Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol168.Issue1.Nov2000:

Mathematical Biosciences 168 (2000) 57±75
www.elsevier.com/locate/mbs

Applying the saddlepoint approximation to bivariate stochastic
processes
Eric Renshaw *
Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower,
26 Richmond Street, Glasgow G1 1XH, UK
Received 16 September 1999; received in revised form 13 June 2000; accepted 3 July 2000

Abstract
The problem of moment closure is central to the study of multitype stochastic population dynamics since
equations for moments up to a given order will generally involve higher-order moments. To obtain a
Normal approximation, the standard approach is to replace third- and higher-order moments by zero,
which may be severely restrictive on the structure of the p.d.f. The purpose of this paper is therefore to
extend the univariate truncated saddlepoint procedure to multivariate scenarios. This has several key advantages: no distributional assumptions are required; it works regardless of the moment order deemed
appropriate; and, we obtain an algebraic form for the associated p.d.f. irrespective of whether or not we
have complete knowledge of the cumulants. The latter is especially important, since no families of distributions currently exist which embrace all cumulants up to any given order. In general the algorithm
converges swiftly to the required p.d.f.; analysis of a severe test case illustrates its current operational
limit. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords: Birth±death migration; Cumulants; Moment closure; Saddlepoint approximation; Spatial processes; Tail

probabilities; Truncation

1. Univariate introduction
Although modelling the growth and dispersal of biological populations is an extremely important and challenging problem (e.g. [8,28]), diculties experienced in handling the associated
non-linear mathematics often force analysts back into using linear approximations. Whilst these
can provide useful information on the initial qualitative growth of a population, ignoring

*

Corresponding author.
E-mail address: eric@stams.strath.ac.uk (E. Renshaw).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 3 7 - 7

58

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

non-linear aspects may have a serious impact on longer-term behavioural prediction. The essence

of this problem lies in the general intractability of the forward Kolmogorov partial di€erential
equation (p.d.e.) for the moment generating function (m.g.f.)
M…h; t† 

1
X
N ˆ0

pN …t†ehN

…1:1†

for the population size probabilities pN …t† at time t. Some progress can be made by rewriting this
equation in terms of the cumulant generating function (c.g.f.)
K…h; t†  ln M…h; t† 

1
X
ji …t†hi
;

i!
iˆ1

…1:2†

since we can use this to generate equations for the cumulants ji …t†…i > 0†. For example, motivated
by a desire to model both the annual catch of an invasion of muskrats in eleven Dutch provinces
between 1968 and 1991, and the rapid colonization by the Africanized honey (Killer) bees of
North and South America (see [17]), Matis et al. [18] consider the power-law logistic process with
population birth and death rates kN ˆ a1 N ÿ b1 N s‡1 and lN ˆ a2 N ‡ b2 N s‡1 , respectively, for
ai ; bi P 0, integer s P 1, and population size N ˆ 0; 1; 2; . . . The associated p.d.e.
oM
oM
os‡1 M
ˆ ‰…eh ÿ 1†a1 ‡ …eÿh ÿ 1†a2 Š
‡ ‰…eh ÿ 1†…ÿb1 † ‡ …eÿh ÿ 1†b2 Š s‡1
ot
oh
oh

…1:3†

easily yields a set of ®rst-order ordinary di€erential equations for the ji …t† (replace M…h; t† by
expfK…h; t†g, expand both sides of the resulting equation in powers of h and equate coecients),
though the di€erential equation for the jth cumulant unfortunately involves terms up to the …j ‡
s†th cumulant. This clearly rules out determining exact solutions to the cumulant equations, and
so Matis et al. [18] adopt a moment closure approach by solving the system of the ®rst …j ‡ s†
cumulant functions with ji  0 for all i > j ‡ s.
This raises two fundamental questions. The ®rst, considered by Matis et al. [18], assesses the
error induced into the cumulants themselves by adopting this truncation procedure. The second,
considered by Renshaw [29], is assuming that the ®rst j ‡ s cumulants are known exactly, what
error is induced into the underlying probability structure by taking all higher-order cumulants to
be zero? He studies this by considering the truncated c.g.f.
Kn …h† 

n
X
ji hi
i!
iˆ1

…1:4†

in tandem with the associated saddlepoint approximation. Easton and Ronchetti [5] proposed this
approach in the context of deriving an approximation to the c.g.f of some statistic Vr …X1 ; . . . ; Xr † of
i.i.d. observations X1 ; . . . ; Xr . They show that it is especially useful in the case of small sample
sizes, and although our own scenario of investigating the structure of a single realisation of a
stochastic process is quite di€erent, their success is certainly encouraging.
A superb account of the derivation of the saddlepoint approximation is provided by Daniels [2]
in terms of the dominant term in the contour-integration formula for the inversion of the c.g.f.
K…h† corresponding to the p.d.f. f …x†. This essentially involves the method of steepest descents,
which nearly always provides a good approximation across the full range of the distribution and is

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

59

intrinsically better than employing either the Central Limit Theorem or Edgeworth-type
approximations. The key result is that for h0 an appropriate root of
x ˆ K 0 …h0 †

…1:5†

(Theorems 6.1 and 6.2 of Daniels [2] guarantee there will be only one), we have the approximation
f …x† ' ‰2pK 00 …h0 †Šÿ1=2 expfK…h0 † ÿ h0 xg:

…1:6†

The power of this approach can be seen immediately on noting that it not only reproduces the
Normal p.d.f. exactly, but also that the saddlepoint approximation for the gamma p.d.f. di€ers
only from the exact result in that C…a† is replaced by Stirling's approximation in the normalising
factor [2]. The truncated saddlepoint approximation, f …n† …x†, is then obtained by substituting (1.4)
into expressions (1.5) and (1.6), yielding
"
#ÿ1=2
(
)
nÿ1
n
nÿ2

X
X
X
ji‡2 hi0
ji hi0
ji‡1 hi0
…n†
ÿ h0 x ; where x ˆ
exp
:
…1:7†
f …x† ˆ 2p
i!
i!
i!
iˆ0
iˆ1
iˆ0

This general form is extremely useful when we wish to examine the structure of the p.d.f. which

corresponds to a given set of cumulants. For although in principle the Kolmogorov equations for
the probabilities fpn …t†g can be solved numerically to any desired degree of accuracy, this may be
computationally far too expensive. In contrast, exploitation of (1.7) is both algebraically tractable
for small n, and numerically fast for all n. Moreover, being a completely general technique it does
not necessitate the preselection of an assumed underlying distribution whose parameters are then
®tted according to some statistical goodness-of-®t criteria. Indeed, for n ˆ 3 (i.e., we incorporate
the mean j1 ˆ l, variance j2 ˆ r2 and third central moment j3 ) we have
f …3† …x† ˆ …4p2 w†ÿ1=4 expfÿ…1=6j23 †‰r6 ÿ 3r2 w ‡ 2w3=2 Šg;

…1:8†

where
h0 ˆ



ÿ r2 ‡

p
w =j3

for

w ˆ r4 ‡ 2j3 …x ÿ l†

…1:9†

[29]. This is a completely general, and algebraically amenable, result, which provides a considerable improvement over the Normal approximation (for which j3 ˆ 0† since it incorporates
skewness. Raising n to 4 and 5 leads to mathematically tractable cubic and quartic equations,
respectively, for h0 , so expressions (1.8) and (1.9) can be re®ned still further.
Renshaw [29] illustrates the application of this approach by comparing the exact Poisson (10)
probabilities with the full saddlepoint probabilities derived through (1.6), and the truncated
saddlepoint probabilities derived through (1.7) with truncation points n ˆ 2; 3; 4 and 6. The full
saddlepoint approximation provides an accuracy to within 3% for i > 2 and to within 1% for
i > 9, in total contrast to the Normal approximation (i.e., (1.7) with n ˆ 2) which behaves badly in
the tails. The third-order case …n ˆ 3† is much better where it exists; for i P 12 it not only out
performs the fourth- and sixth-order approximations, but it also marginally beats the full saddlepoint approximation itself. However, it does collapse in the lower tail of the distribution due to
the necessity of having w P 0 in order for h0 to be real. Note that this is not a serious problem,
since employing the modi®cation proposed by Wang [31], in which j3 and j4 are each scaled down
by expfÿj2 b2 h2 =2g for some appropriate constant b, guarantees full support over i ˆ 0; 1; 2; . . .

60

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

Table 1
Comparison of exact Poisson probabilities with fourth-order truncation for j4 ˆ 5; 10 (true), 20, 50 and 100
x

Exact

0
5
10
15
20
25

0.000045
0.037833

0.125110
0.034718
0.001866
0.000029

j4 ˆ 5

0.000000
0.041460
0.126157
0.035242
0.001826
0.000025

j4 ˆ 10

0.000104
0.037030
0.126157
0.035008
0.001872
0.000028

j4 ˆ 20

0.000299
0.033814
0.126157
0.034594
0.001963
0.000035

j4 ˆ 50

0.000725
0.031068
0.126157
0.033656
0.002211
0.000057

j4 ˆ 100

0.001246
0.029854
0.126157
0.032660
0.002562
0.000096

Extending the study to the logistic birth±death process reinforces the conclusion that the thirdorder approximation (1.8) is best both in terms of transparency and accuracy; here the third-order
approximation covers the full admissible range, whilst the fourth- and sixth-order approximations
do not. Care must be taken though not to dismiss the Normal approximation out of hand, since
with a power-law logistic birth±death process it o€ers the optimal approximation in the tails, and
is only marginally worse than the n ˆ 3, 4 and 6 cases in the centre.
The fact that h0 in (1.9) may be complex highlights the inherent contradiction which underlies
the standard moment closure approach. For the reason behind placing all high-order cumulants
equal to zero stems from the Gaussian result that ji  0 for i P 3. Yet when j3 6ˆ 0 the underlying
distribution cannot be Gaussian, whence ji  0 …i > 3† is hardly an appropriate choice! For the
Poisson (10) example this results in third-order approximations for x P 5 only, an anomaly which
is easily resolved by allowing j4 > 0. For then K40 …h† ÿ x  j1 ‡ j2 h ‡ j3 h2 =2 ‡ j4 h3 =6 ÿ x ˆ 0
solves to give real h for all x P 0. Table 1 shows the e€ect of taking various j4 values on either side
of the true value 10 for x ˆ 0; 5; . . . ; 25. Increasing j4 clearly `¯attens' the resulting p.d.f., though
between j4 ˆ 5 and 20 the central part of the distribution su€ers little or no e€ect. Tail in¯ation is
felt more strongly at smaller values of x than at higher values.
Note that for this speci®c problem, employing recent theoretical developments based on the tail
probability structures of Lugannani and Rice [14] does not yield further improvement. Jensen [13]
provides a full account of this approach (also Reid [22]); Daniels [3,4] gives numerical examples
and reviews [4] a variety of tail-area approximations using the saddlepoint method. Wang [30]
derives the cumulative distribution function of the sample mean of independent bivariate random
vectors, whilst Barndor€-Nielsen and Cox [1] and McCullagh [20] provide further insight into
saddlepoint approximations.

2. Multivariate extension
In terms of straight practical application, extending the truncated saddlepoint approach from
one- to multi-variable scenarios simply involves moving from (1.5)±(1.7) to the m-variable form
f …x1 ; . . . ; xm † '

expfK…h1 ; . . . ; hm † ÿ h1 x1 ÿ


ÿ hm xm g
p
…2p†m=2 j K 00 …h1 ; . . . ; hm † j

…2:1†

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

61

for
oK…h1 ; . . . ; hm †=ohi ˆ xi

…i ˆ 1; . . . ; m†

…2:2†

and j K 00 …
† j the determinant of second derivatives. Although we shall remain with two-variable
processes throughout this paper for reasons of algebraic simplicity, no additional theoretical
diculties occur when m > 2.
To illustrate this procedure, suppose that the variables …X ; Y † follow a bivariate Normal distribution with zero means, variances r21 and r22 , and correlation q. Then the associated c.g.f.


ÿ
…2:3†
K…h1 ; h2 † ˆ …1=2† h21 r21 ‡ 2qr1 r2 h1 h2 ‡ h22 r22 ;
whence (2.2) gives

h1 r21 ‡ qr1 r2 h2 ˆ x

…2:4†

h2 r22 ‡ qr1 r2 h1 ˆ y

which yield, for given co-ordinates …x; y†,
h1 ˆ

…xr2 ÿ yqr1 †
r21 r2 …1 ÿ q2 †

and h2 ˆ

…yr1 ÿ xqr2 †
:
r1 r22 …1 ÿ q2 †

…2:5†

Inserting (2.5) into (2.1) with m ˆ 2 and …x1 ; x2 † ˆ …x; y† then yields the exact p.d.f. of the bivariate
Normal distribution, namely

 2

h
piÿ1
ÿ1
x
2qxy y 2
2
f …x; y† ˆ 2pr1 r2 …1 ÿ q †
ÿ
:
…2:6†
exp
‡
2…1 ÿ q2 † r21 r1 r2 r22

As a second example, suppose that X ; Y are i.i.d. Poisson (1) variables, and let U ˆ X and
V ˆ X ‡ Y . Then
eÿ2
Pr…U ˆ u; V ˆ v† ˆ
;
…2:7†
u!…v ÿ u†!
whence the c.g.f.
K…h1 ; h2 † ˆ eh1 ‡h2 ‡ eh2 ÿ 2:

…2:8†

Solving (2.2) with (2.8) yields eh1 ˆ u=…v ÿ u† and eh2 ˆ v ÿ u, whence (2.1) becomes
evÿ2
f …u; v† '
:
…2:9†
2puu‡1=2 …v ÿ u†vÿu‡1=2
p
On replacing n! in the exact solution (2.7) by …2p†eÿn nn‡1=2 , we immediately recover (2.9). So in
this scenario the success of the saddlepoint approximation is limited only by the accuracy of
Stirling's approximation.
We shall now show how the saddlepoint approach can be applied to a simple stochastic process,
as opposed to a given p.d.f. such as (2.7). Let pairs of (type1, type2) individuals arrive as a Poisson
process at rate a, with individuals dying independently according to two simple death processes
with rates l1 and l2 , respectively. Then on denoting pij …t† ˆ Pr (population is of size …i; j† at time
t), the Chapman±Kolmogorov forward equations are given by
dpij …t†=dt ˆ apiÿ1;jÿ1 …t† ‡ l1 …i ‡ 1†pi‡1;j …t† ‡ l2 …j ‡ 1†pi;j‡1 …t† ÿ …a ‡ il1 ‡ jl2 †pij …t†;

…2:10†

62

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

and the associated equation for the probability generating function (p.g.f.)
G…z1 ; z2 ; t† 

1
X
i;jˆ0

pij …t†zi1 zj2

…2:11†

takes the form
oG
oG
oG
ÿ l1 …1 ÿ z1 †
ÿ l2 …1 ÿ z2 †
ˆ ÿa…1 ÿ z1 z2 †G:
ot
oz1
oz2

…2:12†

Eq. (2.12) solves via the auxiliary equations
dt
ÿdz1
ÿdz2
ÿdG
ˆ
ˆ
ˆ
l1 …1 ÿ z1 † l2 …1 ÿ z2 † aG…1 ÿ z1 z2 †
1

…2:13†

to yield the solution

G…z1 ; z2 ; t† ˆ exp ÿ …a=l1 †…1 ÿ z1 †…1 ÿ eÿl1 t † ÿ …a=l2 †…1 ÿ z2 †…1 ÿ eÿl2 t †
ÿ


‡ ‰a=…l1 ‡ l2 †Š…1 ÿ z1 †…1 ÿ z2 † 1 ÿ eÿ…l1 ‡l2 †t ;

…2:14†

here we assume that the population is initially empty, i.e., G…z1 ; z2 ; 0†  1. On considering each of
the three constituent parts of (2.14), and denoting
ÿ


/1 ˆ …a=l1 †…1 ÿ eÿl1 t †; /2 ˆ …a=l2 †…1 ÿ eÿl2 t †; /12 ˆ ‰a=…l1 ‡ l2 †Š 1 ÿ eÿ…l1 ‡l2 †t ;
we therefore see that …X ; Y † is distributed as the convolution
Poisson…X ; /1 ÿ /12 †  Poisson…Y ; /2 ÿ /12 †  Poisson…XY ; /12 †:

…2:15†

…2:16†

So the exact population size probabilities are given by
pij …t† ˆ

min…i;j†
X
sˆ0

…/1 ÿ /12 †iÿs …/2 ÿ /12 †jÿs /s12 expfÿ/1 ÿ /2 ‡ /12 g
:
…i ÿ s†!…j ÿ s†!s!

…2:17†

To evaluate the associated saddlepoint values we ®rst note that the c.g.f. is given by (2.14) with
K…h1 ; h2 ; t†  G…eh1 ; eh2 ; t†, whence the saddlepoint equations (2.2) become
…/1 ÿ /12 †eh1 ‡ /12 eh1 ‡h2 ˆ x

…2:18†

…/2 ÿ /12 †eh2 ‡ /12 eh1 ‡h2 ˆ y:

For algebraic convenience denote these equations as
aw ‡ bwz ˆ x

and cz ‡ bwz ˆ y;

…2:19†

where
w ˆ eh1 ;

z ˆ eh2 ;

a ˆ /1 ÿ /12 ;

b ˆ /12 ;

c ˆ /2 ÿ /12 :

…2:20†

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

63

Then we have

q
2
w ˆ …1=2ab† ÿ ‰ac ÿ …x ÿ y†bŠ ‡ ‰ac ÿ …x ÿ y†bŠ ‡ 4abcx

q
z ˆ …1=2bc† ÿ ‰ac ÿ …y ÿ x†bŠ ‡ ‰ac ÿ …y ÿ x†bŠ2 ‡ 4abcy :

…2:21†

Now the determinant
j K 00 …h1 ; h2 ; t† jˆ xy ÿ …bwz†2 :

…2:22†

So the full saddlepoint form (2.1) is given by
f …x; y† ˆ

expfÿa…1 ÿ w† ÿ c…1 ÿ z† ÿ b…1 ÿ wz†gwÿx zÿy
q
:
2p xy ÿ …bwz†2

…2:23†

To illustrate this result consider the equilibrium process (i.e., t ˆ 1) with a ˆ 6; l1 ˆ 1 and
l2 ˆ 2. Then from (2.16) we see that the process behaves as the Poisson convolution
Poisson…X ; 4†  Poisson…Y ; 1†  Poisson…XY ; 2†;

…2:24†

with the exact probabilities (2.17) taking the fairly amenable form
pij …1† ˆ eÿ7

min…i;j†
X
rˆ0

4iÿr 2r
:
…i ÿ r†!…j ÿ r†!r!

…2:25†

However, the structure of even this simple multi-termed expression is certainly not transparent,
and contrasts markedly with that of the single-termed saddlepoint approximation (2.23), namely
f …x; y† ˆ

wÿx zÿy e4w‡yÿ7
q ;
2p xy ÿ 4…wz†2

…2:26†

which is clearly much easier to interpret.
To assess the numerical accuracy of (2.26), at the marginal (Poisson) means a=l1 ˆ 6 and
a=l2 ˆ 3 we have p6;3 …1† ˆ 0:041068 (to 6 decimal places), so the saddlepoint value f …6; 3† ˆ
0:042536 (3.57% too high) compares well. Whilst to examine a `worst case' scenario let us take
(1,1); for since Stirling's approximation to 1! is 2.32, we might expect the approximation to
perform poorly. In fact, f …1; 1† ˆ 0:006579 compares quite favourably with p1;1 …1† ˆ 0:005471.
3. Cumulant truncation
Whilst use of the full saddlepoint approximation is clearly ideal, in many situations knowledge
of the full cumulant structure will be unknown. Given the general success of the third-order
truncation scheme in the univariate case [29], it therefore makes sense to examine the equivalent
bivariate situation. Now we have already seen that for our univariate Poisson (10) example,
placing ji ˆ 0 …i > 3† does not work for x < 5. So to keep our (bivariate) options open, let us
consider jij  0 for all i ‡ j > 4. Thus the full c.g.f. is replaced by

64

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

K…h1 ; h2 † ˆ j10 h1 ‡ j01 h2 ‡ j20 h21 =2 ‡ j11 h1 h2 ‡ j02 h22 =2 ‡ j30 h31 =6

‡ j21 h21 h2 =2 ‡ j12 h1 h22 =2 ‡ j03 h32 =6 ‡ j40 h41 =24 ‡ j31 h31 h2 =6
‡ j22 h21 h22 =4 ‡ j13 h1 h32 =6 ‡ j04 h42 =24;

…3:1†

Eqs. (2.2) for …h1 ; h2 † by
g…h1 ; h2 †  j10 ‡ j20 h1 ‡ j11 h2 ‡ j30 h21 =2 ‡ j21 h1 h2 ‡ j12 h22 =2

‡ j40 h31 =6 ‡ j31 h21 h2 =2 ‡ j22 h1 h22 =2 ‡ j13 h32 =6 ÿ x ˆ 0

h…h1 ; h2 †  j01 ‡ j11 h1 ‡ j02 h2 ‡ j21 h21 =2 ‡ j12 h1 h2 ‡ j03 h22 =2

…3:2†

‡ j31 h31 =6 ‡ j22 h21 h2 =2 ‡ j13 h1 h22 =2 ‡ j04 h32 =6 ÿ y ˆ 0;

and jK 00 …h1 ; h2 †j by g1 h2 ÿ g2 h1 where
g1 ˆ og=oh1 ˆ j20 ‡ j30 h1 ‡ j21 h2 ‡ j40 h21 =2 ‡ j31 h1 h2 ‡ j22 h22 =2

g2 ˆ og=oh2 ˆ oh=oh1 ˆ h1 ˆ j11 ‡ j21 h1 ‡ j12 h22 ‡ j31 h21 =2 ‡ j22 h1 h2 ‡ j13 h22 =2

h2 ˆ oh=oh2 ˆ j02 ‡ j12 h1 ‡ j03 h2 ‡

j22 h21 =2

‡ j13 h1 h2 ‡

j04 h22 =2:

…3:3†

Eqs. (3.2) are eciently solved by using bivariate Newton±Raphson, with the iterates


 
…r†
…r†
…r‡1†
…r‡1†
ˆ h1 ‡ ar ; h2 ‡ br ;
h1 ; h2
where

ar ˆ

hg2 ÿ gh2
g1 h2 ÿ g2 h1

and br ˆ

hg1 ÿ gh1
:
g1 h2 ÿ g2 h1

…3:4†

Whence the corresponding saddlepoint approximation is given by
expfK…h1 ; h2 † ÿ h1 x ÿ h2 yg
p
:
f~…x; y† ˆ
2p …g1 h2 ÿ g2 h1 †

…3:5†

For this saddlepoint procedure to yield full support to the underlying process we require the
quadratic surface (3.2) to solve for real …h1 ; h2 † for all appropriate …x; y†-values. An extreme solution is to place j31 ˆ j22 ˆ j13 ˆ 0, for we see from (3.2) that for large jhi j, g  j40 h31 =6 and
h  j04 h32 =6, which independently sweep out all possible x and y values. So a saddlepoint solution
will always exist. However, given that there is no unique way of relaxing the zero condition on the
fourth-order cumulants, a key question arises as to how (say)
h



i




…3:6†
max  f~…x; y† ÿ f …x; y† =f …x; y† or max f~…x; y† ÿ f …x; y†
x;y

x;y

changes with j40 ; . . . ; j04 . At present this is an open problem.
To illustrate how to apply this truncation procedure we shall derive a p.d.f. which possesses the
third-order cumulant structure.

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

…Example A† :

j10 ˆ 10;
j20 ˆ 9;
j30 ˆ 15;

j01 ˆ 10;
j11 ˆ 4;
j21 ˆ 10;

j02 ˆ 16;
j12 ˆ 15;

65

j03 ˆ 25

over 0 6 x; y 6 100. Given that saddlepoint p.d.f.s operate over ÿ1 < x; y < 1, non-negligible
probability mass may accrue outside our chosen bounded region. Thus, not only may we have to
scale our saddlepoint p.d.f. to ensure that probabilities sum to one, but also the ensuing saddlepoint cumulants, j~ij , may di€er from the target cumulants jij . The development of a universally
optimal procedure for achieving this goal is currently an open problem, but the following approach works well. The ®ve free-ranging fourth-order cumulants are used to achieve the ®t, and
here we select the initial values j~40 ˆ j~04 ˆ 100 (which are high enough to ensure that the resulting
iterated p.d.fs have full support over 0 6 x; y 6 100) together with j~31 ˆ j~22 ˆ j~13 ˆ 0. In general,
deriving a p.d.f. to ®t nth-order cumulants would use the n ‡ 2…n ‡ 1†th order cumulants. For fZg
a set of independent uniformly distributed psuedo-random numbers on (ÿ0:5,0.5):
0. set initial saddlepoint cumulants to the target values, i.e., j~10 ˆ j10 ; . . . ; j~03 ˆ j03 , and choose
appropriate values for j~40 ; . . . ; j~04 ;
1. increment a randomly chosen cumulant j~ij by dZ, for appropriate d;
2. (a) evaluate the resulting saddlepoint probabilities f~…x; y† via the iterative
procedure
P
^…x; y† ˆ 1;
f
(3.1)±(3.5), and then (b) rescale them to form f^…x; y† to enable (here) 100
i;jˆ0
3. evaluate the cumulants
j^ij corresponding to f^…x; y†;
P3
jij ÿ jij †2 , and update j~ij if S is reduced and g1 h2 ÿ g2 h1 > 0 (to ensure
4. determine S ˆ i;jˆ0 …^
(3.5) is real);
5. print f^…x; y† when S reaches say 10ÿ6 , then stop;
6. return to 1.
To enable j~40 ; . . . ; j~04 to `bed-in', initial choice of d was kept relatively high at d ˆ 10, but even
such coarse-tuning swiftly led to max j^
jij ÿ jij j < 0:03. Switching to a ®ne-tuning regime with d ˆ
0:1 then quickly produced max j^
jij ÿ jij j < 0:0006. If required, further accuracy could be achieved
with micro-tuning using say d ˆ 0:001; here this gave max j^
jij ÿ jij j  0:00001. Clearly the algorithm is both fast and extremely precise. The resulting j~ij -values are given (to 3 decimal places,
double precision values are used in the computation) by
j~10
j~20
j~30
j~40

ˆ 9:704;
ˆ 8:651;
ˆ 20:157;
ˆ 86:058 …71:715†;

j~01
j~11
j~21
j~31

ˆ 9:810;
ˆ 4:640;
ˆ 10:157;
ˆ 4:988 …13:427†;

j~02 ˆ 16:579;
j~12 ˆ 14:867;
j~22 ˆ 3:542 …11:694†;

j~03 ˆ 20:944
j~31 ˆ 3:526 …16:984†;
j~04 ˆ 118:078 …79:273†;

the associated fourth-order cumulant estimates j^40 ; . . . ; j^04 are given in brackets. Though the two
means are 3.33 and 2.5 standard deviations above zero, the
presence of skewness leads to subP100
stantial probability mass outside 0 6 x; y 6 100 (in step 2a i;jˆ0 f~…x; y† ˆ 0:9151†, so rescaling is
essential. The required p.d.f. f^…x; y† (step 2b) is therefore f~…x; y†=0:9151 over 0 6 x; y 6 100
(Fig. 1(a)), where f~…x; y† is given by (3.5) and gi ; hi by (3.3), with jij being replaced by the above
values of j~ij . Note that of the ®rst nine j~ij -values, seven are in fairly close agreement with jij , with
only j~30 and j~03 being out of line; also, that j~40 and j~04 are substantially larger than j~31 ; j~22 and

66

E. Renshaw / Mathematical Biosciences 168 (2000) 57±75

Fig. 1. Truncated saddlepoint p.d.fs f^…x; y† corresponding to the cumulants of Example A: (a) least squares ®t to
j10 ; . . . ; j03 with free-ranging j40 ; . . . ; j04 ; (b) least squares (Normal) ®t to j10 ; . . . ; j02 with j30 ˆ


ˆ j04 ˆ 0.

j~13 . We shall see in Section 4 that under certain conditions this last observation can prove to be of
crucial importance.
The corresponding rescaled Normal p.d.f. can be obtained from the same algorithm with
j~30 ; . . . ; j~04 ®xed at zero (the second-order saddlepoint approximation is guaranteed to exist).
Initial coarse-tuning is unnecessary, since third- and fourth-order cumulants no longer feature,
and ®ne-tuning swiftly produces excellent accuracy (maximum absolute error of