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼
1 X
eivn P½Nr ðtÞ ¼ n
n¼0
n X
eiuk P½Nr ðsÞ ¼ k9Nr ðtÞ ¼ n
k¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
s s s s ¼ r 1 GNr ðtÞ ðeiv Þ þ r GNr ðtÞ ðeiðu þ vÞ Þ þ ð1rÞGNr ðtÞ 1 þ eiu eiv : t t t t
GNr ðs=tÞ9n ðeiu Þ
ð16Þ
This expression reduces to the ChF of Nr ðsÞ when v ¼0, and to the ChF of Nr ðtÞ when u ¼0. 2.1.4. Covariance function We now derive the covariance of Nr ðsÞ and Nr ðtÞ, as well as that of two consecutive increments, viz., Nr ðsÞ and Nr ðtÞNr ðsÞ for any 0 rs o t r1. First, we observe
VarðNr ðtÞNr ðsÞÞ ¼ VarðNr ðtÞÞ þ VarðNr ðsÞÞ2CovðNr ðsÞ,Nr ðtÞÞ,
370
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
and consequently,
CovðNr ðsÞ,Nr ðtÞÞ ¼ 12½VarðNr ðtÞÞ þ VarðNr ðsÞÞVarðNr ðtÞNr ðsÞÞ: Since, VarðNr ðtÞNr ðsÞÞ ¼ VarðNr ðtsÞÞ as shown earlier in Section 2 and VarðNr ðÞÞ is as given in (12), we get
CovðNr ðsÞ,Nr ðtÞÞ ¼ lsf1lrð1tÞg:
ð17Þ
This leads to an expression for the covariance between two consecutive increments, viz., Nr ðsÞNr ð0Þ Nr ðsÞ and Nr ðtÞNr ðsÞ, as
CovðNr ðsÞ,Nr ðtÞNr ðsÞÞ ¼ CovðNr ðsÞ,Nr ðtÞÞVarðNr ðsÞÞ ¼ lsð1lrð1tÞÞsð1sÞðl þ l2 rÞs2 l ¼ l2 srðt þs2Þ:
ð18Þ
The expression in (18) reveals that two consecutive increments are always negatively correlated. 3. The correlated weighted Poisson process Let Nw be an unobservable weighted Poisson random variable with PMF pw ðn; l, fÞ ¼ P½Nw ¼ n; l, f ¼
wðn; fÞpðn; lÞ , El ½wðN; fÞ
n ¼ 0,1,2, . . . ,
ð19Þ
where wð; fÞ is a non-negative weight function with parameter f 4 0, pð; lÞ is the PMF of a Poisson distribution with parameter l 4 0, and El ½ indicates that the expectation is taken with respect to the variable N following a Poisson distribution with mean parameter l. We shall denote (19) by WPl ðwÞ, which stands for the weighted Poisson distribution with parameter l and weight function wð; fÞ. This concept was first introduced by Fisher (1934), but it was Rao (1965) who studied the weighted distributions in a unified way. Rao (1965) pointed out that in many situations the recorded observations cannot be considered as a random sample from the original distribution for many reasons including non-observability of some events, damage caused to original observations, and adoption of unequal probability sampling. Many weighted distributions have been used in practice and, for example, the weighted distribution with identity weight function is called the length-biased distribution, and it has found many important applications in environmetrics. Let U1 ,U2 , . . . ,UNw be a sequence of equicorrelated standard uniform variables, independently of Nw, i.e., CorrðUi ,Uj Þ ¼ r, 8iaj. Then, the random sum Nrw ðtÞ ¼
Nw X
I½0,t ðUj Þ
for t 2 ½0,1
ð20Þ
j¼1
is said to be a correlated weighted Poisson process on the interval [0,1], fNcw ðtÞ,t 2 ½0,1g, with intensity l and correlation r. Thus, we can define a class of correlated weighted Poisson processes (CWPP) on the interval [0,1] indexed by w 2 W, where W is the set of all non-negative weight functions on Z þ , as follows. Definition 1. For any w 2 W, the process in (20) is said to be a correlated weighted Poisson process with parameter l 40, correlation r, and weight function w, denoted by CWPP½l, r ðwÞ. Remark 3. When r ¼ 0, the process in (20) simply becomes the weighted Poisson process proposed by Balakrishnan and Kozubowski (2008) and if wðN; fÞ ¼ constant, the process in (20) becomes the correlated Poisson process, introduced earlier in Section 2. Since U1 ,U2 , . . . ,UNw are equicorrelated with an identical distribution, I½0,t ðU1 Þ, . . . ,I½0,t ðUNw Þ become exchangeable Bernoulli variables, with CorrðI½0,t ðUi Þ,I½0,t ðUj ÞÞ ¼ r1 , iaj, for some r1 2 ½1=ðn1Þ,1. Lemma 1 shows that when the existing correlation structure between the original r.v’s Ui can be preserved by the r.v’s Zi ¼ I½0,t ðUi Þ for a given value of the correlation coefficient r. Therefore, given Nw ¼ n and for each t 2 ½0,1, the process in (20) is the sum of equicorrelated Bernoulli variables with probability of success t and an equal correlation coefficient r. Thus, by using the expression (6), we readily derive the marginal PGF of Nrw ðtÞ as w
w
GNrw ðtÞ ðzÞ ¼ EðzNr ðtÞ Þ ¼ EðEðzNr ðtÞ 9Nw ÞÞ ¼
1 X
GNrw ðtÞ9n ðzÞP½Nw ¼ n ¼
n¼0
¼ rð1tÞ
1 X
P½Nw ¼ n þ rt
n¼0
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 1
1 X
frð1t þ tzn Þ þ ð1rÞð1t þ tzÞn gP½Nw ¼ n
n¼0 1 X
zn P½Nw ¼ n þ ð1rÞ
n¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ðzÞ
¼ rð1tÞ þ rt GNw ðzÞ þð1rÞGNw ð1t þtzÞ,
1 X
ð1t þtzÞn P½N w ¼ n
n¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ð1t þ tzÞ
ð21Þ
where GNw ðÞ is the probability generating function of the weighted Poisson random variable Nw. In the following lemma (see Rodrigues et al., 2009), we present the probability generating function of the variable Nw.
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
371
Lemma 2. Let f Z 0 and y 2 Y R, and the PMF of the discrete variable Nw be of the form pw ðn; y, fÞ ¼ jðn; fÞ expfynKðy, fÞg,
n ¼ 0,1, . . . :
Then, the corresponding PGF is given by
GNw ðzÞ ¼ expflð1zÞg
Elz ½wðN; fÞ , El ½wðN; fÞ
where l ¼ expfyg and wðn; fÞ ¼ n!jðn; fÞ. Thus, by using Lemma 2, we can express GNrw ðtÞ ðzÞ as
GNrw ðtÞ ðzÞ ¼ rð1tÞ þ rt expflð1zÞg
Elð1t þ tzÞ ½wðN; fÞ Elz ½wðN; fÞ þ ð1rÞ expfltð1zÞg : El ½wðN; fÞ El ½wðN; fÞ
ð22Þ
3.1. Properties of the CWPP In this section, we derive some basic properties of the correlated weighted Poisson processes in (20), such as the marginal and joint distributions, stationarity of the increments, moments, and the covariance function. 3.1.1. Marginal distributions Given the PGF GNrw ðtÞ ðzÞ in (21), we obtain the PMF of the variable Nrw ðtÞ, by taking derivatives of GNrw ðtÞ ðzÞ, as 8 < GNrw ðtÞ ð0Þ, k ¼ 0, gNrw ðtÞ ðkÞ ¼ P½Nrw ðtÞ ¼ k ¼ ðkÞ : GNrw ðtÞ ð0Þ, k ¼ 1,2, . . . , where
GðkÞ Nrw ðtÞ ¼
1 X n k @k w w w ðtÞ ðzÞ G ¼ r tk! P ½N ¼ k þ ð1 r Þ P ½N ¼ nk! t ð1tÞnk : N r @zk k z¼0 n¼k
Thus, we obtain 8 w w k ¼ 0, > < rð1tÞ þ rt GN ð0Þ þ ð1rÞGN ð1tÞ, k l t e ðltÞ wt ðk; fÞ ð23Þ gNrw ðtÞ ðkÞ ¼ w > , k ¼ 1,2, . . . , : rt P½N ¼ k þ ð1rÞ k! elð1tÞ El ½wðN; fÞ P n w where wt ðk; l, fÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þ k; fÞ and P½N ¼ k is as in (19). Furthermore, by using the expression in (8), we also find
EðNrw ðtÞÞ ¼ EðEðNrw ðtÞ9Nw ÞÞ ¼ tEðN w Þ
ð24Þ
and 2
VarðNrw ðtÞÞ ¼ EðVarðNrw ðtÞ9N w ÞÞ þ VarðEðNrw ðtÞ9N w ÞÞ ¼ tð1tÞ½ð1rÞEðNw Þ þ rEðN w Þ þt 2 Var½N w :
ð25Þ
Remark 4. From the expressions of mean and variance of Nrw ðtÞ in (24) and (25), respectively, we readily find the Fisher index of Nrw ðtÞ to be VarðNrw ðtÞÞ VarðNw Þ ¼ 1 þ½t þ rð1tÞ 1 þ rð1tÞEðN w Þ for t 2 ½0,1, w w EðNr ðtÞÞ EðN Þ which reveals that the proposed correlated weighted Poisson process is underdispersed for each t 2 ½0,1 if and only if EðN w Þ ot þ rð1tÞ=rð1tÞ and VarðNw Þ=EðNw Þ o 1rð1tÞ=t þ rð1tÞEðNw Þ. Otherwise, the process is overdispersed. 3.1.2. Distributions of the increments Consider the increment variable Nrw ðtÞNrw ðsÞ for 0 rs o t r1. The derivation of the PGF of Nrw ðtÞNrw ðsÞ is analogous to that of Nrw ðtÞ presented above. In this case, we have Pn 1 X w w ðI ðU ÞI ðU ÞÞ GNrw ðtÞNrw ðsÞ ðzÞ ¼ EðzNr ðtÞNr ðsÞ Þ ¼ Eðz j ¼ 1 ½0,t j ½0,s j ÞP½N w ¼ n: ð26Þ n¼0
Now, since the quantities I½0,t ðUj ÞI½0,s ðUj Þ are equicorrelated Bernoulli variables with probability ts and correlation P coefficient r 2 ½0,1, then, given Nw ¼ n, the increment variable Nrw ðtÞNrw ðsÞ ¼ nj¼ 1 ðI½0,t ðUj ÞI½0,s ðUj ÞÞ CBðn,ts, rÞ, and consequently
GNrw ðtÞNrw ðsÞ ðzÞ ¼ rf1ðtsÞg þ ðtsÞrelð1zÞ þð1rÞelðtsÞð1zÞ :
ð27Þ
372
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
Interestingly, we observe one again that the distribution of the increment is stationary, depending only on the difference ts. 3.1.3. Joint distributions Here, we consider the joint distribution of variables Nrw ðt1 Þ, . . . ,Nrw ðtk Þ for 0 rt1 o otk r1. For simplicity, we derive the expression for the case k¼ 2, and the result for general k can be derived in an analogous manner. We consider in this case the vector ðNrw ðsÞ,Nrw ðtÞÞ for 0 rs o t r1, and derive the joint PMF of this vector. For any 0 rk r n, we have
P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n ¼ P½Nrw ðsÞ ¼ k9Nrw ðtÞ ¼ nP½Nrw ðtÞ ¼ n: Upon noting, by conditioning on Nrw ðtÞ ¼ n, that the variable Nrw ðsÞ CBðn,s=t, rÞ, we readily find k s k=n n s s nk s 1ðk=nÞ P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n ¼ ð1rÞ 1 IA1 ðkÞ þ r 1 IA2 ðkÞ P½Nrw ðtÞ ¼ n, t t t t k
ð28Þ
where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼0,y,n, r 2 ½0,1, and P½Nrw ðtÞ ¼ n is as given earlier in (23). Next, we derive the joint ChF of the vector ðNrw ðsÞ,Nrw ðtÞÞ, for 0 rs o t r1, as follows: w
w
EðeiuNr ðsÞ þ ivNr ðtÞ Þ ¼ ¼
1 X n X
eiuk þ ivn P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n
n¼0k¼0 1 X ivn
n X
n¼0
k¼0
P½Nrw ðtÞ ¼ n
e
eiuk P½Nrw ðsÞ ¼ k9Nrw ðtÞ ¼ n
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ðs=tÞ9n ðeiu Þ
s
r
s s s ¼ r 1 GNrw ðtÞ ðe Þ þ r GNrw ðtÞ ðeiðu þ vÞ Þ þ1rÞGNrw ðtÞ 1 þ eiu eiv : t t t t iv
ð29Þ
Naturally, the expression in (29) reduces to the ChF of Nrw ðsÞ when v ¼0, and to the ChF of Nrw ðtÞ when u ¼0. 3.1.4. Covariance function Here, we derive the covariance of Nrw ðsÞ and Nrw ðtÞ, as well as that of two consecutive increments, viz., Nrw ðsÞ and Nrw ðtÞNrw ðsÞ for any 0 rs o t r1. First, we observe
VarðNrw ðtÞNrw ðsÞÞ ¼ VarðNrw ðtÞÞ þ VarðNrw ðsÞÞ2 CovðNrw ðsÞ,Nrw ðtÞÞ, and consequently,
CovðNrw ðsÞ,Nrw ðtÞÞ ¼ 12½VarðNrw ðtÞÞ þ VarðNrw ðsÞÞVarðNrw ðtÞNrw ðsÞÞ: Since VarðNrw ðtÞNrw ðsÞÞ ¼ VarðNrw ðtsÞÞ as shown earlier, and VarðNrw ðÞÞ is as given in (25), we obtain 2
CovðNrw ðsÞ,Nrw ðtÞÞ ¼ sfð1tÞ½ð1rÞE½Nw þ rEðN w Þ þt VarðN w Þg:
ð30Þ w
w
w
w
w
This leads to an expression for the covariance of two consecutive increments, viz., Nr ðsÞNr ð0Þ Nr ðsÞ and Nr ðtÞNr ðsÞ, as 2
CovðNrw ðsÞ,Nrw ðtÞNrw ðsÞÞ ¼ CovðNrw ðsÞ,Nrw ðtÞÞVarðNrw ðsÞÞ ¼ sðtsÞfVar½Nw ½ð1rÞE½N w þ rE½Nw g:
ð31Þ
The expression in (31) readily reveals that two consecutive increments are positively correlated if only if the variable Nw is 2 overdispersed and Var½N w 4 ð1rÞE½Nw þ rE½N w . Otherwise, the increments are negatively correlated. 4. Some illustrative examples In this section, we present a few specific processes that arise from our general formulation. In Table 1, we present the probability generating function, mean and variance corresponding to these processes. 4.1. Length-biased Poisson process When the weight function of the r.v. Nw is simply wðn; fÞ ¼ n, then the weight function wt ðÞ has an explicit form given by wt ðk; l,tÞ ¼ kþ lð1tÞ. Recall that, in this case, Nw has a Poisson distribution with parameter l shifted up by one, and is underdispersed, since Var½Nw o E½N w . Also, note that the compound Poisson distribution of Nrw ðtÞ is not shifted-Poisson, since its PMF is of the form 8 l lt > < rð1tÞ þ rte þð1rÞe , w k1 l P½Nr ðtÞ ¼ k ¼ l e elt ðltÞk k > þð1rÞ 1t þ , : rt ðk1Þ! k! l
k ¼ 0, k ¼ 1,2, . . . :
Correlated process Length-biased Poisson Exponentially weighted Poisson Negative binomial COM-Poisson
E½Nrw ðtÞ Var½Nrw ðtÞ rð1tÞ þ rt expflð1zÞgþ ð1rÞð1tð1zÞÞ expfltð1zÞg tð1lÞ tð1tÞ½1þ lð1 þ 2r þ rlÞ þ t 2 l rð1tÞ þ rt expfle2f ð1zÞg þ ð1rÞ expfltð1zÞe2f g t l expff þ 2lef g tð1tÞl½expff þ 2lef g þ rl expf2ðf þ lef Þg þ t 2 l½l expf2ðf þ lef Þg þ expff þ 2lef gl expf2ðf þ 2lef Þg tl tð1tÞl½1 þ rlð1 þ fÞ þ t 2 lð1 þ flÞ rð1tÞ þ rtð1 þ flð1zÞÞ1=f þ ð1rÞð1þ fltð1zÞÞ1=f 2 !2 3 " # j Zðlz, fÞ Zðlð1tð1zÞÞ, fÞ 1 j j j P jl 1 1 1 P P P jl j2 l jl þ ð1rÞ rð1tÞ þ rt 24 t 5 ð1 r ð1jÞÞ þ t tð1tÞ Zðl, fÞ Zðl, fÞ f f f f j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ GNrw ðtÞ ðzÞ
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
Table 1 Probability generating function (GNrw ðtÞ ðÞ), mean (E½Nrw ðÞ), and variance (Var½Nrw ðÞ) for different processes.
373
374
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
4.2. Exponentially weighted Poisson process When the weight function of the r.v. Nw is of exponential form, i.e., wðn; fÞ ¼ efm , f 2 R, then the weight function wt ðÞ has an explicit form given by wt ðk; l, f,tÞ ¼ expffklð1tÞð1ef Þg. It is easy to see that Nw has a Poisson distribution with parameter lef and that the variable Nrw ðtÞ has PMF given by 8 le2f > þ ð1rÞ expflte2f g, k ¼ 0, < rð1tÞ þ rte w k k fg f Þ þ ltð1ef Þg P½Nr ðtÞ ¼ k ¼ l expfk f þ l e ð l tÞ expfk f l ð12e > : rt þ ð1rÞ , k ¼ 1,2, . . . : k! k!
4.3. Negative binomial process Let us take the unobservable count variable, Nw, as a variable with a negative binomial distribution with parameters f 4 0 and l 40 (see Piegorsch, 1990; Saha and Paul, 2005), with PMF Gðf1 þ nÞ fl n P½Nw ¼ n; l, f ¼ ð1 þ flÞ1=f , n ¼ 0,1,2, . . . : ð32Þ 1 Gðf Þn! 1 þ fl Note that f ¼ 1 leads to the geometric distribution with parameter 1=ð1 þ lÞ. Comparing (32) with (19), we realize that (32) 1 is a weighted Poisson distribution with parameter fl=ð1 þ flÞ and weight function wðn; fÞ ¼ Gðf þ nÞ. So, we arrive at a closed-form expression for the weight function wt ðk; l, f,tÞ in this case as 2 3k 1 7 flð1tÞ 6 1 6 7 Gðf þkÞ : wt ðk; l, f,tÞ ¼ exp 4 5 flð1tÞ 1 þ fl flð1tÞ 1=f 1 1 1þ fl 1 þ fl Thus, the variable Nrw ðtÞ in this case has PMF as 8 > rð1tÞ þ rtð1 þ flÞ1=f þð1rÞð1 þ fltÞ1=f , > < w P½Nr ðtÞ ¼ k ¼ Gðf1 þ kÞ fl k Gðf1 þ1Þ flt k > r t ð1 þ flÞ1=f þ ð1rÞ ð1þ fltÞ1=f , > 1 1 : Gðf Þk! 1 þ fl Gðf Þk! 1 þ flt
k ¼ 0, k ¼ 1,2, . . . :
4.4. COM-Poisson process Let us take the unobservable count variable Nw as a variable with a COM-Poisson distribution with parameters l 4 0 and f 4 0 (see Shmueli et al., 2005), with PMF n
P½Nw ¼ n; l, f ¼
1 l , Zðl, fÞ ðn!Þf
m ¼ 0,1,2, . . . ,
ð33Þ
P j f where Zðl, fÞ ¼ 1 j ¼ 0 l =ðj!Þ . In particular, when f ¼ 0 and l o 1, the COM-Poisson distribution reduces to the geometric distribution with parameter 1l. The distribution in (33) may also be viewed as a weighted Poisson distribution with weight function wðn; fÞ ¼ ðn!Þ1f ; see Kokonendji et al. (2008). Therefore, for integer f, the weight function wt ðÞ of the variable Nrw is given by wt ðk; l, f,tÞ ¼ expflð1tÞgðk!Þ1f
1 X
½ðk þ1Þðk þ2Þ ðkþ nÞ1f
n¼0
¼ expflð1tÞgðk!Þ
1f
1 F f ðk þ 1; kþ 1,
½lð1tÞn n!
. . . ,k þ 1; lð1tÞÞ,
ð34Þ
where the generalized hypergeometric function is defined by u F v ða1 ,
. . . ,au ; b1 , . . . ,bv ; xÞ ¼
1 X ða1 Þn ða2 Þn ðau Þn xn , ðb1 Þn ðb2 Þn ðbv Þn n! n¼0
with ðaÞn ¼ aða þ1Þ ða þ n1Þ; see Gradshteyn and Ryzhik (2000). Taking k¼0 and t ¼0 in (33), we obtain from (22) a closed-form expression for the COM-Poisson process as
GNrw ðtÞ ðzÞ ¼ rð1tÞ þ rt
1 F f ð1; 1,
. . . ,1; lzÞ
1 F f ð1; 1, . . . ,1; lÞ
þ ð1rÞ
1 F f ð1; 1,
. . . ,1; lð1t þtzÞÞ
1 F f ð1; 1, . . . ,1; lÞ
:
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
In this case, the distribution of the variable Nw is 8 > 1 1 F f ð1; 1, . . . ,1; lð1tÞÞ > > þ ð1rÞ , > rð1tÞ þ rt > < Zðl, fÞ 1 F f ð1; 1, . . . ,1; lÞ P½Nrw ðtÞ ¼ k ¼ > lk ðltÞk 1 F f ð1; 1, . . . ,1; lð1tÞÞ > > rt 1 , þð1rÞ > > Zðl, fÞ f : ðk!Þ ðk!Þf 1 F f ð1; 1, . . . ,1; lÞ
375
k ¼ 0, k ¼ 1,2, . . . :
5. Concluding remarks In this paper, we have introduced correlated Poisson processes and correlated weighted Poisson processes as generalizations of the class of weighted Poisson processes on the interval [0;1] proposed by Balakrishnan and Kozubowski (2008) by incorporating a dependence structure between the standard uniform random variables used in the construction. The dependence is measured by the correlation coefficient which is a natural measure for the underlying exchangeable sequence. We have then studied some basic properties of these processes such as the marginal and joint distributions, stationarity of the increments, moments, and the covariance function. These results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions. Since this class of processes naturally accommodates underdispersion and overdispersion, it will be useful when dealing with data exhibiting greater variability than that allowed by the Poisson process. It will therefore be of great interest to develop inferential methods for the correlated Poisson processes and the correlated weighted Poisson processes. Work in this direction is currently under progress and we hope to report these findings in a future paper.
Acknowledgments The authors express their sincere thanks to two referees for their constructive comments and suggestions on an earlier version of this paper. References Aldous, D., Ibragimov, I.A., Jacod, J., 1985. Exchangeability and related topics. Lectures Notes in Mathematics, vol. 1117; 1985, pp. 2–198. Balakrishnan, N., Kozubowski, T., 2008. A class of weighted Poisson processes. Statistics & Probability Letters 78, 2346–2352. Castillo, J., Pe´rez-Casany, M., 1998. Weighted Poisson distributions for overdispersion and underdispersion situations. Annals of the Institute of Statistical Mathematics 50, 567–585. Castillo, J., Pe´rez-Casany, M., 2005. Overdispersed and underdispersed Poisson generalizations. Journal of Statistical Planning and Inference 134, 486–500. Diniz, C.A.R., Tutia, M.H., Leite, J.G., 2010. Bayesian analysis of a correlated binomial model. Journal of Probability and Statistics 24, 68–77. Fisher, R.A., 1934. The effect of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics 6, 13–25. Gradshteyn, I.S., Ryzhik, I.M., 2000. Table of Integrals, Series, and Products, 6th ed. Academic Press, San Diego. Hinde, J., Demetrio, C.G.B., 1998. Overdispersion: models and estimation. Computational Statistics & Data Analysis 27, 151–170. Kokonendji, C., Mizere, D., Balakrishnan, N., 2008. Connections of the Poisson weight function to overdispersion and underdispersion. Journal of Statistical Planning and Inference 138, 1287–1296. Kozubowski, T.J., Podgo´rski, K., 2009. Distributional properties of the negative binomial Le´vy process. Probability and Mathematical Statistics 9, 43–71. ˜ o, A., 1995. A family of partially correlated Poisson models for overdispersion. Computational Statistics & Data Analysis 20, 511–520. Lucen Piegorsch, W.W., 1990. Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics 46, 863–867. Rao, C.R., 1965. On discrete distributions arising out of methods of ascertainment. Sankhya~ , Series A 27, 311–324. Rodrigues, J., de Castro, M., Cancho, V., Balakrishnan, N., 2009. COM-Poisson cure rate survival models and an application to a cutaneous melanoma data. Journal of Statistical Planning and Inference 139, 3605–3611. Saha, K., Paul, S., 2005. Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter. Biometrics 61, 179–185. Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S., Boatwright, P., 2005. A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society, Series C 54, 127–142. Tallis, G.M., 1962. The use of a generalized multinomial distribution in the estimation of correlation in discrete data. Journal of the Royal Statistical Society, Series B 24, 530–534. Yakovlev, A., Tsodikov, A., 1996. Stochastic Models of Tumor Latency and Their Biostatistical Applications. World Scientific Publishers, Singapore.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
A class of correlated weighted Poisson processes P. Borges a,, J. Rodrigues b, N. Balakrishnan c a
Departamento de Estatı´stica, Universidade do Espı´rito Santo - UFES. Av. Fernando Ferrari 514, Goiabeiras, CEP 29075-910 Vitøria ES, Brazil ~ Carlos, Brazil Departamento de Estatı´stica, Universidade de Sao c Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 b
a r t i c l e in f o
abstract
Article history: Received 14 February 2011 Received in revised form 30 July 2011 Accepted 4 August 2011 Available online 11 August 2011
In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COMPoisson distributions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Correlated Poisson process on [0,1] Standard uniform variables Exchangeable variables Correlation coefficient Correlated weighted Poisson process
1. Introduction In a recent paper, Balakrishnan and Kozubowski (2008) introduced a class of weighted Poisson processes (WPP) on the interval [0,1]. This family assumes that the sequence of standard uniform variables are independent and the unobservable count variable follows a weighted Poisson distribution with parameter l and weight function wðÞ; see, for example, Castillo and Pe´rezCasany (1998, 2005). The probability mass function (PMF) of the WPP with intensity l, fNw ðtÞ,t 2 ½0,1g, is given by
P½Nw ðtÞ ¼ k ¼
elt ðltÞk wt ðkÞ , k! elð1tÞ El ½wðNÞ
k ¼ 0,1, . . . , l 40, t 2 ½0,1,
P n where wt ðkÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þ kÞ. Following their approach, we generalize in this paper the above WPP by considering a dependence structure between the standard uniform variables. In this manner, we obtain a new class of processes that we refer to as correlated weighted Poisson process (CWPP) with intensity l and correlation r, fNrw ðtÞ,t 2 ½0,1g, whose PMF is 8 w w k ¼ 0, > < rð1tÞ þ rt GN ð0Þ þ ð1rÞGN ð1tÞ, k w lt e ð l tÞ w ðkÞ P½Nr ðtÞ ¼ k ¼ t w > , k ¼ 1,2, . . . : rt P½N ¼ k þ ð1rÞ k! elð1tÞ El ½wðN; fÞ for l 4 0, t, r 2 ½0,1, where Nw is a weighted Poisson variable with parameter l and weight function wðÞ, GNw ðÞ is the P n probability generating function of Nw, and wt ðkÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þkÞ. Particular choices of wðÞ lead to some Corresponding author.
E-mail address: [email protected] (P. Borges). 0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.08.002
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
367
important special cases, including CWPP with geometric as well as negative binomial marginal distributions, which frequently arise in applications; see for example, Kozubowski and Podgo´rski (2009) and the references therein. This new class of processes will enable one to tackle the problem of overdispersion and underdispersion inherent in the analysis of count data (see Hinde and Demetrio, 1998) that may arise due to the presence of some correlation between the events. For example, in models of carcinogenesis (see Yakovlev and Tsodikov, 1996), it is usually assumed that cells in a tissue (modeled by a homogeneous Poisson process with intensity l) can give rise to tumors independently of each other, i.e., they are biologically independent as far as carcinogenesis is concerned. Then, statistical independence is used to model biological independence. However, the biological independence assumption may not be true when the dynamics of the cell population of a normal tissue is considered. It is therefore desirable to construct mathematically tractable models of carcinogenesis that can adequately incorporate biological dependence, and this is what motivated the present research work. The rest of this paper is organized as follows. In Section 2, we introduce the correlated Poisson process before formulating the class of correlated weighted Poisson processes in Section 3. In Section 4, the theoretical results are illustrated through some examples. Finally, some concluding remarks are made in Section 5. 2. The correlated Poisson process Let N have a Poisson distribution with parameter l 4 0, and let U1 ,U2 , . . . ,UN be a sequence of independent standard P uniform variables, independently of N. Then, the random sum NðtÞ ¼ N j ¼ 1 I½0,t ðUj Þ, where IA is an indicator of the set A, is a classical Poisson process on the interval [0,1], fNðtÞ,t 2 ½0,1g, with intensity l. We generalize this Poisson process model by relaxing the assumption of mutual independence between the standard uniform variables. To introduce dependence, we assume that the variables are assigned with equal correlation between them. In this process, we obtain another process on the interval [0,1], fNr ðtÞ,t 2 ½0,1g, with intensity l and correlation r, which we refer to as correlated Poisson process (CPP). Thus, for the correlated Poisson process, we have Nr ðtÞ ¼
N X
I½0,t ðUj Þ
for t 2 ½0,1,
ð1Þ
j¼1
where Uj’s are now equicorrelated standard uniform variables, independently of N; i.e.,
CorrðUi ,Uj Þ ¼ r, iaj, i,j ¼ 1,2, . . . ,N ¼ n:
ð2Þ
Remark 1. This model is strongly connected with the concept of exchangeability which restricts the possible correlation structure for the sequences of random variables (r.v.’s) with finite first two moments (Aldous et al., 1985). If ðU1 ,U2 , . . . ,Un Þ is a finite sequence of exchangeable r.v.’s, then 1 ,1 , iaj: ð3Þ CorrðUi ,Uj Þ ¼ r 2 n1 Conversely, every r satisfying (3) serves as a correlation in some exchangeable sequence. To see this, let ðX1 , . . . ,Xn Þ be a sequence of independent r.v.’s with EðXi Þ ¼ 0, VðXi Þ ¼ 1, and define Ui ¼ Xi þd
n X
Xk ,
i ¼ 1,2, . . . ,n,
k¼1
for a constant d Z 1=n. Thus, ðU1 , . . . ,Un Þ is an exchangeable sequence and
r ¼ EðUi Uj Þ ¼ 1
1 2
nd þ2d þ1
,
iaj:
Varying d, it is possible to obtain all values of r 2 ½1=ðn1Þ,1. For example, r ¼ 1=ðn1Þ when d ¼ 1=n, and r ¼ 0 when d ¼0. For infinite exchangeable sequences, we of course have r 2 ½0,1. Now, let us note that distributionally, the variables Zj ¼ I½0,t ðUj Þ have the same Bernoulli distribution with parameter t ¼ P½Uj o t 2 ½0,1 and corresponding probability generating function (PGF) as GZ ðsÞ ¼ ð1tÞ þ ts, s 2 ½0,1. However, now U1 , . . . ,Un are equicorrelated (CorrðUi ,Uj Þ ¼ r for iaj) with identical marginal distributions. Therefore, Z1 ,Z2 , . . . ,Zn are exchangeable Bernoulli variables with CorrðZi ,Zj Þ ¼ r1 , for some r1 2 ½1=ðn1Þ,1 and iaj, as noted above in Remark 1. The following lemma gives a necessary and sufficient condition for r1 ¼ r. Lemma 1. Let CorrðUi ,Uj Þ ¼ r for iaj, where t 1t 1 , , ,1 : r 2 max 1t t n1
368
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
Then, CorrðZi ,Zj Þ ¼ r iff
P½Zi ¼ 19Zj ¼ 1 ¼ t þ ð1tÞr and P½Zi ¼ 09Zj ¼ 0 ¼ ð1tÞ þ tr:
ð4Þ
Proof. The r.v.’s Zi (i¼1,2,y) are identically distributed as Bernoulli with parameter t, and so EðZÞ ¼ t ¼ PðUi o tÞ and VarðZÞ ¼ tð1tÞ. Then, CorrðZi ,Zj Þ ¼ r for iaj iff CovðZi ,Zj Þ ¼ r VarðZÞ, i.e., when EðZi Zj Þt 2 ¼ tð1tÞr. On the other hand, we simply have
EðZi Zj Þ ¼ P½Zi ¼ 1,Zj ¼ 1: As a result, we have P½Zi ¼ 1,Zj ¼ 1 ¼ t 2 þ tð1tÞr, and by the conditional probability formula, we then obtain P½Zi ¼ 19Zj ¼ 1 ¼ t þ ð1tÞr. The second relation in (4) follows from the fact that Zin ¼ 1Zi ,
i ¼ 1,2, . . .
are also Bernoulli r.v.’s with P½Zin ¼ 0 ¼ 1ð1tÞ ¼ t, P½Zin ¼ 1 ¼ 1t, and consequently
CorrðZin ,Zjn Þ ¼
E½ð1Zin Þð1Zjn Þð1tÞ2 tð1tÞ
¼ r:
The conditional probabilities should be non-negative, and therefore we have the restriction that t 1t 1 , , ,1 : & r 2 max 1t t n1
The indicator r.v.’s Zi are equicorrelated, being exchangeable. Then, the conditional probabilities in (4) show when the existing correlation structure between the original r.v.’s Ui can be preserved by the r.v.’s Zi given the value of the correlation coefficient r. Hence, the process in (1) is a sum of equicorrelated Bernoulli variables (i.e., CorrðZi ,Zj Þ ¼ r for iaj). 2.1. Properties of the CPP Here, we establish some basic properties of the correlated Poisson process in (1). Let us take N to be a Poisson variable with PMF n
pn ¼ P½N ¼ n ¼
el l , n!
l 4 0 and n ¼ 0,1,2, . . . :
ð5Þ
The mean and variance of this model are known to be EðNÞ ¼ VarðNÞ ¼ l. 2.1.1. Marginal distributions P Given N ¼n and for each t 2 ½0,1, the process Nr ðtÞ ¼ nj¼ 1 I½0,t ðUj Þ is the sum of equicorrelated Bernoulli variables with probability of success t and an equal correlation coefficient r. Then, Tallis (1962) showed that the conditional PGF takes on the form
GNr ðtÞ9ðN ¼ nÞ ðzÞ ¼ EðzNr ðtÞ9ðN ¼ nÞ Þ ¼ rð1t þ tzn Þ þ ð1rÞð1t þ tzÞn :
ð6Þ
The corresponding distribution has been referred to in the literature as correlated binomial with parameters n, t and r ˜ o, 1995), denoted by CBðn,t, rÞ, i.e., Nr ðtÞ9ðN ¼ nÞ CBðn,t, rÞ. The PMF, the mean and the variance of Nr ðtÞ9ðN ¼ nÞ (Lucen (Diniz et al., 2010) are given, respectively, by n k P½Nr ðtÞ ¼ k9n,t, r ¼ ð1rÞ ð7Þ t ð1tÞnk IA1 ðkÞ þ rt k=n ð1tÞ1ðk=nÞ IA2 ðkÞ, k where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼0,y,n, and r 2 ½0,1, and
EðNr ðtÞ ¼ k9n,t, rÞ ¼ nt and VarðNr ðtÞ9n,t, rÞ ¼ tð1tÞfn þ rnðn1Þg:
ð8Þ
Note that the CBðn,t, rÞ model is equivalent to the binomial model when r ¼ 0. Now, by unconditioning with respect to N, we can readily derive the marginal PGF, PMF, mean and variance of Nr ðtÞ. For example, we obtain the PGF as
GNr ðtÞ ðzÞ ¼ EðzNr ðtÞ Þ ¼ EðEðzNr ðtÞ 9NÞÞ ¼
1 X
GNr ðtÞ9n ðzÞP½N ¼ n ¼
n¼0 lð1zÞ
¼ rð1tÞ þ t re
ltð1zÞ
þ ð1rÞe
1 X
frð1t þ tzn Þ þ ð1rÞð1t þ tzÞn g
n¼0
:
ln el n! ð9Þ
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
369
Next, we obtain the PMF of Nr ðtÞ in a similar manner as
8 l lt > < rð1tÞ þ rte þð1rÞe , k P½Nr ðtÞ ¼ k9n,t, rP½N ¼ n ¼ rtl el ð1rÞðt lÞk elt gNr ðtÞ ðkÞ ¼ P½Nr ðtÞ ¼ k ¼ > þ , : n¼k k! k! 1 X
k ¼ 0, k ¼ 1,2, . . . :
ð10Þ
Furthermore, we also find
EðNr ðtÞÞ ¼ EðEðNr ðtÞ9NÞÞ ¼ EðNTÞ ¼ lt
ð11Þ
VarðNr ðtÞÞ ¼ EðVarðNr ðtÞ9NÞÞ þ VarðEðNr ðtÞ9NÞÞ ¼ tð1tÞðl þ l2 rÞ þt2 l:
ð12Þ
and
Remark 2. From the expressions of mean and variance of Nr ðtÞ in (11) and (12), respectively, we readily find the Fisher index of Nr ðtÞ to be
VarðNr ðtÞÞ ¼ 1 þ lrð1tÞ for t 2 ½0,1, EðNr ðtÞÞ which reveals that the proposed correlated Poisson process accommodates overdispersion naturally. 2.1.2. Distributions of the increments Consider the increment variable Nr ðtÞNr ðsÞ for 0 r s o t r 1. The computation of the PGF of Nr ðtÞNr ðsÞ is analogous to that of Nr ðtÞ presented above. In this case, we have Pn 1 X ðI ðU ÞI ðU ÞÞ GNr ðtÞNr ðsÞ ðtÞ ¼ EðzNr ðtÞNr ðsÞ Þ ¼ Eðz j ¼ 1 ½0,t j ½0,s j ÞP½N ¼ n: ð13Þ n¼0
Since this time the quantities I½0,t ðUj ÞI½0,s ðUj Þ are equicorrelated Bernoulli variables with probability t s and correlation coefficient r 2 ½0,1, then, given N ¼n, the increment variable Nr ðtÞNr ðsÞ ¼
n X
ðI½0,t ðUj ÞI½0,s ðUj ÞÞ CBðn,ts, rÞ:
j¼1
Hence,
GNr ðtÞNr ðsÞ ðtÞ ¼ rf1ðtsÞg þðtsÞrelð1zÞ þ ð1rÞelðtsÞð1zÞ :
ð14Þ
Interestingly, we observe that the distribution of the increment is stationary, depending only on the difference t s. 2.1.3. Joint distributions We now consider the joint distribution of variables Nr ðt1 Þ, . . . ,Nr ðtk Þ for 0 r t1 o o tk r 1. For simplicity, we present here the expression for the case k¼2. We consider in this case the vector ðNr ðsÞ,Nr ðtÞÞ for 0 r s ot r 1, and derive the PMF of this vector. For any 0 rk rn, we have
P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼ P½Nr ðsÞ ¼ k9Nr ðtÞ ¼ nP½Nr ðtÞ ¼ n: Upon noting that, conditioned on Nr ðtÞ ¼ n, the variable Nr ðsÞ CBðn,s=t, rÞ, we readily find k s k=n n s s nk s 1ðk=nÞ P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼ ð1rÞ 1 IA1 ðkÞ þ r 1 IA2 ðkÞ P½Nr ðtÞ ¼ n, t t t t k
ð15Þ
where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼ 0,y,n, r 2 ½0,1, and P½Nr ðtÞ ¼ n is as given earlier in (10). Next, we derive the characteristic function (ChF) of the vector ðNr ðsÞ,Nr ðtÞÞ, for 0 rs o t r1, as follows:
EðeiuNr ðsÞ þ ivNr ðtÞ Þ ¼
1 X n X n¼0k¼0
eiuk þ ivn P½Nr ðsÞ ¼ k,Nr ðtÞ ¼ n ¼
1 X
eivn P½Nr ðtÞ ¼ n
n¼0
n X
eiuk P½Nr ðsÞ ¼ k9Nr ðtÞ ¼ n
k¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
s s s s ¼ r 1 GNr ðtÞ ðeiv Þ þ r GNr ðtÞ ðeiðu þ vÞ Þ þ ð1rÞGNr ðtÞ 1 þ eiu eiv : t t t t
GNr ðs=tÞ9n ðeiu Þ
ð16Þ
This expression reduces to the ChF of Nr ðsÞ when v ¼0, and to the ChF of Nr ðtÞ when u ¼0. 2.1.4. Covariance function We now derive the covariance of Nr ðsÞ and Nr ðtÞ, as well as that of two consecutive increments, viz., Nr ðsÞ and Nr ðtÞNr ðsÞ for any 0 rs o t r1. First, we observe
VarðNr ðtÞNr ðsÞÞ ¼ VarðNr ðtÞÞ þ VarðNr ðsÞÞ2CovðNr ðsÞ,Nr ðtÞÞ,
370
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
and consequently,
CovðNr ðsÞ,Nr ðtÞÞ ¼ 12½VarðNr ðtÞÞ þ VarðNr ðsÞÞVarðNr ðtÞNr ðsÞÞ: Since, VarðNr ðtÞNr ðsÞÞ ¼ VarðNr ðtsÞÞ as shown earlier in Section 2 and VarðNr ðÞÞ is as given in (12), we get
CovðNr ðsÞ,Nr ðtÞÞ ¼ lsf1lrð1tÞg:
ð17Þ
This leads to an expression for the covariance between two consecutive increments, viz., Nr ðsÞNr ð0Þ Nr ðsÞ and Nr ðtÞNr ðsÞ, as
CovðNr ðsÞ,Nr ðtÞNr ðsÞÞ ¼ CovðNr ðsÞ,Nr ðtÞÞVarðNr ðsÞÞ ¼ lsð1lrð1tÞÞsð1sÞðl þ l2 rÞs2 l ¼ l2 srðt þs2Þ:
ð18Þ
The expression in (18) reveals that two consecutive increments are always negatively correlated. 3. The correlated weighted Poisson process Let Nw be an unobservable weighted Poisson random variable with PMF pw ðn; l, fÞ ¼ P½Nw ¼ n; l, f ¼
wðn; fÞpðn; lÞ , El ½wðN; fÞ
n ¼ 0,1,2, . . . ,
ð19Þ
where wð; fÞ is a non-negative weight function with parameter f 4 0, pð; lÞ is the PMF of a Poisson distribution with parameter l 4 0, and El ½ indicates that the expectation is taken with respect to the variable N following a Poisson distribution with mean parameter l. We shall denote (19) by WPl ðwÞ, which stands for the weighted Poisson distribution with parameter l and weight function wð; fÞ. This concept was first introduced by Fisher (1934), but it was Rao (1965) who studied the weighted distributions in a unified way. Rao (1965) pointed out that in many situations the recorded observations cannot be considered as a random sample from the original distribution for many reasons including non-observability of some events, damage caused to original observations, and adoption of unequal probability sampling. Many weighted distributions have been used in practice and, for example, the weighted distribution with identity weight function is called the length-biased distribution, and it has found many important applications in environmetrics. Let U1 ,U2 , . . . ,UNw be a sequence of equicorrelated standard uniform variables, independently of Nw, i.e., CorrðUi ,Uj Þ ¼ r, 8iaj. Then, the random sum Nrw ðtÞ ¼
Nw X
I½0,t ðUj Þ
for t 2 ½0,1
ð20Þ
j¼1
is said to be a correlated weighted Poisson process on the interval [0,1], fNcw ðtÞ,t 2 ½0,1g, with intensity l and correlation r. Thus, we can define a class of correlated weighted Poisson processes (CWPP) on the interval [0,1] indexed by w 2 W, where W is the set of all non-negative weight functions on Z þ , as follows. Definition 1. For any w 2 W, the process in (20) is said to be a correlated weighted Poisson process with parameter l 40, correlation r, and weight function w, denoted by CWPP½l, r ðwÞ. Remark 3. When r ¼ 0, the process in (20) simply becomes the weighted Poisson process proposed by Balakrishnan and Kozubowski (2008) and if wðN; fÞ ¼ constant, the process in (20) becomes the correlated Poisson process, introduced earlier in Section 2. Since U1 ,U2 , . . . ,UNw are equicorrelated with an identical distribution, I½0,t ðU1 Þ, . . . ,I½0,t ðUNw Þ become exchangeable Bernoulli variables, with CorrðI½0,t ðUi Þ,I½0,t ðUj ÞÞ ¼ r1 , iaj, for some r1 2 ½1=ðn1Þ,1. Lemma 1 shows that when the existing correlation structure between the original r.v’s Ui can be preserved by the r.v’s Zi ¼ I½0,t ðUi Þ for a given value of the correlation coefficient r. Therefore, given Nw ¼ n and for each t 2 ½0,1, the process in (20) is the sum of equicorrelated Bernoulli variables with probability of success t and an equal correlation coefficient r. Thus, by using the expression (6), we readily derive the marginal PGF of Nrw ðtÞ as w
w
GNrw ðtÞ ðzÞ ¼ EðzNr ðtÞ Þ ¼ EðEðzNr ðtÞ 9Nw ÞÞ ¼
1 X
GNrw ðtÞ9n ðzÞP½Nw ¼ n ¼
n¼0
¼ rð1tÞ
1 X
P½Nw ¼ n þ rt
n¼0
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 1
1 X
frð1t þ tzn Þ þ ð1rÞð1t þ tzÞn gP½Nw ¼ n
n¼0 1 X
zn P½Nw ¼ n þ ð1rÞ
n¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ðzÞ
¼ rð1tÞ þ rt GNw ðzÞ þð1rÞGNw ð1t þtzÞ,
1 X
ð1t þtzÞn P½N w ¼ n
n¼0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ð1t þ tzÞ
ð21Þ
where GNw ðÞ is the probability generating function of the weighted Poisson random variable Nw. In the following lemma (see Rodrigues et al., 2009), we present the probability generating function of the variable Nw.
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
371
Lemma 2. Let f Z 0 and y 2 Y R, and the PMF of the discrete variable Nw be of the form pw ðn; y, fÞ ¼ jðn; fÞ expfynKðy, fÞg,
n ¼ 0,1, . . . :
Then, the corresponding PGF is given by
GNw ðzÞ ¼ expflð1zÞg
Elz ½wðN; fÞ , El ½wðN; fÞ
where l ¼ expfyg and wðn; fÞ ¼ n!jðn; fÞ. Thus, by using Lemma 2, we can express GNrw ðtÞ ðzÞ as
GNrw ðtÞ ðzÞ ¼ rð1tÞ þ rt expflð1zÞg
Elð1t þ tzÞ ½wðN; fÞ Elz ½wðN; fÞ þ ð1rÞ expfltð1zÞg : El ½wðN; fÞ El ½wðN; fÞ
ð22Þ
3.1. Properties of the CWPP In this section, we derive some basic properties of the correlated weighted Poisson processes in (20), such as the marginal and joint distributions, stationarity of the increments, moments, and the covariance function. 3.1.1. Marginal distributions Given the PGF GNrw ðtÞ ðzÞ in (21), we obtain the PMF of the variable Nrw ðtÞ, by taking derivatives of GNrw ðtÞ ðzÞ, as 8 < GNrw ðtÞ ð0Þ, k ¼ 0, gNrw ðtÞ ðkÞ ¼ P½Nrw ðtÞ ¼ k ¼ ðkÞ : GNrw ðtÞ ð0Þ, k ¼ 1,2, . . . , where
GðkÞ Nrw ðtÞ ¼
1 X n k @k w w w ðtÞ ðzÞ G ¼ r tk! P ½N ¼ k þ ð1 r Þ P ½N ¼ nk! t ð1tÞnk : N r @zk k z¼0 n¼k
Thus, we obtain 8 w w k ¼ 0, > < rð1tÞ þ rt GN ð0Þ þ ð1rÞGN ð1tÞ, k l t e ðltÞ wt ðk; fÞ ð23Þ gNrw ðtÞ ðkÞ ¼ w > , k ¼ 1,2, . . . , : rt P½N ¼ k þ ð1rÞ k! elð1tÞ El ½wðN; fÞ P n w where wt ðk; l, fÞ ¼ 1 n ¼ 0 ð½ð1tÞl =n!Þwðn þ k; fÞ and P½N ¼ k is as in (19). Furthermore, by using the expression in (8), we also find
EðNrw ðtÞÞ ¼ EðEðNrw ðtÞ9Nw ÞÞ ¼ tEðN w Þ
ð24Þ
and 2
VarðNrw ðtÞÞ ¼ EðVarðNrw ðtÞ9N w ÞÞ þ VarðEðNrw ðtÞ9N w ÞÞ ¼ tð1tÞ½ð1rÞEðNw Þ þ rEðN w Þ þt 2 Var½N w :
ð25Þ
Remark 4. From the expressions of mean and variance of Nrw ðtÞ in (24) and (25), respectively, we readily find the Fisher index of Nrw ðtÞ to be VarðNrw ðtÞÞ VarðNw Þ ¼ 1 þ½t þ rð1tÞ 1 þ rð1tÞEðN w Þ for t 2 ½0,1, w w EðNr ðtÞÞ EðN Þ which reveals that the proposed correlated weighted Poisson process is underdispersed for each t 2 ½0,1 if and only if EðN w Þ ot þ rð1tÞ=rð1tÞ and VarðNw Þ=EðNw Þ o 1rð1tÞ=t þ rð1tÞEðNw Þ. Otherwise, the process is overdispersed. 3.1.2. Distributions of the increments Consider the increment variable Nrw ðtÞNrw ðsÞ for 0 rs o t r1. The derivation of the PGF of Nrw ðtÞNrw ðsÞ is analogous to that of Nrw ðtÞ presented above. In this case, we have Pn 1 X w w ðI ðU ÞI ðU ÞÞ GNrw ðtÞNrw ðsÞ ðzÞ ¼ EðzNr ðtÞNr ðsÞ Þ ¼ Eðz j ¼ 1 ½0,t j ½0,s j ÞP½N w ¼ n: ð26Þ n¼0
Now, since the quantities I½0,t ðUj ÞI½0,s ðUj Þ are equicorrelated Bernoulli variables with probability ts and correlation P coefficient r 2 ½0,1, then, given Nw ¼ n, the increment variable Nrw ðtÞNrw ðsÞ ¼ nj¼ 1 ðI½0,t ðUj ÞI½0,s ðUj ÞÞ CBðn,ts, rÞ, and consequently
GNrw ðtÞNrw ðsÞ ðzÞ ¼ rf1ðtsÞg þ ðtsÞrelð1zÞ þð1rÞelðtsÞð1zÞ :
ð27Þ
372
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
Interestingly, we observe one again that the distribution of the increment is stationary, depending only on the difference ts. 3.1.3. Joint distributions Here, we consider the joint distribution of variables Nrw ðt1 Þ, . . . ,Nrw ðtk Þ for 0 rt1 o otk r1. For simplicity, we derive the expression for the case k¼ 2, and the result for general k can be derived in an analogous manner. We consider in this case the vector ðNrw ðsÞ,Nrw ðtÞÞ for 0 rs o t r1, and derive the joint PMF of this vector. For any 0 rk r n, we have
P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n ¼ P½Nrw ðsÞ ¼ k9Nrw ðtÞ ¼ nP½Nrw ðtÞ ¼ n: Upon noting, by conditioning on Nrw ðtÞ ¼ n, that the variable Nrw ðsÞ CBðn,s=t, rÞ, we readily find k s k=n n s s nk s 1ðk=nÞ P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n ¼ ð1rÞ 1 IA1 ðkÞ þ r 1 IA2 ðkÞ P½Nrw ðtÞ ¼ n, t t t t k
ð28Þ
where A1 ¼ f0,1, . . . ,ng, A2 ¼ f0,ng, k¼0,y,n, r 2 ½0,1, and P½Nrw ðtÞ ¼ n is as given earlier in (23). Next, we derive the joint ChF of the vector ðNrw ðsÞ,Nrw ðtÞÞ, for 0 rs o t r1, as follows: w
w
EðeiuNr ðsÞ þ ivNr ðtÞ Þ ¼ ¼
1 X n X
eiuk þ ivn P½Nrw ðsÞ ¼ k,Nrw ðtÞ ¼ n
n¼0k¼0 1 X ivn
n X
n¼0
k¼0
P½Nrw ðtÞ ¼ n
e
eiuk P½Nrw ðsÞ ¼ k9Nrw ðtÞ ¼ n
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} GNw ðs=tÞ9n ðeiu Þ
s
r
s s s ¼ r 1 GNrw ðtÞ ðe Þ þ r GNrw ðtÞ ðeiðu þ vÞ Þ þ1rÞGNrw ðtÞ 1 þ eiu eiv : t t t t iv
ð29Þ
Naturally, the expression in (29) reduces to the ChF of Nrw ðsÞ when v ¼0, and to the ChF of Nrw ðtÞ when u ¼0. 3.1.4. Covariance function Here, we derive the covariance of Nrw ðsÞ and Nrw ðtÞ, as well as that of two consecutive increments, viz., Nrw ðsÞ and Nrw ðtÞNrw ðsÞ for any 0 rs o t r1. First, we observe
VarðNrw ðtÞNrw ðsÞÞ ¼ VarðNrw ðtÞÞ þ VarðNrw ðsÞÞ2 CovðNrw ðsÞ,Nrw ðtÞÞ, and consequently,
CovðNrw ðsÞ,Nrw ðtÞÞ ¼ 12½VarðNrw ðtÞÞ þ VarðNrw ðsÞÞVarðNrw ðtÞNrw ðsÞÞ: Since VarðNrw ðtÞNrw ðsÞÞ ¼ VarðNrw ðtsÞÞ as shown earlier, and VarðNrw ðÞÞ is as given in (25), we obtain 2
CovðNrw ðsÞ,Nrw ðtÞÞ ¼ sfð1tÞ½ð1rÞE½Nw þ rEðN w Þ þt VarðN w Þg:
ð30Þ w
w
w
w
w
This leads to an expression for the covariance of two consecutive increments, viz., Nr ðsÞNr ð0Þ Nr ðsÞ and Nr ðtÞNr ðsÞ, as 2
CovðNrw ðsÞ,Nrw ðtÞNrw ðsÞÞ ¼ CovðNrw ðsÞ,Nrw ðtÞÞVarðNrw ðsÞÞ ¼ sðtsÞfVar½Nw ½ð1rÞE½N w þ rE½Nw g:
ð31Þ
The expression in (31) readily reveals that two consecutive increments are positively correlated if only if the variable Nw is 2 overdispersed and Var½N w 4 ð1rÞE½Nw þ rE½N w . Otherwise, the increments are negatively correlated. 4. Some illustrative examples In this section, we present a few specific processes that arise from our general formulation. In Table 1, we present the probability generating function, mean and variance corresponding to these processes. 4.1. Length-biased Poisson process When the weight function of the r.v. Nw is simply wðn; fÞ ¼ n, then the weight function wt ðÞ has an explicit form given by wt ðk; l,tÞ ¼ kþ lð1tÞ. Recall that, in this case, Nw has a Poisson distribution with parameter l shifted up by one, and is underdispersed, since Var½Nw o E½N w . Also, note that the compound Poisson distribution of Nrw ðtÞ is not shifted-Poisson, since its PMF is of the form 8 l lt > < rð1tÞ þ rte þð1rÞe , w k1 l P½Nr ðtÞ ¼ k ¼ l e elt ðltÞk k > þð1rÞ 1t þ , : rt ðk1Þ! k! l
k ¼ 0, k ¼ 1,2, . . . :
Correlated process Length-biased Poisson Exponentially weighted Poisson Negative binomial COM-Poisson
E½Nrw ðtÞ Var½Nrw ðtÞ rð1tÞ þ rt expflð1zÞgþ ð1rÞð1tð1zÞÞ expfltð1zÞg tð1lÞ tð1tÞ½1þ lð1 þ 2r þ rlÞ þ t 2 l rð1tÞ þ rt expfle2f ð1zÞg þ ð1rÞ expfltð1zÞe2f g t l expff þ 2lef g tð1tÞl½expff þ 2lef g þ rl expf2ðf þ lef Þg þ t 2 l½l expf2ðf þ lef Þg þ expff þ 2lef gl expf2ðf þ 2lef Þg tl tð1tÞl½1 þ rlð1 þ fÞ þ t 2 lð1 þ flÞ rð1tÞ þ rtð1 þ flð1zÞÞ1=f þ ð1rÞð1þ fltð1zÞÞ1=f 2 !2 3 " # j Zðlz, fÞ Zðlð1tð1zÞÞ, fÞ 1 j j j P jl 1 1 1 P P P jl j2 l jl þ ð1rÞ rð1tÞ þ rt 24 t 5 ð1 r ð1jÞÞ þ t tð1tÞ Zðl, fÞ Zðl, fÞ f f f f j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ j ¼ 0 ðj!Þ Zðl, fÞ GNrw ðtÞ ðzÞ
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
Table 1 Probability generating function (GNrw ðtÞ ðÞ), mean (E½Nrw ðÞ), and variance (Var½Nrw ðÞ) for different processes.
373
374
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
4.2. Exponentially weighted Poisson process When the weight function of the r.v. Nw is of exponential form, i.e., wðn; fÞ ¼ efm , f 2 R, then the weight function wt ðÞ has an explicit form given by wt ðk; l, f,tÞ ¼ expffklð1tÞð1ef Þg. It is easy to see that Nw has a Poisson distribution with parameter lef and that the variable Nrw ðtÞ has PMF given by 8 le2f > þ ð1rÞ expflte2f g, k ¼ 0, < rð1tÞ þ rte w k k fg f Þ þ ltð1ef Þg P½Nr ðtÞ ¼ k ¼ l expfk f þ l e ð l tÞ expfk f l ð12e > : rt þ ð1rÞ , k ¼ 1,2, . . . : k! k!
4.3. Negative binomial process Let us take the unobservable count variable, Nw, as a variable with a negative binomial distribution with parameters f 4 0 and l 40 (see Piegorsch, 1990; Saha and Paul, 2005), with PMF Gðf1 þ nÞ fl n P½Nw ¼ n; l, f ¼ ð1 þ flÞ1=f , n ¼ 0,1,2, . . . : ð32Þ 1 Gðf Þn! 1 þ fl Note that f ¼ 1 leads to the geometric distribution with parameter 1=ð1 þ lÞ. Comparing (32) with (19), we realize that (32) 1 is a weighted Poisson distribution with parameter fl=ð1 þ flÞ and weight function wðn; fÞ ¼ Gðf þ nÞ. So, we arrive at a closed-form expression for the weight function wt ðk; l, f,tÞ in this case as 2 3k 1 7 flð1tÞ 6 1 6 7 Gðf þkÞ : wt ðk; l, f,tÞ ¼ exp 4 5 flð1tÞ 1 þ fl flð1tÞ 1=f 1 1 1þ fl 1 þ fl Thus, the variable Nrw ðtÞ in this case has PMF as 8 > rð1tÞ þ rtð1 þ flÞ1=f þð1rÞð1 þ fltÞ1=f , > < w P½Nr ðtÞ ¼ k ¼ Gðf1 þ kÞ fl k Gðf1 þ1Þ flt k > r t ð1 þ flÞ1=f þ ð1rÞ ð1þ fltÞ1=f , > 1 1 : Gðf Þk! 1 þ fl Gðf Þk! 1 þ flt
k ¼ 0, k ¼ 1,2, . . . :
4.4. COM-Poisson process Let us take the unobservable count variable Nw as a variable with a COM-Poisson distribution with parameters l 4 0 and f 4 0 (see Shmueli et al., 2005), with PMF n
P½Nw ¼ n; l, f ¼
1 l , Zðl, fÞ ðn!Þf
m ¼ 0,1,2, . . . ,
ð33Þ
P j f where Zðl, fÞ ¼ 1 j ¼ 0 l =ðj!Þ . In particular, when f ¼ 0 and l o 1, the COM-Poisson distribution reduces to the geometric distribution with parameter 1l. The distribution in (33) may also be viewed as a weighted Poisson distribution with weight function wðn; fÞ ¼ ðn!Þ1f ; see Kokonendji et al. (2008). Therefore, for integer f, the weight function wt ðÞ of the variable Nrw is given by wt ðk; l, f,tÞ ¼ expflð1tÞgðk!Þ1f
1 X
½ðk þ1Þðk þ2Þ ðkþ nÞ1f
n¼0
¼ expflð1tÞgðk!Þ
1f
1 F f ðk þ 1; kþ 1,
½lð1tÞn n!
. . . ,k þ 1; lð1tÞÞ,
ð34Þ
where the generalized hypergeometric function is defined by u F v ða1 ,
. . . ,au ; b1 , . . . ,bv ; xÞ ¼
1 X ða1 Þn ða2 Þn ðau Þn xn , ðb1 Þn ðb2 Þn ðbv Þn n! n¼0
with ðaÞn ¼ aða þ1Þ ða þ n1Þ; see Gradshteyn and Ryzhik (2000). Taking k¼0 and t ¼0 in (33), we obtain from (22) a closed-form expression for the COM-Poisson process as
GNrw ðtÞ ðzÞ ¼ rð1tÞ þ rt
1 F f ð1; 1,
. . . ,1; lzÞ
1 F f ð1; 1, . . . ,1; lÞ
þ ð1rÞ
1 F f ð1; 1,
. . . ,1; lð1t þtzÞÞ
1 F f ð1; 1, . . . ,1; lÞ
:
P. Borges et al. / Journal of Statistical Planning and Inference 142 (2012) 366–375
In this case, the distribution of the variable Nw is 8 > 1 1 F f ð1; 1, . . . ,1; lð1tÞÞ > > þ ð1rÞ , > rð1tÞ þ rt > < Zðl, fÞ 1 F f ð1; 1, . . . ,1; lÞ P½Nrw ðtÞ ¼ k ¼ > lk ðltÞk 1 F f ð1; 1, . . . ,1; lð1tÞÞ > > rt 1 , þð1rÞ > > Zðl, fÞ f : ðk!Þ ðk!Þf 1 F f ð1; 1, . . . ,1; lÞ
375
k ¼ 0, k ¼ 1,2, . . . :
5. Concluding remarks In this paper, we have introduced correlated Poisson processes and correlated weighted Poisson processes as generalizations of the class of weighted Poisson processes on the interval [0;1] proposed by Balakrishnan and Kozubowski (2008) by incorporating a dependence structure between the standard uniform random variables used in the construction. The dependence is measured by the correlation coefficient which is a natural measure for the underlying exchangeable sequence. We have then studied some basic properties of these processes such as the marginal and joint distributions, stationarity of the increments, moments, and the covariance function. These results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions. Since this class of processes naturally accommodates underdispersion and overdispersion, it will be useful when dealing with data exhibiting greater variability than that allowed by the Poisson process. It will therefore be of great interest to develop inferential methods for the correlated Poisson processes and the correlated weighted Poisson processes. Work in this direction is currently under progress and we hope to report these findings in a future paper.
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