PROBABILITY
AND
MATHEMATICAL STATISTICS
Vol. 29, Fasc. 2 (2009), pp. 205–226
COMPOUND NEGATIVE BINOMIAL APPROXIMATIONS
FOR SUMS OF RANDOM VARIABLES
BY
P. V E L L A I S A M Y
AND
N. S. U PA D H Y E∗ (M UMBAI )
Abstract. The negative binomial approximations arise in telecommunications, network analysis and population genetics, while compound negative binomial approximations arise, for example, in insurance mathematics.
In this paper, we first discuss the approximation of the sum of independent, but not identically distributed, geometric (negative binomial) random
variables by a negative binomial distribution, using Kerstan’s method and
the method of exponents. The appropriate choices of the parameters of the
approximating distributions are also suggested. The rates of convergence
obtained here improve upon, under certain conditions, some of the known
results in the literature. The related Poisson convergence result is also studied. We then extend Kerstan’s method to the case of compound negative
binomial approximations and error bounds for the total variation metric are
obtained. The approximation by a suitable finite signed measure is also studied. Some interesting special cases are investigated in detail and a few examples are discussed as well.
2000 AMS Mathematics Subject Classification: Primary: 60E05;
Secondary: 62E17, 60F05.
Key words and phrases: Compound negative binomial approximation, compound Poisson approximation, finite signed measure, Kerstan’s
method, method of exponents, total variation distance.
1. INTRODUCTION
P
Let {Xi } be a sequence of discrete random variables (rv’s) and Sn = nj=1 Xj .
When the Xi are independent B(pi ) variables, it is known that (see Khintchine [15]
or Le Cam [16])
(1.1)
¡
¢
dT V Sn , P (λ) ¬
n
X
i=1
∗
p2i ,
The author thanks CSIR for the award of a research fellowship.
206
P. Vellai sa my and N . S. Up a d hy e
where, for any two rv’s X and Y ,
dT V (X, Y ) = sup |P (X ∈ A) − P (Y ∈ A)|
A
denotes the total
Pn variation metric and P (λ) denotes the Poisson variable with parameter λ = i=1 pi . Kerstan [14] improved the above bound to
n
n
¢ ¡X
¢
¡
¢
¡X
p2i /
pi
dT V Sn , P (λ) ¬ 1.05
i=1
i=1
when pi ¬ 1/4. Barbour and Hall [3] further improved the bound to
¡
¢
dT V Sn , P (λ) ¬
n
X
i=1
p2i min{1, λ−1 },
where λ−1 is known as the ‘magic factor’. See, also Barbour et al. [4] for more
details and related results. The negative binomial (NB) approximation to the sum
of indicator rv’s is studied by Brown and Phillips [5]. They showed that NB distribution arises also as the limiting distribution of Pólya distribution. Also, Brown
and Xia [6] considered the NB approximation to the number of two-runs.
In this
we first consider the problem of approximating the distribution
Ppaper,
n
of Sn = i=1 Xi , where Xi ’s are independent geometric Ge(pi ) variables, by an
N B(r, p) distribution. In Section 2, we obtain the upper bounds similar to (1.1),
using Kerstan’s method (Kerstan [14]) and generalize the results to the sum of
independent negative binomial random variables using the method of exponents
L
(Čekanavičius and Roos [8]–[10]). Also, the conditions under which Sn → P (λ)
are investigated. Kerstan’s method is originally due to Kerstan [14], which was
later modified and used by several authors, say, for example, Daley and Vere-Jones
([11], pp. 187–190), Witte [30] and Roos [19], [20], where the problems of Poisson,
multivariate Poisson and compound Poisson (CP) approximations are considered.
Recently, Barbour [2] obtained a bound, using Stein’s method, for multivariate
Poisson-binomial distribution. This bound is comparable to the one obtained by
Roos [19] using Kerstan’s method. Also, Roos [20] studied CP approximations for
the sums of independent discrete valued rv’s using Kerstan’s method. Čekanavičius
[7] considered approximation of compound distributions using Le Cam’s [16] operator theoretic method.
Note that the CP distribution plays an important role in risk theory. Consider
the model for the total claim amount in a portfolio of insurance policies. Let N
denote the number of claims arising from policies in a given period of time, and let
Xi denote the claimP
size amount of the i-th claim. In a collective risk model, the
random sum SN = N
i=1 Xi denotes the aggregate claim, where it is assumed that
Xi ’s are iid and N is independent of the Xi . In many applications, the number of
claims N is assumed to follow Poisson distribution, and thus the distribution of the
207
CNB approximations for sums of random variables
random sum SN is a CP distribution. However, there are many situations (see, for
example, Panjer and Willmot [18], Drekic and Willmot [12]) and, in particular, in
non-life insurance modeling, the CP distribution may not serve as a suitable model
for the insurance data. In such cases, and especially when V(N ) > E(N ), an NB
model is usually suggested. Thus, the study of the compound negative binomial
(CNB) distribution arises naturally, and hence the CNB approximation problems
are of importance. In Section 3, we use Kerstan’s method to obtain the first-order
result for the total variation distance. Also, we study the approximation of Sn by a
finite signed measure, which leads to the improvements in the constant that appears
in the first-order result. In Section 4, we apply our results to the distributions which
can be written as infinite mixtures and obtain results analogous to the ones given in
Section 3. Finally, an application of our results to life and health insurance is also
pointed out.
2. NEGATIVE BINOMIAL AND POISSON APPROXIMATIONS
Throughout the paper, Z+ = {0, 1, . . .} denotes the set of non-negative integers. Let Xi ∼ Ge(pi ), 1 ¬ i ¬ n, be independent geometric rv’s with
P (Xi = k) = pi (1 − pi )k
for k ∈ Z+ , 0 < pi < 1.
We are interested in approximating the distribution of the sum Sn =
an N B(r, p) rv N with
µ
¶
r+k−1 r
P (N = k) =
p (1 − p)k for k ∈ Z+ ,
k
Pn
j=1 Xj
to
where r > 0 and 0 < p < 1, and also choosing the appropriate parameters r and
p so that the error is minimum. We adopt Kerstan’s method (see Roos [19]) and
the method of exponents (see Čekanavičius and Roos [8]–[10]) for finding upper
bounds for dT V (Sn , N ).
P∞
m
2.1. Kerstan’s method. For the power series
P∞ f (z) = m=0 am z , where
am ∈ R and z ∈ C, define the norm kf (z)k = m=0 |am |. Then, it is well known
that kf1 (z)f2 (z)k ¬ kf1 (z)kkf2 (z)k. For a Z+ -valued rv X, the probability generating function (pgf)
E(z X ) =
∞
X
z m P (X = m),
m=0
where z ∈ C and |z| ¬ 1, exists. Then, for X ­ 0 and Y ­ 0 (see Roos [19]),
(2.1)
1
dT V (X, Y ) = kE(z X ) − E(z Y )k.
2
We now obtain the norm estimates for γ(z) = E(z Sn ) − E(z N ) for the case r = n.
208
P. Vellai sa my and N . S. Up a d hy e
Note that
(2.2)
(2.3)
(2.4)
where
¶n
p
γ(z) =
−
1 − qz
i=1
µ
¶n
n
£Q
¤
p
=
(1 + Li ) − 1
1 − qz
i=1
·
µ
¶n/j ¸
n
j
Q
X
X
p
=
Lis
,
1 − qz
j=1 1¬i1 <...<ij ¬n s=1
n
Q
µ
pi
1 − qi z
¶
µ
pi (1 − qz)
Li =
−1=
p(1 − qi z)
µ
¶
pi
(z − 1)
1−
.
p (1 − qi z)
Therefore, using Lemma 3.1 of Čekanavičius and Roos [8] and k1/(1 − qi z)k =
1/pi , we get
¯
¯°
° µ
¶n/j °
¶°
µ ∞ m
°
¯
¯°
°
°
X q
p
m
° ¬ 1 ¯1 − pi ¯ °(z − 1) exp n
°Li
(z − 1) °
°
¯
¯
°
°
°
1 − qz
pi
p
j m=1 m
s
¯
¯
¯
¯
¾
½
¡
¢°
1 ¯¯
2j
pi ¯¯ °
pi ¯¯
1 ¯¯
°
°
¬ ¯1 − ¯ (z − 1) exp (nq/j)(z − 1) ¬ ¯1 − ¯ min 2,
,
pi
p
pi
p
nqe
since
°
©
kh(z)k = ° exp (n/j)
P∞
∞
X
ª°
q m (z m − 1)/m ° = 1.
m=2
For, if λm = nq m /(jm), λ = m=2 λm and Q({m}) = λm /λ, then h(z) =
E(z X ), where X ∼ CP (λ, Q). Hence, from (2.4) we obtain
°
¶n/j °
µ
n
j °
°
Q
X
X
p
°
°
(2.5)
kγ(z)k ¬
°Lis 1 − qz
°
s=1
j=1 1¬i1 <i2 <...<ij ¬n
· n ° µ
¶n/j °¸j
n
°
°
X
1 X
p
°Li
°
¬
°
°
1 − qz
j=1 j! i=1
¯
¯
· n
½ s
¾¸j
n
X
1 ¯¯
pi ¯¯
2j
1 X
¬
¯1 − p ¯ min 2, nqe
j=1 j! i=1 pi
√
n
X
( jαn )j
,
¬
j!
j=1
where
(2.6)
¯
¯
½ r
¾
pi ¯¯
1 ¯¯
2
αn =
¯1 − p ¯ min 2, nqe .
i=1 pi
n
X
Thus, from (2.1)–(2.5), we have the following result.
209
CNB approximations for sums of random variables
T HEOREM 2.1. LetP
Xi , 1 ¬ i ¬ n, be a sequence of independent geometric
Ge(pi ) variables, Sn = nj=1 Xj , and N ∼ N B(n, p). Then
(2.7)
dT V (Sn , N ) ¬ min{1.37αn , 1},
and, when αn < e,
αn /e
dT V (Sn , N ) ¬ √
,
2 2π(1 − αn /e)
(2.8)
where αn is given by (2.6).
The first estimate (referred to as a practical estimate), given in (2.7), follows
from the fact that
¾
½
αn
,1 ,
dT V (Sn , N ) ¬ min{f (αn ), 1} ¬ min
x0
P
√ j
where f (x) = 12 ∞
j=1 (x j) /j! and, numerically, it can be seen that 0.73 <
x0 < 0.74 is the unique solution of f (x) = 1. The later estimate follows by an application of Stirling’s approximation formula and is less than one when αn < 2.26.
R EMARK 2.1. (i) If pi = p, then it follows from (2.7) that dT V (Sn , N ) = 0,
which holds iff Sn ∼ N B(n, p), as expected.
(ii) Observe that the bound given in (2.7) contains a ‘magic factor’ (nq)−1/2
which improves theP
estimate for large n.
(iii) Let µn = ni=1 (qi /pi ). One way to choose p is such that E(Sn ) = E(N ),
which leads to the choice p = n/(n + µn ) and the upper bound for this case can
be obtained from the practical estimate given in (2.7). The other choices of p for
better accuracy are p = mini pi and p = maxi pi .
Next, we obtain some improvements over the above results along with generalizations for the sum of independent NB variables.
2.2. The method of exponents.
In this subsection, we obtain an N B(r, p) apP
proximation result for Sn = nj=1 Xj , where Xj ∼ N B(αj , pj ), using the method
¢
¡ Pn
P
of exponents. Here, we assume that r = α = ni=1 αi and p =
i=1 αi pi /α
and write the pgf of Sn as
(2.9)
E(z
Sn
)=
n
Q
µ
j=1
pj
1 − qj z
µ
= exp
¶αj
µ
= exp
n
X
j=1
µ
αj ln
pj
1 − qj z
¶¶
¶
n
1 X
αj qjm (z m − 1) := exp(F ).
m=1 m j=1
∞
X
210
P. Vellai sa my and N . S. Up a d hy e
Similarly,
(2.10)
Now
µ
E(z ) = exp α
N
∞
X
qm m
(z − 1)
m=1 m
¶
:= exp(A).
°
R1 ³
¡
¢´0 °
°
°
k exp(F ) − exp(A)k = ° exp(A)
exp x(F − A) dx°
0
¬
R1 °
¡
¢°
°(F − A) exp xF + (1 − x)A °dx,
0
(0 )
where the prime denotes the derivative with respect to x. Substituting the power
series expansion for F and A from (2.9) and (2.10), respectively, we obtain, for
0 < x < 1,
xF + (1 − x)A = αq(z − 1) + T,
where
T =
∞
X
n
¢
1¡X
xαj qjm + (1 − x)αq m (z m − 1).
m=2 m j=1
Also, observe that exp(T ) forms a compound distribution, as all the multipliers of
(z m − 1) in T are non-negative, and hence k exp(T )k = 1. Therefore,
°
°
¡
¢°
° exp(F ) − exp(A)k ¬ °(F − A) exp αq(z − 1) °
n
∞
¢°
¡
¢°
X
1¡X
αj qjm − αq m °(z m − 1) exp αq(z − 1) °
¬
m=2 m j=1
¶
µ n
2
¡
¢°
X αj qj
αq 2 °
°(z − 1) exp αq(z − 1) °
−
¬
p
j=1 pj
¶
½ r
¾
µ n
2
X αj qj
αq 2
2
−
min 2,
,
¬
p
αqe
j=1 pj
where
in the second inequality is implied by the fact that αq m
Pn the non-negativity
¬ j=1 αj qjm , which in turn follows from a simple application of Jensen’s inequality. Also, the last inequality follows from Lemma 3.1 of Čekanavičius and
Roos [8]. Thus, we obtain the following result from (2.1).
T HEOREM 2.2. For αi > 0, let Xi ∼P
N B(αi , pi ), 1 ¬ i ¬ n, be a sequence
n
of independent
random
variables,
S
=
n
i=1 Xi , and N ∼ N B(α, p), where
Pn
Pn
α = i=1 αi and p = i=1 αi pi /α. Then
µ n
¶
½
¾
2
X αj qj
αq 2
1
(2.11)
dT V (Sn , N ) ¬
−
min 1, √
.
p
2αqe
j=1 pj
CNB approximations for sums of random variables
211
R EMARK 2.2. Observe that the bound in (2.11), obtained using the method of
exponents, also involves the ‘magic factor’ (αq)−1/2 and is comparable to the one
given in (2.7) obtained by Kerstan’s method.
2.3. NB to Poisson approximation. Let N ∼ N B(r, p), where r > 0 and
0 < p < 1, and Y ∼ P (λ). Here, we follow the approach of expansion in the exponents. Write the pgf of N (see (2.10)) as
µ
E(z ) = exp r
N
∞
X
qm m
(z − 1)
m=1 m
¶
= exp(A).
¡
¢
Also, the pgf of Y is exp λ(z − 1) := exp(B). Then, as seen earlier,
k exp(A) − exp(B)k ¬
R1 °
¡
¢°
°(A − B) exp xA + (1 − x)B °dx.
0
Moreover,
xA + (1 − x)B = rq(z − 1) +
∞
X
¢
1¡ m m
xq (z − 1) := rq(z − 1) + M.
m=2 m
Then k exp(M )k = 1, as all the multipliers of (z m − 1) are non-negative. Letting
rq = λ and following the arguments similar to the derivation of Theorem 2.2, we
obtain
T HEOREM 2.3. Let N ∼ N B(r, p) and Y ∼ P (λ), where λ = rq and q =
1 − p. Then
(2.12)
½
¾
rq 2
1
dT V (N, Y ) ¬
min 1, √
.
p
2rqe
R EMARK 2.3. (i) The bound given above contains the ‘magic factor’ λ−1/2
which reduces the bound considerably when λ = rq is large.
(ii) The best available bound in the literature for this case is given by Roos
[22] (see also Roos [21]), namely,
½
¾
3p
rq
(2.13)
,1 .
dT V (N, Y ) ¬ 2 min
p
4rqe
It
seen
that our bound in (2.12) is better than the above one, when r ¬
¡ can be
¢
3/(4q 2 e) min{1, 3/(2q)}.
Finally, we give below the rate of convergence result for Poisson approximation to Sn , obtained using again the method of exponents.
212
P. Vellai sa my and N . S. Up a d hy e
T HEOREM 2.4. P
Let Xi ∼ N B(αi , pi ), 1 ¬ i ¬ n, be independent random
variables, and Sn = ni=1 Xi . Then
¡
¢
dT V Sn , P (λ) ¬
(2.14)
where λ =
Pn
i=1 αi qi
½
¾
αj qj2
1
min 1, √
,
2λe
j=1 pj
n
X
= αq.
R EMARK 2.4. (i) It is interesting to note that the bound in (2.14) is nothing
but the sum of the bounds in (2.11) and (2.12).
(ii) A comparison of bounds (2.11) and (2.14) shows that an NB approximation is better than Poisson approximation in the case of sum of independent
N B(αi , pi ) variables. This motivates our study of approximation by compound
distributions and, in particular, to CNB distribution in the next section.
of Ge(pni ) variables, and Sn =
Pn C OROLLARY 2.1. Let {Xni } be a sequence
Pn
i=1 Xni .
If max1¬i¬n qni → 0 so that
i=1 qni
→ λ > 0 as n → ∞, then
L
Sn → P (λ).
The corollary follows from (2.14) and the fact that
2
n
qnj
X
qnj
¬ lim ( max qnj ) lim
= 0.
n→∞
n→∞ 1¬j¬n
n→∞
j=1 pnj
j=1 pnj
0 ¬ lim
n
X
The above result is essentially due to Wang [29].
3. APPROXIMATION BY COMPOUND DISTRIBUTIONS
In this section, we study CNB approximation to Sn , where Sn =
and Xi ∼ Fi , a discrete real-valued distribution.
Pn
i=1 Xi ,
3.1. Preliminary results. Let µ be a finite signed measure on (R, B). A measurable set B is said to be a positive set with respect to µ, denoted by B ­ 0, if
µ(A ∩ B) ­ 0 for every A ∈ B. Similarly, a set C is called a negative set with
respect to µ, denoted by C ¬ 0, if µ(C ∩ A) ¬ 0 for every A ∈ B. Also, a pair
(B, C) is said to be the Hahn decomposition of R if B ∪ C = R, where B ­ 0 and
C ¬ 0. Also, the total variation norm of µ is defined (see, for example, Aliprantis
and Burkinshaw [1]; Rudin [23])) as
(3.1)
kµk = |µ|(R) = µ+ (R) + µ− (R) = µ(B) − µ(C)
= 2µ(B) − µ(R),
CNB approximations for sums of random variables
213
where µ+ and µ− are positive and negative variations of µ. It is well known that
the total variation distance dT V between the distribution of two discrete rv’s X and
Y is
(3.2)
dT V (X, Y ) = sup |P (X ∈ A) − P (Y ∈ A)|
A
= P (X ∈ D) − P (Y ∈ D)
(3.3)
= (PX − PY )(D) := µX,Y (D),
©
ª
where D = {m : P (X = m) ­ P (Y = m)} = m : µX,Y {m} ­ 0 is a positive set of µX,Y (Wang [28] or Vellaisamy and Chaudhuri [25]). Also, by (3.1), we
have
(3.4)
kµX,Y k = 2µX,Y (D).
Thus, from (3.3) and (3.4) we obtain the relation
1
dT V (X, Y ) = µX,Y (D) = kµX,Y k,
2
where D is a positive set of µX,Y .
3.2. CNB and CP approximations. Let Yi be iid real-valued rv’s with distribution Q, N ∼ N B(r, p), and M ∼ P (λ), λ > 0. Assume that the Yi , N and M
P
are independent. Then the distribution of TN = N
j=1 Yj is the CNB distribution
with parameters r, p and Q, and is denoted by CN B(r, p, Q). Since
P (TN ∈ A) =
∞
X
P (N = k)Q∗k (A),
k=0
we get
(3.5)
µ
CN B(r, p, Q) =
pδ0
δ0 − qQ
¶r
,
where q = 1 − p, Q∗k denotes the k-fold convolution of Q, and δ0 is the Dirac
measure at 0. Similarly, the distribution of TM , denoted by CP
¡ (λ, Q), is ¢the CP
distribution with parameters λ and Q, and its distribution is exp λ(Q − δ0 ) . First,
we obtain the error bounds in the approximation of CN B(r, p, Q) to CP (λ, Q).
T HEOREM 3.1. Let r > 0, Q be any distribution on R and rq = λ. Then
°
°µ
¶r
½ r ¾
°
¡
¢° rq 2
pδ0
2
°
°
− exp λ(Q − δ0 ) ° ¬
min 2,
.
sup °
δ0 − qQ
p
λe
Q
214
P. Vellai sa my and N . S. Up a d hy e
P r o o f. First note that for every Q
°µ
°
¶r
°
¡
¢°
pδ
0
°
°
° δ0 − qQ − exp λ(Q − δ0 ) °
¯
¯
¶
∞ ¯µ
X
r + m − 1 r m e−λ λm ¯¯
m
¯
¬
p q −
¯ kQk
¯
m!
m
m=0
°
°
¶r
µ
°
¡
¢°
pδ0
°
=°
− exp λ(δ1 − δ0 ) °
° = 2dT V (N, Y ),
δ0 − qδ1
and now the result follows from Theorem 2.3.
¥
R EMARK 3.1. The above result follows easily also from Theorem 2.3 and
Lemma 3.1 of Vellaisamy and Chaudhuri [24].
Next we consider the approximation of finite sum Sn =
PNj
Pn
j=1 Zj ,
where
Zj = i=1 Xi , and Nj ∼ N B(αj , pj ), to the distributions
CN B(n,
¡
¢α p, Q) and
CP (λ, Q). Note that Zj ∼ CN B(αj , pj , Q) = pj δ0 /(δ0 − qj Q) j , the compound negative binomial distribution with parameters αj > 0, pj and Q.
P
T HEOREM 3.2. Let Sn = nj=1 Zj , where Zj ’s are independent with distributions CN B(αj , pj , Q). Then, for any distribution Q on R, we have
°n µ
¶αi µ
¶α °
°Q
°
pi δ 0
pδ0
°
(3.6) sup °
−
°
δ0 − qQ °
Q
i=1 δ0 − qi Q
¶
½ r
¾
µ n
X αi q 2
αq 2
2
i
−
min 2,
;
¬
p
αqe
i=1 pi
°n µ
°
¶αi
°Q
¡
¢°
pi δ 0
°
(3.7) sup °
− exp λ(Q − δ0 ) °
°
Q
i=1 δ0 − qi Q
½ r ¾
n
X
2
αi qi2
¬
min 2,
,
λe
i=1 pi
where α =
Pn
i=1 αi ,
p=
Pn
i=1 αi pi /α,
q = 1 − p and λ = αq.
P r o o f. The bound in (3.6) follows from Theorem 2.2 and the fact that for
every Q
°n µ
¶αi µ
¶α ° ° n µ
¶αi µ
¶α °
°Q
° °Q
°
p
δ
p
δ
pδ
pδ
i
0
i
0
0
0
°
°¬°
°.
−
−
°
°
°
δ0 − qQ
δ0 − qδ1 °
i=1 δ0 − qi δ1
i=1 δ0 − qi Q
Similarly, the bound in (3.7) follows from Theorem 2.4 and for every Q
°n µ
° °n µ
°
¶αi
¶αi
°Q
¡
¢° ° Q
¡
¢°
p
δ
p
δ
i
0
i
0
°
°
− exp λ(Q − δ0 ) °
− exp λ(δ1 − δ0 ) °
°
°¬°
°.
i=1 δ0 − qi Q
i=1 δ0 − qi δ1
Then the proof is completed.
¥
CNB approximations for sums of random variables
215
R EMARK 3.2. A comparison of the bounds given in (3.6) and (3.7) shows that
CN B approximations may be preferred over CP approximations.
3.3. CNB approximation to Sn by Kerstan’s method. In this section, we consider the sum Sn of n independent rv’s X1 , X2 , . . . , Xn taking values in R. Also,
let pi = P (Xi 6= 0), qi = P (Xi = 0) and Qi (·) = P (Xi ∈ · | Xi 6= 0) denote the
conditional probability measures. Then, for any Borel measurable set A ⊂ R,
(3.8)
P (Xi ∈ A) = qi P (Xi ∈ A | Xi = 0) + pi Qi (A) = (qi δ0 + pi Qi )(A),
and hence the distribution of Sn is
(3.9)
L(Sn ) = L(X1 ) ∗ L(X2 ) ∗ . . . ∗ L(Xn ) =
n ¡
Q
¢
δ0 + pj (Qj − δ0 ) .
j=1
R EMARK 3.3. Note that for the representation in (3.9), it suffices that the Xi
are independent of Si−1 for every 1 ¬ i ¬ n. However, the independence of Xi
and Si−1 , 1 ¬ i ¬ n, does not imply independence of the Xi (see Example 2.1
of Vellaisamy and Upadhye [27]). We refer to such models as previous-sum independent models. Recently, Vellaisamy and Sankar [26] have used such models for
modeling dependent production processes.
Our aim in this section is to approximate the distribution L(Sn ) to a suitable
CN B(n, p, Q) = L(TN ). We choose the parameter p and the distribution Q such
that E(Sn ) = E(TN ) which may possibly reduce dT V (Sn , TN ). Let now
(3.10)
Q=
1
Xn
n
X
p
i=1 i i=1
pi Qi
be the probability distribution of Y1 so that Q is a finite mixture of Q1 , Q2 , . . . , Qn ,
where Qj ’s are as given in (3.8). Since
E(Xi ) =
R
xd(qi δ0 + pi Qi ) =
R
we have
E(Y1 ) =
R
xd(pi Qi ),
R
R
n
¡X
¢
E(Sn )
xd
pi Qi = Xn
.
p
p
i=1
i=1 i R
i=1 i
1
Xn
This leads to
E(TN ) =
E(N )E(Sn )
Xn
.
p
i=1 i
Hence,
E(Sn ) = E(TN ) ⇐⇒ E(N ) =
n
X
i=1
pi ⇐⇒
n
X
nq
=
pi .
p
i=1
216
P. Vellai sa my and N . S. Up a d hy e
Solving for p, we get
(3.11)
p=
n
n+
Xn
p
i=1 i
.
Henceforth, all products represent the convolutions. Substituting (3.11) and (3.10)
in (3.5), and using (3.9), we obtain
(3.12) L(Sn ) − CN B(n, p, Q)
µ
n ¡
¢
Q
=
δ0 + pj (Qj − δ0 ) −
j=1
=
=
n ¡
Q
¢
δ0 + pj (Qj − δ0 ) −
j=1
n
¡Q
µ
pδ0
δ0 − qQ
¶n
nδ0
Xn
nδ0 − i=1 pi (Qi
¶n
− δ0 )
¢
(δ0 + Lj ) − δ0 CN B(n, p, Q)
j=1
=
n
X
X
j ¡
Q
j=1 1¬i1 <...<ij ¬n s=1
where
(3.13)
¢
Lis CN B(n/j, p, Q) ,
µ
¡
¢
Li = δ0 + pi (Qi − δ0 ) δ0 −
¶
pj
(Qj − δ0 ) − δ0
j=1 n
n
X
n
¡
¢X
pj
= pi (Qi − δ0 ) − δ0 + pi (Qi − δ0 )
(Qj − δ0 ).
j=1 n
Therefore,
(3.14) kL(Sn ) − CN B(n, p, Q)k ¬
n
¢j
1¡X
kLi CN B(n/j, p, Q)k ,
j=1 j! i=1
n
X
where Li is as defined in (3.13).
3.4. Norm estimates.
L EMMA 3.1. Let Q and p be defined in (3.10) and (3.11), respectively. Then
for any r > 0
µ
¶1∧r
pi
k(Qi − δ0 )CN B(r, p, Q)k ¬ 2 1 −
(i)
,
n+λ
µ µ
¶1∧r n
µ
¶1∧r ¶
X pj
pj
pi
(ii) kLi CN B(r, p, Q)k ¬ 2 pi 1 −
+
1−
,
n+λ
n+λ
j=1 n
where λ =
Pn
i=1 pi
and q = 1 − p.
217
CNB approximations for sums of random variables
P
P r o o f. (i) Let R1 = (pi /λ)Qi , and R2 = nj=1, j6=i pj Qj /λ, so that Q =
R1 + R2 . Note first that
µ
¶r
pδ0
δ0 − qR1
(Qi − δ0 )CN B(r, p, Q) = (Qi − δ0 )
δ0 − qR1 δ0 − q(R1 + R2 )
which leads to
(3.15)
°
µ
°
k(Qi − δ0 )CN B(r, p, Q)k ¬ °
(Q
−
δ
)
0
° i
pδ0
δ0 − qR1
¶r °
°
°kRk,
°
¡
¢r
where R = (δ0 − qR1 )/(δ0 − qQ) . Now
°µ
¶r °
°
°
δ0 − qR1
°
kRk = °
° δ0 − q(R1 + R2 ) °
°µ
¶r °
°
°
δ0
°
°
=°
δ0 − qR2 /(δ0 − qR1 ) °
° ∞ µ
¶m °
¶µ
° X r+m−1
°
qR2
°
°
=°
°
δ0 − qR1
m
m=0
µ
¶
µ
¶
∞
∞
X
r+m−1 m
m X m+s−1
¬
q kR2 k
q s kR1 ks
m
s
m=0
s=0
¶ ∞ µ
¶ µ ¶
¶ µ
∞ µ
X
m + s − 1 s pi s
r + m − 1 m λ − pi m X
q
¬
q
λ
λ
s
m
s=0
m=0
¶m µ
¶m
µ
¶ µ
∞
X
λ
r + m − 1 m λ − pi
=
q
λ
λ − qpi
m
m=0
¶r
µ
λ − qpi
=
.
λp
Substituting the values of p and q, we get
µ
¶
λ − pi r
(3.16)
.
kRk ¬ 1 +
n
Consider next
°
µ
°
°(Qi − δ0 )
°
¶r °
°
pδ0
°
δ0 − qR1 °
°
°
¶
∞ µ
°
°
X
r+k−1 r
k°
°
p (qR1 ) °
= °(Qi − δ0 )
k
k=0
°
µ
¶ ³
´k °
∞
°
°
X r+k−1
r qpi
°
p
= °(Qi − δ0 )
Qki °
°
k
λ
° ∞ µ k=0
°
¶
µ
¶
∞
°
X
r + k − 1 k k°
r° X r + k − 1
k k+1
=p °
µ Qi −
µ Qi °
°,
k
k
k=0
k=0
218
P. Vellai sa my and N . S. Up a d hy e
where µ = qpi /λ. Now, putting k + 1 = m in the first summation and k = m in
the second summation, we obtain
°
¶r °
µ
°
°
pδ0
°
°
(3.17) °(Qi − δ0 )
δ0 − qR1 °
° ∞ µµ
°
¶
µ
¶ ¶
°
°
r + m − 2 m−1
r+m−1 m
r° X
m
=p °
µ
−
µ
Q i − δ0 °
°
m−1
m
m=1
¯
¯
µ
µ
¶
µ
¶
¶
∞ ¯
X
r + m − 2 m−1
r + m − 1 m ¯¯
r
m
¯
¬p 1+
µ
−
µ ¯ kQi k
¯
m−1
m
m=1
¯
¯¶
µ
¶
∞ µ
X
r + m − 2 m−1 ¯¯ m(1 − µ) − (r − 1)µ ¯¯
r
=p 1+
µ
¯ .
¯
m−1
m
m=1
When r ­ 1, we get from (3.17)
(3.18)
°
¶r °
µ
°
°
pδ
0
°(Qi − δ0 )
°
°
δ0 − qR1 °
¶¶
µ
¶
µ
∞ µ
X
r + m − 2 m−1
µ
r
¬p 1+
µ
(1 − µ) + (r − 1)
m
m−1
m=1
r
2p
=
.
(1 − µ)r−1
Similarly, when 0 < r < 1, we obtain
°
¶r °
µ
°
°
pδ
0
°(Qi − δ0 )
° ¬ 2pr .
(3.19)
°
δ0 − qR1 °
Substituting the values of p and µ in (3.18) and (3.19), we finally get
µ
k(Qi − δ0 )CN B(r, p, Q)k ¬ 2 1 −
pi
n+λ
¶1∧r
,
which proves part (i).
(ii) Observe that, from (3.13),
kLi CN B(r, p, Q)k ¬ pi k(Qi − δ0 )CN B(r, p, Q)k
°
°
n
°¡
°
¢X
pj
°
+°
δ
+
p
(Q
−
δ
)
(Q
−
δ
)CN
B(r,
p,
Q)
i
i
0
j
0
° 0
°
j=1 n
µ µ
¶1∧r
µ
¶1∧r ¶
n
X
pi
pk
pk
¬ 2 pi 1 −
+
1−
,
n+λ
n+λ
k=1 n
using (3.8) and part (i). Hence, the lemma follows.
¥
219
CNB approximations for sums of random variables
3.5. The ¡first-order result. We
¢ obtain the error bounds on the total variation
distance dT V Sn , CN B(n, p, Q) for the choices of p and Q discussed in Subsection 3.4.
T HEOREM 3.3. Let Xi , 1 ¬ i ¬ n, be independent real-valued rv’s with pi =
P (Xi 6= 0), and Qi (·) = P (Xi ∈ · | Xi 6= 0). Also, let
λ=
n
X
pi ,
n
1 X
pi Qi
λ i=1
Q=
i=1
and
p=
n
.
n+λ
Then
¡
¢
dT V Sn , CN B(n, p, Q) ¬ min
(3.20)
½ µ n j¶ ¾
1 X βn
,1
2 j=1 j!
(3.21)
¬ min {0.911βn , 1},
¡
¢
P
where λ2 = ni=1 p2i , and βn = 4 λ − λ2 /(n + λ) .
P r o o f. Using part (ii) of Lemma 3.1, we obtain
(3.22)
µ
¬2
n
X
i=1
n
X
kLi CN B(n/j, p, Q)k
µ
pi 1 −
i=1
pi
n+λ
¶
+
n
X
µ
pk 1 −
k=1
pk
n+λ
¶¶
µ
=4 λ−
λ2
n+λ
¶
= βn .
Using (3.14) and (3.22), we get
(3.23)
kL(Sn ) − CN B(n, p, Q)k ¬
¬
n
¢j
1¡X
kLi CN B(n/j, p, Q)k
j=1 j! i=1
n
X
βnj
,
j=1 j!
n
X
and hence (3.20) follows.
From part (i) of Lemma 3.1 we infer that
¡
¢
dT V Sn , CN B(n, p, Q) ¬ min{f (βn ), 1},
where
f (x) =
∞
1 X
ex − 1
xj
=
.
2 j=1 j!
2
Let x0 = ln(3). Then f (x) is increasing, f (x0 ) = 1, and f (x) ¬ x/x0 for x ∈
(0, x0 ). Hence, min{f (x), 1} ¬ min{x/x0 , 1}, and so (3.21) follows. ¥
220
P. Vellai sa my and N . S. Up a d hy e
3.6. Approximation by a finite signed measure. We consider here the approximation of the distribution of the sum Sn by a finite signed measure defined by
¡
W := δ0 −
(3.24)
n
X
¢
pi (Qi − δ0 ) CN B(n, p, Q),
i=1
which is a variant of CN B(n, p, Q) and has the property that W (R) = 1. The
choice of this measure is motivated by the expansion of Li , defined in (3.13), and
to remove the first term in the expansion of L(Sn ) − CN B(n, p, Q).
T HEOREM 3.4. Let the assumptions of Theorem 3.3 hold, and W be as defined in (3.24). Then
(3.25)
dT V (Sn , W ) ¬ min
(3.26)
½ µ
¶ ¾
n
X
1 βn
βnj
+
,1
2 2
j=2 j!
¬ min{0.775βn , 1}.
P r o o f. Consider first
(3.27) kL(Sn ) − W k
°
= °L(Sn ) − CN B(n, p, Q) −
n
X
°
pi (Qi − δ0 )CN B(n, p, Q)°
i=1
n
°X
=°
j ¡
Q
X
j=1 1¬i1 <...<ij ¬n s=1
−
n
X
¢
Lis CN B(n/j, p, Q)
°
pj (Qj − δ0 )CN B(n, p, Q)°,
j=1
using (3.12). Writing the term corresponding to j = 1 separately, we get
(3.28)
n ¡
°X
¢
kL(Sn ) − W k = °
Li − pi (Qi − δ0 ) CN B(n, p, Q)
+
i=1
n
X
X
j ¡
Q
j=2 1¬i1 <...<ij ¬n s=1
¬
n
X
°¡
°
¢
° Li − pi (Qi − δ0 ) CN B(n, p, Q)°
i=1
n
X
+
¢°
Lis CN B(n/j, p, Q) °
n
¢j
1¡X
kLi CN B(n/j, p, Q)k .
j=2 j! i=1
221
CNB approximations for sums of random variables
Since
n
¡
¢X
pj
Li − pi (Qi − δ0 ) = − δ0 + pi (Qi − δ0 )
(Qj − δ0 ),
j=1 n
using Lemma 3.1 (i), we get
°¡
°
¢
° Li − pi (Qi − δ0 ) CN B(n, p, Q)° ¬
n
X
pj
k(Qj − δ0 )CN B(n, p, Q)k
j=1 n
¶
n µ
p2j
2 X
¬
pj −
.
n j=1
n+λ
Hence,
µ
n ¡
°X
°
¢
°
Li − pi (Qi − δ0 ) CN B(n, p, Q)° ¬ 2 λ +
i=1
λ2
n+λ
¶
=
βn
.
2
Therefore,
(3.29)
kL(Sn ) − W k ¬
n
X
βn
βnj
+
,
2
j=2 j!
where βn is as defined in (3.22). Since W (R) = 1 and kL(Sn ) − W k ¬ 2, the
result in (3.25) follows.
To prove (3.26), note that
dT V (Sn , W ) ¬ min{g(βn ), 1},
where g(x) =
∞
x 1 X
xj
+
.
4 2 j=2 j!
Note also that min{g(βn ), 1} ¬ βn /x0 , where x0 ∈ (0, ∞) is the unique solution
of g(x) = 1. Numerically, it can be seen that 1.29 < x0 < 1.3. Therefore,
¡
¢
dT V L(Sn ), W ¬ min{0.775βn , 1}.
This proves the theorem.
¥
R EMARK 3.4. Comparing the practical estimates (3.21) and (3.26), we note
that the approximation by a finite signed measure improves the constant of approximation.
4. SOME SPECIAL CASES
In this section, let Sn be defined as in (3.9) and we assume that
P for every i,
1 ¬ i ¬ n, there exists a probability distribution {qi,j } on N (i.e., ∞
j=1 qi,j = 1)
222
P. Vellai sa my and N . S. Up a d hy e
so that Qi is a mixture of {Uj }, a sequence of probability measures concentrated
on R\{0}. That is,
(4.1)
Qi =
∞
X
qi,j Uj .
j=1
For instance, qi,j = δi,j , the Kronecker delta, and Uj = Qj corresponds to the
trivial case. Another example due to Roos [20] is the following:
Let {Bj }j­1 be a partition of R\{0}. Assume P (Xi ∈ · | Xi ∈ Bj ) = Uj is
the same for all Xi , 1 ¬ i ¬ n. Then
(4.2)
Qi (·) =
∞
X
P (Xi ∈ Bj | Xi 6= 0)P (Xi ∈ · | Xi ∈ Bj ) =
j=1
∞
X
qi,j Uj ,
j=1
where qi,j = P (Xi ∈ Bj | Xi 6= 0).
We now require the following lemma.
P
L
4.1. Let Q = ni=1 pi Qi /λ, where Qi is of the form in (4.1), and
PEMMA
λ = ni=1 pi . Then for any r > 0 we have
(4.3)
k(Ul − δ0 )CN B(r, p, Q)k ¬ 2(1 − qql )1∧r ,
(4.4)
∞
X
kLi CN B(r, p, Q)k ¬ 2pi
qi,l (1 − qql )1∧r
l=1
n
∞
X
pj X
+2
j=1
where ql =
Pn
i=1 pi qi,l /λ
n
qj,m (1 − qqm )1∧r ,
m=1
and q = λ/(n + λ).
P r o o f. The proof of the lemma follows along the linesP
similar to those of
Lemma 3.1, except that we now choose R1 = ql Ul and R2 = ∞
t=1, t6=l qt Ut . ¥
T HEOREM 4.1. Assume the conditions of Lemma 4.1 hold. Let
ql =
n
X
pi qi,l /λ
and
q=
i=1
λ
.
n+λ
Then
(4.5)
¡
¢
dT V Sn , CN B(n, p, Q) ¬ min
(4.6)
¡
¢
P
2
where ζ = 4λ 1 − q ∞
l=1 ql .
½
¾
n
ζj
1 X
,1
2 j=1 j!
¬ min {0.911ζ, 1} ,
223
CNB approximations for sums of random variables
P r o o f. The result essentially follows from Lemma 4.1 and the arguments
given in the proof of Theorem 3.3. Note that from (4.3) and (4.4), we have
n
X
kLi CN B(n/j, p, Q)k ¬ 4
i=1
∞ X
n
X
pi qi,l (1 − qql )
l=1 i=1
µ
= 4λ 1 −
∞
λ X
q2
n + λ l=1 l
¶
= ζ.
The practical estimate in (4.6) also follows in a similar manner. ¥
Next, we present an analogous result to Theorem 3.4 for the case under consideration, and the proof is omitted.
T HEOREM 4.2. Let W be the signed measure as defined in (3.24). Also, let Q
and Qi be defined in (3.10) and (4.1), respectively. Then
(4.7)
¶ ¾
½ µ
n
X
1 ζ
ζj
+
,1
dT V (Sn , W ) ¬ min
2 2 j=2 j!
(4.8)
¬ min{0.775ζ, 1},
¢
¡
P
2
where ζ = 4λ 1 − q ∞
l=1 ql .
Next,
P as applications of the above results, we discuss two examples where
Qi = ∞
j=1 qi,j Uj exists in discrete and continuous cases. Also, we analyze the
conditions under which the bounds are optimal.
E XAMPLE 4.1 (Discrete case). Let L(Yi ) = Qi ∼ Ge(ηi ), 1 ¬ i ¬ n, the
geometric distribution with probability distribution (a number of trials for the first
success)
P (Yi = k) = (1 − ηi )k−1 ηi ,
k = 1, 2, . . .
¢
¡
P
Let Sn = nj=1 Xj , where L(Xi ) = δ0 + pi L(Yi ) − δ0 and pi = P (Xi 6= 0).
P
Our aim is to approximate Sn = nj=1 Xj to CN B(n, p, Q), where p and Q are
as defined in (3.11) and (3.10). Let now
η > ηmax = max ηi
1¬i¬n
and
qi,j = (1 − bi )j−1 bi
for j ­ 1,
where bi = ηi /η. Choose Uj = N B(j, η) with probability distribution
µ
¶
x−1 j
Uj (x) =
η (1 − η)x−j
j−1
for x = j, j + 1, . . .
224
P. Vellai sa my and N . S. Up a d hy e
Then it can be easily seen that Qi =
P∞
j=1 qi,j Uj .
Using now (4.6), we get
¡
¢
dT V Sn , CN B(n, p, Q)
∞
©
¡
¢ ª
X
¬ min 3.65λ 1 − q
ql2 , 1
½
µ
= min 3.65λ 1 −
l=1
¶2 ¶ ¾
n
1 X
l−1
pi (1 − ηi /η) ηi /η
,1
λ i=1
½
µ
¶ ¾
∞ X
n
X
1
p2i (η − ηi )2(l−1) ηi2
¬ min 3.65λ 1 −
,1
(n + λ)λ l=1 i=1
η 2l
µ
½
µ
¶¶ ¾
n
X
1
ηi
2
= min 3.65λ 1 −
p
,1 .
(n + λ)λ i=1 i 2η − ηi
∞
λ X
n + λ l=1
µ
Note that the above bound is decreasing in η, and so attains the minimum when
η = ηmax .
E XAMPLE 4.2 (Continuous case). Let Qi ∼ E(ti ), the exponential distribution with density
½ −t x
ti e i
for x > 0,
fQi (x) =
0
otherwise,
and t > max1¬i¬n ti . Let qi,j = (1 − bi )j−1 bi , where bi = ti /t, and Uj ∼ G(t, j),
the gamma distribution with density

 tj
e−tx xj−1 for x > 0,
fUj (x|t, j) = (j − 1)!

0
otherwise.
Then, it follows that Qi =
P∞
j=1 qi,j Uj .
Consequently, from (4.6) we get
¡
¢
dT V Sn , CN B(n, p, Q)
½
µ
¬ min 3.65λ 1 −
¶ ¾
∞
λ X
2
q ,1
n + λ l=1 l
½
µ
¶¶ ¾
µ
n
X
1
ti
2
¬ min 3.65λ 1 −
,1 ,
p
(n + λ)λ i=1 i 2t − ti
following the arguments in Example 4.1.
Finally, we point out an application of our results to the individual risk model,
which is widely used in life and health insurance. Consider a portfolio with n
policies with associated non-negative risks, say, X1 , . . . , Xn . Assume that the risk
i produces a claim with probability pi , and let Qi denote its conditional claim
CNB approximations for sums of random variables
225
P
amount. Then Sn = nj=1 Xj denotes the total claim in the individual model. In
general, the distribution of Sn is complicated. When all pi ’s are small, one may
approximate L(Sn ) to a suitable compound distribution (see Roos [20]). If some
of the pi ’s are not small, it is natural to approximate L(Sn ) to
(4.9)
CN B(r, p, Q) =
∞
X
πk (r, p)Qk ,
k=0
where
µ
¶
r+k−1 r k
πk (r, p) =
p q
k
for k = 0, 1, 2, . . . ,
and
Q=
n
∞
X
1 X
pi Qi =
ql Ul .
λ i=1
l=1
Observe that (4.9) is indeed a random sum, and represents the total claim amount
in the collective risk model (Grandell [13] or Mikosch [17]). Our results in Theorems 3.3 and 4.1 are helpful to obtain the error estimates in such cases.
Acknowledgments. The authors are grateful to Professors V. Čekanavičius
and B. Roos for several comments and encouragements, and also to the referee
for his critical comments and suggestions which led to improvements.
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Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai-400076
E-mail: [email protected]
E-mail: [email protected]
Received on 1.2.2008;
revised version on 11.11.2008