On the uniform strong consistency of an estimator of the offspring mean in a branching process with immigration

Statistics & Probability North-Holland

August

Letters 12 (1991) 151-155

1991

On the uniform strong consistency of an estimator of the offspring mean in a branching process with immigration T.N. Sriram Department

of Statistics,

Uniuersity of Georgia, Athens,

GA 30602, USA

Received May 1990 Revised June 1990

Abstract: For the critical and sub-critical shown to be strongly consistent uniformly

branching process with immigration, the natural estimator of the offspring mean ‘m’ is over a whole class of offspring distributions with m E (0, I] and bounded variance.

AMS

60580.

1980 Subject Classification:

Keywords:

Branching

process

Primary

with immigration,

uniform

strong

consistency,

martingale,

almost

supermartingale.

1. Introduction Suppose Z,, denote the n th generation size of a branching process with immigration. representation z, = zc’tH-I,,

+ Y,,

n=l,2

,..-,

We then have the

(1.1)

k=l

where &_ ,,k is the number of offspring of the k th individual belonging to the (n - l)th generation, and Y, denotes the number of immigrants in the n th generation. Throughout the paper, we will assume that (5,-l,k}, n=l,L..., k=L2 ,... , and { Y,, }, n = 1, 2,. . . , are two independent sequences of i.i.d. random variables and that the immigration process {Y,} is observable. The offspring distribution F and the immigration distribution are assumed to be unspecified with finite means m and A, and finite variances u2 and a;, respectively. The initial state Z, is a random variable (not depending on m) which is assumed to be independent of { 5,,, } and { I: }, and has an arbitrary distribution. In this paper, we confine ourselves to the sub-critical case (m < 1) and the critical case (m = 1). It is clear from (1.1) that a natural estimate of m is

In fact, for power-series offspring, and immigration distributions it can be shown that Gin and x,, = n-‘C:=,I: are the maximum likelihood estimates of m and A, respectively. Eventhough we will not make 0167-7152/91/$03.50

0 1991 - Elsevier Science Publishers

B.V. (North-Holland)

151

Volume 12,

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STATISTICS & PROBABILITY

LETTERS

August 1991

any specific distributional assumptions regarding 5 and Y in this paper, we will still consider &, in (1.2) as a reasonable estimate with which to work. The estimation problem for branching processes with immigration has been discussed extensively in the literature. Heyde and Seneta (1972, 1974) proposed an estimator of m based on the partial information on {Z,} alone and showed that it is strongly consistent for m, if m c 1. Later, Klimko and Nelson (1978) proposed an estimator of m based on conditional least squares method and established its strong consistency when m < 1. Recently, Wei and Winnicki (1990) have proposed an estimator of m based on weighted conditional least squares method and established its strong consistency (when m # 1) and weak consistency (when m = 1). Our estimate fiiz,, however, is based on the full information on both generation sizes {Z, } and the immigration process {Y}, i = 1, 2,. . . , n. In the literature, the estimation of m by A, has been considered by Nanthi (1979, 1983) and Venkataraman and Nanthi (1982). It is also shown in Nanthi (1979) that if m < 1, then A,, is strongly consistent for m. Suppose we define Fn to be the u-field generated by {Z,,, [i_i9i, Y:, 1 < i G n, j > l}, then it can be easily seen that {C:, ,( Z, - mZ, _ , - q), Fn } is a martingale. This and the strong law of large numbers for martingales (see Hall and Heyde, 1980, Theorem 2.18) yields that &, is strongly consistent for each fixed m E (0, 11. The purpose of this paper, however, is to show that the strong consistency of $r, holds uniformly over a whole class of offspring distributions with m E (0, 11 and bounded variance. See the theorem below for the definition of uniform strong consistency. The main result is stated as a theorem in Section 2 and is proved using two lemmas. The key idea here is to exploit the fact that the processes involved in &,, - m possess a common almost supermartingale property (see Robbins and Siegmund, 1971, for the definition) and then use a maximal inequality result for ‘almost supermartingales’ from Robbins and Siegmund (1971). The uniformity results are then established by exploiting the properties of the underlying process {Z,} in (1.1). Next, we state the main theorem.

2. Uniform

strong consistency

Let GO,= {F: EF([,,i) = m E (0, l] and Var,(t,,i) = u2 E (0, u:]} with 0 < u, -C cc Throughout the rest of the paper, we will write ‘sup P’ to mean sup{ PF, FE G,,}. Theorem.

Let A,, be as in (1.2). Then &,, is uniformly lim sup P{ IA, - m/>CSforsomen>k} kAcc

and

known.

strongly consistent for m: for all 6 > 0, =O.

(2.1)

The proof of the Theorem depends on two lemmas, the first of which is similar to Lemma 2.2 of Lai and Siegmund (1983). However, it is worth pointing out that the results here hold uniformly over a whole class of offspring distributions with m E (0, l] and bounded variance, where as in Lai and Siegmund (1983) the results hold when the distribution of error terms in AR(l) set up is fixed and only the autoregressive parameter varies. The proofs of Lemma A.1 and A.2 stated below are given in the appendix. Lemma A.l. c, + 00,

Assume

the model (1.1). For each y > +, S > 0 and increasing

lim sup P

k-m

i il

(Z,-mZ,_,

- I:)

ldi3

max(c,,,

(iiZ,_,j’j

sequence

forsomenrk)

of positive

=O.

constants

(2.2)

i=l

Lemma A.2. Assume the model (1.1). For p > 3, lim supP k+m

152

~(Zi_l+l)~Snpforsomen>k i

i=l

=O. I

(2.3)

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LETI-ERS

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Proof of the Theorem. Note that

IM 2

(Z,-mz,_, (G”

[

-m)2=

i=l e

=

i=l 5

-

1

~(z,-t?zz,-l-r;, i[

Since the conclusion have that

I:)

1

z,_,

I(

2,n&,+1)

1=l

of Lemma

2

[ II 2

e&z,-,+w

1=l

b-1

r=l

.

I=1

(2.4)

A.1 holds with E:=, (Z, _ , + 1) in place of Ey=, Z, _ ,, by Lemma

I

A.2 we

2

P

sup

2

(z,-mz,_,-

r,)

il i=l

i=l


>Sni

i(z,-mz,-,-y)

(Z,_l+l)

forsomen2k

1=1

I 1=1 2>Sni:(Z,-,+1),

e(Z,_,+l)<&t”

forsomen>,k

i=l


P


P

iI,&(z,-mz,-, I il

K) ~~81~‘~p[~l~Z,_~+l)]1+1’~forsome~~k~+o~~~ 2

i (Z,-mZ,_,i=l

y)

>8’-‘/p

max[rzl+‘jP,

[ il

(Z,_,

+ l)J+t”j

for some n >, k +O by Lemma

ask-co

+ o(l) (2.5)

A.l. Also, since Z, 2 y for all i > 1, by SLLN we have

n 2

(z,_,

+ l>/

i

[;zl

z-I

‘I;~(~i~~~y.l(l+(,,/~~~~))

r=l +

XP’(1 + X-‘)

almost surely as n + 00

uniformly over FE G,,. Here, the uniformity holds since the distribution Hence the required result follows from (2.4) (2.5) (2.6). q

(2.6)

of y does not depend

on F.

Appendix Proof of Lemma A.l. Let I, = C:= ,Z,_, show that

and C,, = c!,‘~. Since [F, + Z,,12’ 6 22y max(ci,

I

I,“),

it suffices

to

2

sup P

i

i[

r=l

(Z,-mZ,_,

-

q)

>S2[~n+Z,,]2y

forsome

nak

(A.11 153

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as k-t

LETTERS

August 1991

cc. Let

1 2

x,,=

5

[

i=l

(z,-mz,_,-

r,)

Then X,, is an almost supermartingale

/[F~,+ZJ2’,

5,

=

“2wt~n+,

+

L+,l”‘.

in the sense that

E{X,+,(e}
(A-2)

which satisfies condition (1) of Robbins and Siegmund (1971) Proposition 2 of Robbins and Siegmund (1971),

with {n in place of E,. Therefore,

by

But, since y > $,

u,-‘E : 5,~

[I,,, -

E f

n=k

GE

JI&m(Fk+, + x)-” m


/0 (

= (2yuniformly over

+ L+,l"'

Ll/[~k+,

n=k

dx

Fk+,+x)-*’ l)-rF;~:y-‘)+O

dx as k+

co

(A.4)

FE Go,. Also, k-l

EX,=E

<

c (Z,-mZ,_,,=l k-l

E

c

1 : 1 i

y,) *+o*Zk_,

(Z, - mZ,_,

-

y)

i=l

*,[?k + I,_,,”

/[&+zk]2’

+ Ema2Z,-,/[&

+ z,]”

a,$ Z,_,/[&+z,]*’ I=1

/os(Fk+x)-2Y

=sq 2

dx-+O

as k-,

co

(A-5)

uniformly over FE G,,. The conclusion (A.l) now follows from (A.3), (A.4) and (A.5). Proof

of Lemma

0

A.2. Since C:= 1(Z, _ , + 1) < Z, + C:=, (Z, + 1) and Z, does not depend on m, it suffices

to show that lim supP

k-m

154

i

i /=1

(Z,+1)>6nP

forsomenak

= 0.

(A-6)

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Once again note that ( n ~“I:= ,( Z, + l), %n } is an almost

which satisfies condition (1) of Robbins and Siegmund Robbins and Siegmund (1971) yields that

August

LETT’ERS

supermartingale,

1991

that is,

(1971). One more application

of Proposition

2 of

It is easy to see that EZ, = O(i) for all FE Go,. Hence

=O(kp(p-2))

uniformly

over FE G,,. Also, by Toeplitz

2 Z,/nP= fl=l since ZJn Therefore,

-+O

as k+

co,

(A.9)

Lemma

F (Z,/nP-‘-6)K(‘+s)< ?I=1

cc

a.s.

p-‘-6

-+ 0 a.s. for all 6 E (0, p - 3) by the Borel-Cantelli

E 5

Z,/nP=

n=k

5 n=k

EZ,,/nP

< K, E

n-(p-”

+ 0

lemma

and the fact that EZ, = O(i).

ask-+oo,

(A.10)

n=k

uniformly over FE Go,, where K, is a generic constant from (A.8) (A.9) and (A.lO). 0

independent

of F. The required

result now follows

References

Hall, P. and CC. Heyde (1980). Martingale Lmit Theory and irs Application (Academic Press, New York). Heyde, C.C. and E. Seneta (1972), Estimation theory for growth and immigration rates in a multiplicative process, J. Appl. Probab. 9, 235-256. Heyde, CC. and E. Seneta (1974). Notes on “Estimation theory for growth and immigration rates in a multiplicative process”, Appl. Probub. 11, 572-577. Klimko, L.A. and P.I. Nelson (1978). On conditional least squares estimation for stochastic processes, Ann. Stafist. 6, 629642. Lai, T.L. and D. Siegmund (1983), Fixed accuracy estimation of an autoregressive parameter, Ann. Srarisr. 11(2), 478-485. Nanthi, K. (1979). Some limit theorems of statistical relevance

on branching processes, Ph.D. Thesis, Univ. of Madras (Madras, Tamil Nadu). Nanthi, K. (1983). Statistical estimation for stochastic processes, Queen’s papers in Pure and Appl. Math. 62. Robbins, H.E. and D. Siegmund (1971). A convergence theorem for nonnegative almost supermartingales and some apphcations. in: J.S. Rustagi, ed., Optimizing Methods in Srarrsfrcs (Academic Press, New York) pp. 223-258. Venkataraman. K.N. and K. Nanthi (1982). A limit theorem on a subcritical GaltonWatson process with immigration, Ann. Probnb. 10, 1069-1074. Wei, C.Z. and J. Wmnicki (1990). Estimation of the means in the branching process with immigration, to appear in: Ann. Statist. 155