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Estimation in continuous state-space branching processes
Journal of Statistical Planning and Inference 92 (2001) 111–119
www.elsevier.com/locate/jspi
Estimation in continuous state-space branching processes a Department
Mohan Kalea , S.R. Deshmukhb; ∗ of Statistics, Sir Parashurambhau College, Pune 411030, India of Statistics, University of Pune, Pune 411007, India
b Department
Received 12 November 1997; accepted 13 March 2000
Abstract Extinction probability of a continuous-time continuous state-space branching process is a function of ÿrst two moments of the process. These moments are estimated on the basis of a discrete skeleton of the process. Estimators of smooth functions of the ÿrst two moments and in particular of extinction probability are proposed. Martingale limit theory is utilised to establish the optimal c 2001 Elsevier Science B.V. All rights reserved. properties of the estimators.
1. Introduction Bartoszynski (1975) presented a model to describe growth of rabies virus in human host infected by a rabid animal. Buhler and Keller (1978) have observed that the stochastic process described by Bartoszynski (1975) is a continuous state-space branching process in continuous time. This process has also been used to model the growth of placenta in a pregnant woman. The probabilistic behaviour of such a process has been studied by Lamperti (1967), Seneta and Vere-Jones (1969), Grey (1974), Kallenberg (1979) and Pakes (1988) after its introduction by Jirina (1958). The present article proposes a procedure for the estimation of extinction probability q which is a function of ÿrst two moments of the so-called o spring distribution. In epidemiology it is of interest to estimate probability (1 − q) of major outbreak in the spread of any epidemic. Let {Z(t); t¿0} be a continuous-time continuous state-space branching process (cf. Athreya and Ney, 1972, p. 259) with E(Z(t) | Z(0) = z) = ze t and 2 z + tz if = 0; (1.1) E(Z 2 (t) | Z(0) = z) = z 2 e2 t + −1 z(e2 t − e t ) if 6= 0: Here plays a role of criticality parameter similar to that of the o spring mean in a discrete-time and discrete state-space branching process. The extinction probability ∗
Corresponding author.
c 2001 Elsevier Science B.V. All rights reserved. 0378-3758/01/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 1 4 1 - 5
112
M. Kale, S.R. Deshmukh / Journal of Statistical Planning and Inference 92 (2001) 111–119
q is one if 60 and is equal to exp(−2 −1 ) if ¿ 0 and z = 1. Estimator of q is proposed as a function of estimators of and which are based on the discrete skeleton {Z(hn); n¿1} of the process {Z(t); t¿0} with h ¿ 0. It is proved in Section 2 that {Z(hn); n¿1} is a discrete-time continuous state-space branching process (cf. Kallenberg, 1979). The parameters and are functions of the o spring mean m and the o spring variance 2 of {Z(hn)¿1}. Consequently, the estimator of q is proposed as a function of estimators of m and 2 . The asymptotic properties of all these estimators are established using martingale limit theory in the supercritical case, i.e. when m ¿ 1 and on the non-extinction set A. 2. Estimation of mean m and variance 2 Let {Z(t); t¿0} be a continuos-time and continuous state-space branching process. The following theorem proves that {Z(hn); n¿1} is a discrete-time continuous statespace branching process. Theorem 2.1. The process {Z(hn); n¿1} for each h ¿ 0 is a discrete-time continuous state-space branching process. Proof. Let t () = −log E(exp(−Z(t))) be the cumulant generating function (c.g.f.) of Z(t), then the Markov and branching property result in the identity t+s () = t ( s ()) (cf. Athreya and Ney, 1972). Since it is true for any t and s, it follows that {Z(hn); n¿1} is a discrete-time continuous state-space branching process. With h = 1, let Z(hn) = Zn , then the observed data are Z1 ; : : : ; Zn . In this process E(Zn+1 | Fn ) = mZn and Var(Zn+1 | Fn ) = 2 Zn where, Fn is a ÿeld generated by {Z1 ; Z2 ; : : : ; Zn }. As in the Binayme–Galton–Watson (BGW) process estimators of m and 2 are proposed as n n−1 −1 P P Zi Zi−1 mˆ n = i=1
i=1
and ˆ2n =
n 1P −1 (Zi − mˆ n Zi−1 )2 Zi−1 : n i=1
Form of these estimators do resemble that of the estimators of o spring mean and o spring variance of the discrete state-space branching process. The properties of these estimators have been studied by Heyde (1974) in discrete state-space branching process, wherein Zn can be expressed as the random sum of independent and identically distributed random variables each having the same distribution as of Z1 . This summation structure is heavily used in establishing the properties of the estimators. It is to be noted that such a structure is not available for the continuous state-space branching process and we have to adopt techniques di erent from those used by Heyde (1974)
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113
and by Wei and Winniki (1989). In this set up the asymptotic distributions of mˆ n and ˆ2n are obtained using the martingale central limit theorem (CLT) due to Scott (1978) (cf. Guttorp, 1991) and due to Brown (1971), respectively. The following theorem gives results on mˆ n . Theorem 2.2. On A as n → ∞, (i) mˆ n converges almost surely (a.s.) to m. (ii) Pn asymptotic distribution of −1 ( i=1 Zi−1 )1=2 (mˆ n − m) is standard normal. Proof. The ÿrst result is proved using Toeplitz’s lemma and the fact that {Zn m−n ; n¿1} is a martingale with respect to Fn and converges a.s. to W ¿ 0 (cf. Kallenberg, 1979). To prove the second result, Scott’s (1978) CLT is used. It is noted that n n n P P P Zi−1 (mˆ n − m) = (Zi − mZi−1 ) = Ui = Sn∗ i=1
i=1
i=1
is a zero mean martingale with respect to Fn in view of the fact that, E[(Zi − Pn Pn mZi−1 ) | Fi−1 ] = 0. Let Vn2 = i=1 E(Ui2 | Fi−1 ) = 2 Sn−1 , with Sn−1 = i=1 Zi−1 and P n let sn2 = i=1 E(Ui2 ) = 2 (mn − 1)(m − 1)−1 which is ÿnite for m ¿ 1. Further, n −1 n P P Zi−1 E(Zi ) converges a:s: to W ¿ 0 (2.1) Vn2 sn−2 = i=1
i=1
with E(W ) ¡ ∞ which follows by the convergence of the martingale {Zn m−n ; n¿1} and Toeplitz’s lemma. Moreover, for m ¿ 1 and r¿0, 2 = sn−2 sn−r
2 (mn−r − 1)(m − 1) (m − 1)2 (mn − 1)
= (m−r − m−n )(1 − m−n )−1 → m−r as n → ∞:
(2.2)
To verify the third condition of Scott’s theorem the crucial result is E(exp(−sZn+1 ) | Z1 ; : : : ; Zn ) = exp(−h(s)Zn ) a:s:;
(2.3) −sZ1
where s may be complex with Re(s) 6= 0 and h(s) = −log E(e 1979, p. 15). With a2 = (m − 1)=m consider, En = E(exp(itUn =aVn ) | Fn−1 ) =E
it(Zn − mZn−1 ) p | Fn−1 exp a Sn−1
−itmZn−1 p = exp a Sn−1 −itmZn−1 p = exp a Sn−1
! E
!!
itZ pn | Fn−1 exp a Sn−1
! exp −h
). (cf. Kallenberg,
−it p a Sn−1
!!
!
! Zn−1
a:s: (using(2:3))
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"
" −itmZn−1 p = exp + Zn−1 log E 1 + a Sn−1
itZ p1 a Sn−1
!
t2Z 2 + ··· − 2 21 2 a Sn−1
#
−t 2 Zn−1 ∗ + n ; = exp 2a2 Sn−1
k=2 ) for k¿3. Now by the a.s. conwhere ∗n contains terms of the type (EZ1k )(Zn−1 =Sn−1 (k=2−1) a:s: −1 a:s: 2 −n → a on A and Sn−1 →0 vergence of Zn m to W and by Toeplitz lemma Zn−1 Sn−1 as n → ∞ on A. Hence ∗n → 0 a.s. on A. Thus
En → exp(−t 2 =2)
a:s: on A:
(2.4)
In view of (2.1), (2.2) and (2.4) Scott’s CLT is applicable. To establish the asymptotic normality of mˆ n let, Z(n−i) − mZ(n−(i−1)) ; i¡n ni = −1=2 m (m − 1)1=2 sn and ∗ni = sn ni Vn−1 ; i ¡ n and zero for i¿n, then n 1=2 n P −1 P Zi−1 (mˆ n − m) = Vn−1 (Zi+1 − mZi ) i=1
=Vn−1 =a
∞ P i=0
i=1
n−1 P
n−1 P
i=0
i=0
(Z(n−i) − mZ(n−(i−1)) ) = Vn−1 (asn )
∗ni = a lim sup k→∞
k−1 P i=0
ni
∗ni = ah(∗n0 ; : : :);
Pk−1 where h(x0 ; : : : :)=lim sup( j=0 xj ) is a continuous function. Let ∗n ={∗n0 ; ∗n1 ; ∗n2 ; : : : ; } and ∗ = (Y0 ; Y1 ; : : :), where Y0 ; Y1 ; : : : are independent random variables with Yi R L N(0; m−i ). By Scott’s CLT ∗n → ∗ and by continuous mapping theorem (cf. Billingsley, 1968) h(∗n ) converges in law to h(∗ ). Now to identify the limit law, note that Pn i=0 Yi is a zero mean martingale, further, " 2 #1=2 n n P P Y E Y 6 E i=0
( =
i=0
n P
i=0
EY
2
PP i6=j
)1=2 E(Yi Yj )
¡∞
Pn as variance of Yi is m−i and m ¿ 1. By the martingale convergence theorem i=0 Yi Pn−1 converges a.s. and hence in law to Y . Further, i=0 Yi has normal distribution with mean zero and variance n2 = (1 − m−n )(1 − m−1 )−1 which converges to (1 − m−1 )−1 = Pn−1 a−2 . For any real x; P[ i=0 Yi 6x] = (x n−1 ); where denotes the distribution function of the standard normal. By continuity of ; (x n−1 ) converges to (xa). Hence, Pn−1 Pn−1 L L −2 i=0 Yi → N(0; a ) and a i=0 Yi → N(0; 1) and the proof is complete. The asymptotic properties of ˆ2n are studied by studying asymptotic properties of Pn −1 (Zi − mZi−1 )2 in view of the following lemma. n∗2 = (1=n) i=1 Zi−1
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Lemma 2.1. On A; for ¿ 0; n1− (n∗2 − ˆ2n ) converges in probability to zero as n → ∞. Proof. Observe that
n P −1 (mˆ n − m) − (mˆ 2n − m2 )] n1− (n∗2 − ˆ2n ) = n− Zi−1 [2Zi Zi−1 i=1 n − P = (n ) Zi−1 (mˆ n − m)2 : i=1
Pn In view of Theorem 2.2, −2 ( i=1 Zi−1 )(mˆ n − m)2 has asymptotically chi-square distribution with one degree of freedom, for ¿ 0; n− → 0 as n → ∞ and hence the result follows. In the next theorem consistency of n∗2 is established. Theorem 2.3. On A; n∗2 converges a.s. to 2 as n → ∞. Pn −1 − 2 , and Wn = i= Xi . It can be shown that Proof. Let Xi = (Zi − mZi−1 )2 Zi−1 {Wn ; Fn ; n¿1} is a zero mean martingale. Now, from Eq. (2.3), it follows that, Var(Xi ) −1 ) where k is the kurtosis of Z1 . Then on the non-extinction is 24 − (3 − k)4 E(Zi−1 set A n ∞ 1 P P −1 4 −1 4 i−2 E(Xi2 | Fi−1 ) = (22 − 3Zi−1 + kZi−1 ) 2 i i=1 i=1 ∞ 1 P −1 (22 + k4 Zi−1 ) ¡ ∞: ¡ 2 i i=1 Thus lim n−1 Wn is zero a.s. on A (cf. Hall and Heyde, 1980, Theorem 2:18). Hence n2∗ converges to 2 a.s. on A. To establish the asymptotic normality we need to show that EZn−1 → 0 as n → ∞. In BGW process it is trivially true since on A, Zn ¿1. However, in continuous state-space process we do not have such inequality. We prove the required result in the following lemma, using properties of c.g.f. and the bounded convergence theorem. Lemma 2.2. On A; EZn−1 → 0 as n → ∞. Proof. On nonextinction set A, Zn ¿ 0 ∀n, and Zn−1 is deÿned ∀ n¿1. Let Fn and n denote the distribution function and the Laplace transform respectively of Zn , then Z ∞ Z ∞ Z ∞ x−1 dFn (x) = e−sx ds dFn (x) EZn−1 = Z =
0
∞ 0
Z 0
0
∞
0
Z −sx e dFn (x) ds =
0
∞
n (s) ds:
The interchange of integrals follows from Fubini’s theorem, in view of the non-negativity of the integrand. Now, Z1 is inÿnitely divisible (i.d.), hence Zn , is i.d. for each
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n¿1 (cf. Kallenberg, 1979, p. 49), consequently, n (s) = e−hn (s) , where hn (s) is the c.g.f. of Zn ; hn (s) is strictly increasing in s ¿ 0 ∀n¿1. Further, in the supercritical case and when r = P[lim Zn = 0] = 0; h(s) ¿ s ∀s ¿ 0 and limn→∞ hn (s) = ∞. Thus hn−1 (h(s))¿hn−1 (s). So hn (s) ¿ hn−1 (s) ∀s ¿ 0 i.e. hn is strictly increasing in n¿1; ∀s ¿ 0. Consequently gn (s) = exp(−hn (s)) is strictly decreasing in n¿1. Thus 06gn (s)6g1 (s); ∀s ¿ 0. Now, Z ∞ Z ∞ Z ∞ g1 (s) ds = e−h(s) ds6 e−s ds = 1: 0
0
0
−hn (s)
→ 0 as n → ∞. Hence by the Lebesgue bounded convergence Further gn (s) = e theorem and treating all integrals w.r.t. the Lebesgue measure Z ∞ −1 e−hn (s) ds = 0: lim EZn = lim n→∞
n→∞
0
−1 In view of the above lemma Var(Xi ) = 24 − (3 − k)4 E(Zi−1 ) is ÿnite. Thus the increments of the martingale {(Wn ; Fn ); n¿1} have ÿnite variance. Following lemma summarizes some result about the martingale di erences {Xi ; i¿1}. Let Vn∗2 = Pn 2 ∗2 ∗2 i=1 E(Xi |Fi−1 ) and sn = EVn .
Lemma 2.3. On A as n→∞ (i) sn∗−2 Vn∗2 → 1 a.s.; (ii) sn∗2 =n→24 ; (iii) sn∗−2 I [|Xi |¿sn ]] → 0; ¿ 0.
Pn
i=1
E[Xi2
Proof. Note that, n P (3 − k) 4 ∗2 4 2 − Vn = Zi−1 i=1 and sn∗2 = EVn∗2 =
n P
24 − (3 − k)4 E
1
: Zi−1 Pn −1 → 0 a.s. by the Toeplitz lemma Therefore sn∗−2 Vn∗2 → 1 a.s. since (1=n) i=1 Zi−1 Pn −1 and (1=n) i=1 E(Zi−1 ) → 0 as n → ∞ by Lemma 2.2 and the Toeplitz lemma. Further, Pn −1 Vn2∗ n−1 = 24 − (3 − k)4 (1=n) i=1 Zi−1 converges a.s. to 24 and hence sn∗2 n−1 also converges to 24 . To prove the third result consider, √ E[Xn2 I [|Xn | ¿ c n]] i=1
−2 −1 − 2Zn−1 (Zn − mZn−1 )2 2 + 4 } =E[{(Zn − mZn−1 )4 Zn−1 √ −1 (Zn − mZn−1 )2 − 2 | ¿ c n] ×I [|Zn−1 √ −2 −1 (Zn − mZn−1 )4 I [|Zn−1 (Zn − mZn−1 )2 | ¿ c n] 6E[Zn−1 √ −1 (Zn − mZn−1 )2 | ¿ c n] + 4 E[I [|Zn−1 −2 −2 (Zn − mZn−1 )4 I [Zn−1 (Zn − mZn−1 )4 ¿ c2 n]) = E(Zn−1 √ −1 ¿ c n]: + 4 P[(Zn − mZn−1 )2 Zn−1
(2.5)
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117
To prove the convergence of the ÿrst term in (2.5) we establish the uniform inte−2 4 grability of Yn2 = −4 Zn−1 (Zn − mZn−1 )p . It is shown in (2.4) that conditional characteristic function of (Zn − mZn−1 )=a Sn−1 converges a.s. to that of the standard normal distribution and hence by the Lebesgue dominated convergence theorem unconditional characteristic function also converges to that of the standard normal. Also √ (Sn−1 =Zn−1 )1=2 converges a.s. to a−1 , as n → ∞. Therefore ((Zn − mZn−1 )= Zn−1 )2 converges in distribution to Y which has chi-square distribution with one degree of freedom and Yn2 converges in law to Y 2 (cf. Parzen, 1972, p. 424). Now, −2 (Zn − mZn−1 )2 −4 ) E(Yn2 ) = E(Zn−1
−1 )]−4 → 3 = [34 − (3 − k)4 E(Zn−1
as n → ∞:
2
Since Y is a square of chi-square random variable with one degree of freedom, E(Y 2 )= Var(Y ) + (E(Y ))2 = 2 + 1 = 3. Thus E(Yn2 ) → EY 2 = 3. −1 Hence Zn−1 (Zn − mZn−1 )4 is a sequence of uniformly integrable random variables (cf. Billingsley, 1968, Theorem 5:4). Hence, the ÿrst term in (2.5) converges to zero as n → ∞. Using the Markov inequality, second term in (2.5) and√hence (2.5) converges to zero. We have sn∗2 =n → 24 as n → ∞, hence choosing =c= 24 ; E[Xn2 I [Xn |¿sn∗ ] Pn converges to zero as n → ∞, and by the Toeplitz lemma (1=n) i=1 E(Xi2 I [|Xi |¿ sn∗ ]) → 0 as n → ∞: Now, n n P 1P sn∗−2 E[Xi2 I [|Xi |¿sn∗ ]] = nsn∗−2 E[Xi2 I [|Xi |¿sn∗ ]] → 0 n i=1 i=1 as n → ∞ by (ii) and proof of the lemma is complete. In the next theorem the asymptotic distribution of n∗2 is derived using the results established in the above lemma. Theorem 2.4. On A as n → ∞ the asymptotic distribution of (n=24 )1=2 (n∗2 − 2 ) is standard normal. Proof. Note that, (n=24 )1=2 (n∗2 − 2 ) is rewritten as 1=2 ∗ n 1=2 1 P n 1 s −1 2 2 √n sn∗−1 Wn ; (Z (Zi − mZi−1 ) − = 24 n i=1 i−1 24 n Pn where Wn = i=1 Xi is a zero mean martingale with ÿnite variance for increments. Using Lemma 2.3 and martingale CLT due to Brown (1971), (cf. Hall and Heyde, 1980) it follows that sn∗−1 Wn converges in law to standard normal distribution as n → ∞. Hence proof of the theorem is complete using the second result in Lemma 2.3. In view of Lemma 2.1, asymptotic properties of ˆ2n and 2 ∗n are the same except the fact that ˆ2n is weakly consistent for 2 because the di erence between the two converges to zero in probability as proved in Lemma 2.1. The next theorem states the asymptotic properties of ˆ2n .
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Theorem 2.5. On A; (i) ˆ2n converges to 2 in probability; (ii) the asymptotic distribution of (n=24 )1=2 (ˆ2n − 2 ) is standard normal. In the next section, we propose the estimator for a smooth function of m and 2 hence for q and establish their asymptotic properties. 3. Estimation of extinction probability It is observed that the c.g.f. extinction probability, population drift, etc., of the process are functions of m and 2 . Therefore, a general procedure of estimation is proposed ÿrst and then, as a special case, the estimation of extinction probability is discussed when data are Z1 ; : : : ; Zn . It is assumed that the function ‘f’ is di erentiable up to order 2 and the ÿrst and second derivatives are continuous. By Taylor’s series expansion @f 2 2 f(mˆ n ; ˆn ) = f(m; ) + (mˆ n − m) @m m=mˆ n ;2 =ˆ2n @f + (ˆ2n − 2 ) 2 ˆ2n ;mˆ n @ " 2 2 1 2 2 @ f 2 2 @ f (mˆ n − m) + + (ˆn − ) 2 @m2 (m˜ n ;˜2 ) @4 ˜2 ;m˜ n n n # 2 f @ + 2(mˆ n − m)(ˆ2n − 2 ) ; (3.1) @m@2 m˜ n ;˜2 n
∗
∗
˜2n
ˆ2n
2
= + (1 − ) ; 0 ¡ ∗ ¡ 1; 0 ¡ ¡ 1. where m˜ n = mˆ n + (1 − )m; Hence, using consistency and asymptotic normality of mˆ n and ˆ2n established in √ Section 2, it follows immediately that on A, with the norming factor n, second and the fourth terms in (3.1) converge to zero in probability and hence, −2 ! √ @f L 2 2 4 n(f(mˆ n ; ˆn ) − f(m; )) → N 0; 2 as n → ∞: (3.2) @2 Now with z = 1 and t = 1 in 1.1, = loge m; = 2 exp(− ){exp( ) − 1}−1 and q = exp{−2 = }. Hence estimators of these are obtained by replacing m by mˆ n and 2 √ by ˆ2 . Asymptotic distributions are easily derived from (3.2). In particular, n(qˆn − q) converges in law to normal distribution with mean 0 and variance 2 2 =(2 2 q2 ). In summary, a discrete skeleton of the process produces a reasonably good estimator of the extinction probability. Acknowledgements Both the authors thank the referee for his suggestions and also thank Council for Scientiÿc and Industrial Research, New Delhi, India for ÿnancial support to carry out this work.
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References Athreya, K.B., Ney, P., 1972. Branching Processes. Springer, Berlin. Bartoszynski, P., 1975. A model for the risk of rabies. ISI, Invited paper No. 9, ISI 1–9, Warsaw. Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New York. Brown, B.M., 1971. Martingale central limit theorems. Ann. Math. Statist. 42, 59–66. Buhler, W.J., Keller, G., 1978. Some remarks on Bartoszynski’s Rabies Model. Math Biosci. 39, 273–279. Grey, D.R., 1974. Asymptotic behaviour of continuous time and continuous state space branching processes. J. Appl. Probab. 11, 669–677. Guttorp, P., 1991. Statistical Inference for Branching Processes. Wiley, New York. Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and its Applications. Academic Press, London. Jirina, M., 1958. Stochastic branching process with continuous state space. Czechoslovak Math. 8, 292–313. Kallenberg, P.J.M., 1979. Branching processes with continuous state space. Mathematical Centre Tracts, No. 117, Amsterdam. Lamperti, J., 1967. The limit of a sequence of branching processes. ZW 7, 271–288. Pakes, A.G., 1988. Some limit theorems for continuous state branching process. J. Austal. Math. Soc. Ser. A 44, 71–87. Parzen, E., 1972. Modern Probability Theory and its Applications. Wiley Eastern, New Delhi. Scott, D.J., 1978. A central limit theorem for martingales and applications to branching processes. Stochastic Process Appl. 6, 241–252. Seneta, E., Vere-Jones, D., 1969. On a problem of M. Jirina concerning continuous state space branching processes. Zech. Math. J. 19 (94), 277–283.
www.elsevier.com/locate/jspi
Estimation in continuous state-space branching processes a Department
Mohan Kalea , S.R. Deshmukhb; ∗ of Statistics, Sir Parashurambhau College, Pune 411030, India of Statistics, University of Pune, Pune 411007, India
b Department
Received 12 November 1997; accepted 13 March 2000
Abstract Extinction probability of a continuous-time continuous state-space branching process is a function of ÿrst two moments of the process. These moments are estimated on the basis of a discrete skeleton of the process. Estimators of smooth functions of the ÿrst two moments and in particular of extinction probability are proposed. Martingale limit theory is utilised to establish the optimal c 2001 Elsevier Science B.V. All rights reserved. properties of the estimators.
1. Introduction Bartoszynski (1975) presented a model to describe growth of rabies virus in human host infected by a rabid animal. Buhler and Keller (1978) have observed that the stochastic process described by Bartoszynski (1975) is a continuous state-space branching process in continuous time. This process has also been used to model the growth of placenta in a pregnant woman. The probabilistic behaviour of such a process has been studied by Lamperti (1967), Seneta and Vere-Jones (1969), Grey (1974), Kallenberg (1979) and Pakes (1988) after its introduction by Jirina (1958). The present article proposes a procedure for the estimation of extinction probability q which is a function of ÿrst two moments of the so-called o spring distribution. In epidemiology it is of interest to estimate probability (1 − q) of major outbreak in the spread of any epidemic. Let {Z(t); t¿0} be a continuous-time continuous state-space branching process (cf. Athreya and Ney, 1972, p. 259) with E(Z(t) | Z(0) = z) = ze t and 2 z + tz if = 0; (1.1) E(Z 2 (t) | Z(0) = z) = z 2 e2 t + −1 z(e2 t − e t ) if 6= 0: Here plays a role of criticality parameter similar to that of the o spring mean in a discrete-time and discrete state-space branching process. The extinction probability ∗
Corresponding author.
c 2001 Elsevier Science B.V. All rights reserved. 0378-3758/01/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 1 4 1 - 5
112
M. Kale, S.R. Deshmukh / Journal of Statistical Planning and Inference 92 (2001) 111–119
q is one if 60 and is equal to exp(−2 −1 ) if ¿ 0 and z = 1. Estimator of q is proposed as a function of estimators of and which are based on the discrete skeleton {Z(hn); n¿1} of the process {Z(t); t¿0} with h ¿ 0. It is proved in Section 2 that {Z(hn); n¿1} is a discrete-time continuous state-space branching process (cf. Kallenberg, 1979). The parameters and are functions of the o spring mean m and the o spring variance 2 of {Z(hn)¿1}. Consequently, the estimator of q is proposed as a function of estimators of m and 2 . The asymptotic properties of all these estimators are established using martingale limit theory in the supercritical case, i.e. when m ¿ 1 and on the non-extinction set A. 2. Estimation of mean m and variance 2 Let {Z(t); t¿0} be a continuos-time and continuous state-space branching process. The following theorem proves that {Z(hn); n¿1} is a discrete-time continuous statespace branching process. Theorem 2.1. The process {Z(hn); n¿1} for each h ¿ 0 is a discrete-time continuous state-space branching process. Proof. Let t () = −log E(exp(−Z(t))) be the cumulant generating function (c.g.f.) of Z(t), then the Markov and branching property result in the identity t+s () = t ( s ()) (cf. Athreya and Ney, 1972). Since it is true for any t and s, it follows that {Z(hn); n¿1} is a discrete-time continuous state-space branching process. With h = 1, let Z(hn) = Zn , then the observed data are Z1 ; : : : ; Zn . In this process E(Zn+1 | Fn ) = mZn and Var(Zn+1 | Fn ) = 2 Zn where, Fn is a ÿeld generated by {Z1 ; Z2 ; : : : ; Zn }. As in the Binayme–Galton–Watson (BGW) process estimators of m and 2 are proposed as n n−1 −1 P P Zi Zi−1 mˆ n = i=1
i=1
and ˆ2n =
n 1P −1 (Zi − mˆ n Zi−1 )2 Zi−1 : n i=1
Form of these estimators do resemble that of the estimators of o spring mean and o spring variance of the discrete state-space branching process. The properties of these estimators have been studied by Heyde (1974) in discrete state-space branching process, wherein Zn can be expressed as the random sum of independent and identically distributed random variables each having the same distribution as of Z1 . This summation structure is heavily used in establishing the properties of the estimators. It is to be noted that such a structure is not available for the continuous state-space branching process and we have to adopt techniques di erent from those used by Heyde (1974)
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and by Wei and Winniki (1989). In this set up the asymptotic distributions of mˆ n and ˆ2n are obtained using the martingale central limit theorem (CLT) due to Scott (1978) (cf. Guttorp, 1991) and due to Brown (1971), respectively. The following theorem gives results on mˆ n . Theorem 2.2. On A as n → ∞, (i) mˆ n converges almost surely (a.s.) to m. (ii) Pn asymptotic distribution of −1 ( i=1 Zi−1 )1=2 (mˆ n − m) is standard normal. Proof. The ÿrst result is proved using Toeplitz’s lemma and the fact that {Zn m−n ; n¿1} is a martingale with respect to Fn and converges a.s. to W ¿ 0 (cf. Kallenberg, 1979). To prove the second result, Scott’s (1978) CLT is used. It is noted that n n n P P P Zi−1 (mˆ n − m) = (Zi − mZi−1 ) = Ui = Sn∗ i=1
i=1
i=1
is a zero mean martingale with respect to Fn in view of the fact that, E[(Zi − Pn Pn mZi−1 ) | Fi−1 ] = 0. Let Vn2 = i=1 E(Ui2 | Fi−1 ) = 2 Sn−1 , with Sn−1 = i=1 Zi−1 and P n let sn2 = i=1 E(Ui2 ) = 2 (mn − 1)(m − 1)−1 which is ÿnite for m ¿ 1. Further, n −1 n P P Zi−1 E(Zi ) converges a:s: to W ¿ 0 (2.1) Vn2 sn−2 = i=1
i=1
with E(W ) ¡ ∞ which follows by the convergence of the martingale {Zn m−n ; n¿1} and Toeplitz’s lemma. Moreover, for m ¿ 1 and r¿0, 2 = sn−2 sn−r
2 (mn−r − 1)(m − 1) (m − 1)2 (mn − 1)
= (m−r − m−n )(1 − m−n )−1 → m−r as n → ∞:
(2.2)
To verify the third condition of Scott’s theorem the crucial result is E(exp(−sZn+1 ) | Z1 ; : : : ; Zn ) = exp(−h(s)Zn ) a:s:;
(2.3) −sZ1
where s may be complex with Re(s) 6= 0 and h(s) = −log E(e 1979, p. 15). With a2 = (m − 1)=m consider, En = E(exp(itUn =aVn ) | Fn−1 ) =E
it(Zn − mZn−1 ) p | Fn−1 exp a Sn−1
−itmZn−1 p = exp a Sn−1 −itmZn−1 p = exp a Sn−1
! E
!!
itZ pn | Fn−1 exp a Sn−1
! exp −h
). (cf. Kallenberg,
−it p a Sn−1
!!
!
! Zn−1
a:s: (using(2:3))
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"
" −itmZn−1 p = exp + Zn−1 log E 1 + a Sn−1
itZ p1 a Sn−1
!
t2Z 2 + ··· − 2 21 2 a Sn−1
#
−t 2 Zn−1 ∗ + n ; = exp 2a2 Sn−1
k=2 ) for k¿3. Now by the a.s. conwhere ∗n contains terms of the type (EZ1k )(Zn−1 =Sn−1 (k=2−1) a:s: −1 a:s: 2 −n → a on A and Sn−1 →0 vergence of Zn m to W and by Toeplitz lemma Zn−1 Sn−1 as n → ∞ on A. Hence ∗n → 0 a.s. on A. Thus
En → exp(−t 2 =2)
a:s: on A:
(2.4)
In view of (2.1), (2.2) and (2.4) Scott’s CLT is applicable. To establish the asymptotic normality of mˆ n let, Z(n−i) − mZ(n−(i−1)) ; i¡n ni = −1=2 m (m − 1)1=2 sn and ∗ni = sn ni Vn−1 ; i ¡ n and zero for i¿n, then n 1=2 n P −1 P Zi−1 (mˆ n − m) = Vn−1 (Zi+1 − mZi ) i=1
=Vn−1 =a
∞ P i=0
i=1
n−1 P
n−1 P
i=0
i=0
(Z(n−i) − mZ(n−(i−1)) ) = Vn−1 (asn )
∗ni = a lim sup k→∞
k−1 P i=0
ni
∗ni = ah(∗n0 ; : : :);
Pk−1 where h(x0 ; : : : :)=lim sup( j=0 xj ) is a continuous function. Let ∗n ={∗n0 ; ∗n1 ; ∗n2 ; : : : ; } and ∗ = (Y0 ; Y1 ; : : :), where Y0 ; Y1 ; : : : are independent random variables with Yi R L N(0; m−i ). By Scott’s CLT ∗n → ∗ and by continuous mapping theorem (cf. Billingsley, 1968) h(∗n ) converges in law to h(∗ ). Now to identify the limit law, note that Pn i=0 Yi is a zero mean martingale, further, " 2 #1=2 n n P P Y E Y 6 E i=0
( =
i=0
n P
i=0
EY
2
PP i6=j
)1=2 E(Yi Yj )
¡∞
Pn as variance of Yi is m−i and m ¿ 1. By the martingale convergence theorem i=0 Yi Pn−1 converges a.s. and hence in law to Y . Further, i=0 Yi has normal distribution with mean zero and variance n2 = (1 − m−n )(1 − m−1 )−1 which converges to (1 − m−1 )−1 = Pn−1 a−2 . For any real x; P[ i=0 Yi 6x] = (x n−1 ); where denotes the distribution function of the standard normal. By continuity of ; (x n−1 ) converges to (xa). Hence, Pn−1 Pn−1 L L −2 i=0 Yi → N(0; a ) and a i=0 Yi → N(0; 1) and the proof is complete. The asymptotic properties of ˆ2n are studied by studying asymptotic properties of Pn −1 (Zi − mZi−1 )2 in view of the following lemma. n∗2 = (1=n) i=1 Zi−1
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Lemma 2.1. On A; for ¿ 0; n1− (n∗2 − ˆ2n ) converges in probability to zero as n → ∞. Proof. Observe that
n P −1 (mˆ n − m) − (mˆ 2n − m2 )] n1− (n∗2 − ˆ2n ) = n− Zi−1 [2Zi Zi−1 i=1 n − P = (n ) Zi−1 (mˆ n − m)2 : i=1
Pn In view of Theorem 2.2, −2 ( i=1 Zi−1 )(mˆ n − m)2 has asymptotically chi-square distribution with one degree of freedom, for ¿ 0; n− → 0 as n → ∞ and hence the result follows. In the next theorem consistency of n∗2 is established. Theorem 2.3. On A; n∗2 converges a.s. to 2 as n → ∞. Pn −1 − 2 , and Wn = i= Xi . It can be shown that Proof. Let Xi = (Zi − mZi−1 )2 Zi−1 {Wn ; Fn ; n¿1} is a zero mean martingale. Now, from Eq. (2.3), it follows that, Var(Xi ) −1 ) where k is the kurtosis of Z1 . Then on the non-extinction is 24 − (3 − k)4 E(Zi−1 set A n ∞ 1 P P −1 4 −1 4 i−2 E(Xi2 | Fi−1 ) = (22 − 3Zi−1 + kZi−1 ) 2 i i=1 i=1 ∞ 1 P −1 (22 + k4 Zi−1 ) ¡ ∞: ¡ 2 i i=1 Thus lim n−1 Wn is zero a.s. on A (cf. Hall and Heyde, 1980, Theorem 2:18). Hence n2∗ converges to 2 a.s. on A. To establish the asymptotic normality we need to show that EZn−1 → 0 as n → ∞. In BGW process it is trivially true since on A, Zn ¿1. However, in continuous state-space process we do not have such inequality. We prove the required result in the following lemma, using properties of c.g.f. and the bounded convergence theorem. Lemma 2.2. On A; EZn−1 → 0 as n → ∞. Proof. On nonextinction set A, Zn ¿ 0 ∀n, and Zn−1 is deÿned ∀ n¿1. Let Fn and n denote the distribution function and the Laplace transform respectively of Zn , then Z ∞ Z ∞ Z ∞ x−1 dFn (x) = e−sx ds dFn (x) EZn−1 = Z =
0
∞ 0
Z 0
0
∞
0
Z −sx e dFn (x) ds =
0
∞
n (s) ds:
The interchange of integrals follows from Fubini’s theorem, in view of the non-negativity of the integrand. Now, Z1 is inÿnitely divisible (i.d.), hence Zn , is i.d. for each
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n¿1 (cf. Kallenberg, 1979, p. 49), consequently, n (s) = e−hn (s) , where hn (s) is the c.g.f. of Zn ; hn (s) is strictly increasing in s ¿ 0 ∀n¿1. Further, in the supercritical case and when r = P[lim Zn = 0] = 0; h(s) ¿ s ∀s ¿ 0 and limn→∞ hn (s) = ∞. Thus hn−1 (h(s))¿hn−1 (s). So hn (s) ¿ hn−1 (s) ∀s ¿ 0 i.e. hn is strictly increasing in n¿1; ∀s ¿ 0. Consequently gn (s) = exp(−hn (s)) is strictly decreasing in n¿1. Thus 06gn (s)6g1 (s); ∀s ¿ 0. Now, Z ∞ Z ∞ Z ∞ g1 (s) ds = e−h(s) ds6 e−s ds = 1: 0
0
0
−hn (s)
→ 0 as n → ∞. Hence by the Lebesgue bounded convergence Further gn (s) = e theorem and treating all integrals w.r.t. the Lebesgue measure Z ∞ −1 e−hn (s) ds = 0: lim EZn = lim n→∞
n→∞
0
−1 In view of the above lemma Var(Xi ) = 24 − (3 − k)4 E(Zi−1 ) is ÿnite. Thus the increments of the martingale {(Wn ; Fn ); n¿1} have ÿnite variance. Following lemma summarizes some result about the martingale di erences {Xi ; i¿1}. Let Vn∗2 = Pn 2 ∗2 ∗2 i=1 E(Xi |Fi−1 ) and sn = EVn .
Lemma 2.3. On A as n→∞ (i) sn∗−2 Vn∗2 → 1 a.s.; (ii) sn∗2 =n→24 ; (iii) sn∗−2 I [|Xi |¿sn ]] → 0; ¿ 0.
Pn
i=1
E[Xi2
Proof. Note that, n P (3 − k) 4 ∗2 4 2 − Vn = Zi−1 i=1 and sn∗2 = EVn∗2 =
n P
24 − (3 − k)4 E
1
: Zi−1 Pn −1 → 0 a.s. by the Toeplitz lemma Therefore sn∗−2 Vn∗2 → 1 a.s. since (1=n) i=1 Zi−1 Pn −1 and (1=n) i=1 E(Zi−1 ) → 0 as n → ∞ by Lemma 2.2 and the Toeplitz lemma. Further, Pn −1 Vn2∗ n−1 = 24 − (3 − k)4 (1=n) i=1 Zi−1 converges a.s. to 24 and hence sn∗2 n−1 also converges to 24 . To prove the third result consider, √ E[Xn2 I [|Xn | ¿ c n]] i=1
−2 −1 − 2Zn−1 (Zn − mZn−1 )2 2 + 4 } =E[{(Zn − mZn−1 )4 Zn−1 √ −1 (Zn − mZn−1 )2 − 2 | ¿ c n] ×I [|Zn−1 √ −2 −1 (Zn − mZn−1 )4 I [|Zn−1 (Zn − mZn−1 )2 | ¿ c n] 6E[Zn−1 √ −1 (Zn − mZn−1 )2 | ¿ c n] + 4 E[I [|Zn−1 −2 −2 (Zn − mZn−1 )4 I [Zn−1 (Zn − mZn−1 )4 ¿ c2 n]) = E(Zn−1 √ −1 ¿ c n]: + 4 P[(Zn − mZn−1 )2 Zn−1
(2.5)
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To prove the convergence of the ÿrst term in (2.5) we establish the uniform inte−2 4 grability of Yn2 = −4 Zn−1 (Zn − mZn−1 )p . It is shown in (2.4) that conditional characteristic function of (Zn − mZn−1 )=a Sn−1 converges a.s. to that of the standard normal distribution and hence by the Lebesgue dominated convergence theorem unconditional characteristic function also converges to that of the standard normal. Also √ (Sn−1 =Zn−1 )1=2 converges a.s. to a−1 , as n → ∞. Therefore ((Zn − mZn−1 )= Zn−1 )2 converges in distribution to Y which has chi-square distribution with one degree of freedom and Yn2 converges in law to Y 2 (cf. Parzen, 1972, p. 424). Now, −2 (Zn − mZn−1 )2 −4 ) E(Yn2 ) = E(Zn−1
−1 )]−4 → 3 = [34 − (3 − k)4 E(Zn−1
as n → ∞:
2
Since Y is a square of chi-square random variable with one degree of freedom, E(Y 2 )= Var(Y ) + (E(Y ))2 = 2 + 1 = 3. Thus E(Yn2 ) → EY 2 = 3. −1 Hence Zn−1 (Zn − mZn−1 )4 is a sequence of uniformly integrable random variables (cf. Billingsley, 1968, Theorem 5:4). Hence, the ÿrst term in (2.5) converges to zero as n → ∞. Using the Markov inequality, second term in (2.5) and√hence (2.5) converges to zero. We have sn∗2 =n → 24 as n → ∞, hence choosing =c= 24 ; E[Xn2 I [Xn |¿sn∗ ] Pn converges to zero as n → ∞, and by the Toeplitz lemma (1=n) i=1 E(Xi2 I [|Xi |¿ sn∗ ]) → 0 as n → ∞: Now, n n P 1P sn∗−2 E[Xi2 I [|Xi |¿sn∗ ]] = nsn∗−2 E[Xi2 I [|Xi |¿sn∗ ]] → 0 n i=1 i=1 as n → ∞ by (ii) and proof of the lemma is complete. In the next theorem the asymptotic distribution of n∗2 is derived using the results established in the above lemma. Theorem 2.4. On A as n → ∞ the asymptotic distribution of (n=24 )1=2 (n∗2 − 2 ) is standard normal. Proof. Note that, (n=24 )1=2 (n∗2 − 2 ) is rewritten as 1=2 ∗ n 1=2 1 P n 1 s −1 2 2 √n sn∗−1 Wn ; (Z (Zi − mZi−1 ) − = 24 n i=1 i−1 24 n Pn where Wn = i=1 Xi is a zero mean martingale with ÿnite variance for increments. Using Lemma 2.3 and martingale CLT due to Brown (1971), (cf. Hall and Heyde, 1980) it follows that sn∗−1 Wn converges in law to standard normal distribution as n → ∞. Hence proof of the theorem is complete using the second result in Lemma 2.3. In view of Lemma 2.1, asymptotic properties of ˆ2n and 2 ∗n are the same except the fact that ˆ2n is weakly consistent for 2 because the di erence between the two converges to zero in probability as proved in Lemma 2.1. The next theorem states the asymptotic properties of ˆ2n .
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Theorem 2.5. On A; (i) ˆ2n converges to 2 in probability; (ii) the asymptotic distribution of (n=24 )1=2 (ˆ2n − 2 ) is standard normal. In the next section, we propose the estimator for a smooth function of m and 2 hence for q and establish their asymptotic properties. 3. Estimation of extinction probability It is observed that the c.g.f. extinction probability, population drift, etc., of the process are functions of m and 2 . Therefore, a general procedure of estimation is proposed ÿrst and then, as a special case, the estimation of extinction probability is discussed when data are Z1 ; : : : ; Zn . It is assumed that the function ‘f’ is di erentiable up to order 2 and the ÿrst and second derivatives are continuous. By Taylor’s series expansion @f 2 2 f(mˆ n ; ˆn ) = f(m; ) + (mˆ n − m) @m m=mˆ n ;2 =ˆ2n @f + (ˆ2n − 2 ) 2 ˆ2n ;mˆ n @ " 2 2 1 2 2 @ f 2 2 @ f (mˆ n − m) + + (ˆn − ) 2 @m2 (m˜ n ;˜2 ) @4 ˜2 ;m˜ n n n # 2 f @ + 2(mˆ n − m)(ˆ2n − 2 ) ; (3.1) @m@2 m˜ n ;˜2 n
∗
∗
˜2n
ˆ2n
2
= + (1 − ) ; 0 ¡ ∗ ¡ 1; 0 ¡ ¡ 1. where m˜ n = mˆ n + (1 − )m; Hence, using consistency and asymptotic normality of mˆ n and ˆ2n established in √ Section 2, it follows immediately that on A, with the norming factor n, second and the fourth terms in (3.1) converge to zero in probability and hence, −2 ! √ @f L 2 2 4 n(f(mˆ n ; ˆn ) − f(m; )) → N 0; 2 as n → ∞: (3.2) @2 Now with z = 1 and t = 1 in 1.1, = loge m; = 2 exp(− ){exp( ) − 1}−1 and q = exp{−2 = }. Hence estimators of these are obtained by replacing m by mˆ n and 2 √ by ˆ2 . Asymptotic distributions are easily derived from (3.2). In particular, n(qˆn − q) converges in law to normal distribution with mean 0 and variance 2 2 =(2 2 q2 ). In summary, a discrete skeleton of the process produces a reasonably good estimator of the extinction probability. Acknowledgements Both the authors thank the referee for his suggestions and also thank Council for Scientiÿc and Industrial Research, New Delhi, India for ÿnancial support to carry out this work.
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