Discussion Paper: 2010/08
Saddlepoint approximation of expected shortfall
for transformed means
Simon A. Broda and Marc S. Paolella
www.feb.uva.nl/ke/UvA-Econometrics
Amsterdam School of Economics
Department of Quantitative Economics
Roetersstraat 11
1018 WB AMSTERDAM
The Netherlands
Saddlepoint Approximation of Expected Shortfall for Transformed
Means
Simon A. Brodaa, ∗
a
Marc S. Paolellab, c
Department of Quantitative Economics, University of Amsterdam, The Netherlands
b
Swiss Banking Institute, University of Zurich, Switzerland
c
Swiss Finance Institute
December 2010
Abstract
Expected shortfall, as a coherent risk measure, has received a substantial amount of attention in the literature recently. For many distributions of practical interest however, it cannot be obtained in explicit form, and numerical techniques must be employed. The present
manuscript derives a saddlepoint approximation for the expected shortfall associated with
certain random variables that permit a stochastic representation in terms of some underlying random variables possessing a moment generating function. The new approximation can
be evaluated quickly and reliably, and provides excellent accuracy. The doubly noncentral t
distribution is considered as an example.
Key Words: Expected Shortfall, Noncentral t Distribution, Saddlepoint Approximation
∗
Address correspondence to Simon Broda, Department of Quantitative Economics, 1018WB Amsterdam, The
Netherlands, or email [email protected]. Part of the research of M. Paolella has been carried out within the
National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK),
which is a research program supported by the Swiss National Science Foundation.
1
Introduction
The question of how to adequately measure risk is central to risk management, and has seen
much controversy lately: beginning with the seminal paper of Artzner, Delbaen, Eber, and Heath
(1999), an enormous body of literature has accumulated which criticizes the most widely used risk
measure, Value at Risk (VaR), for its failure to be coherent. In particular, VaR is not subadditive,
thus violating one of the four axioms defining a coherent risk measure. The expected shortfall,
as a coherent alternative, appears to be emerging as a new standard.
Suppose the return on a financial position is represented by a continuous random variable
X with density fX (x). Then the expected shortfall (ES) is defined, for a given confidence level
q ∈ (0, 1), as
ES
(q)
1
(X) ≡ −E [X|X ≤ xq ] = −
q
Z
xq
xfX (x)dx,
(1)
−∞
where xq is the corresponding VaR, i.e., the 100q% quantile of X, see Acerbi and Tasche (2002a,b).
Often q is set to 1%. Note that here, we have defined the expected shortfall in terms of relative
loss, as is common. For many distributions of interest, (1) must be evaluated numerically. This
time-consuming task potentially precludes the use of certain distributions in computationally demanding tasks such as mean-ES portfolio allocation, so that a reliable approximation is desirable.
The saddlepoint approximation, introduced next, is a natural candidate.
Saddlepoint approximations for densities result from expanding the Fourier-Mellin integral of
the moment generating function (hereafter mgf) in an asymptotic series. Since their introduction
by Daniels (1954), they have been found to deliver excellent accuracy in countless applications; see
Butler (2007), Broda and Paolella (2011), and the references therein. The cumulative distribution
function could be approximated by numerically integrating the approximate density; however,
Lugannani and Rice (1980) derived a direct approximation to the distribution function which
circumvents this need. Both Daniels’s and Lugannani and Rice’s approximation require that the
random variable to which the approximation is to be applied possess an mgf. Daniels and Young
(1991) have generalized these results to marginal distributions of nonlinear functions of bivariate
random vectors; see also Jing and Robinson (1994).
Concerning the expected shortfall as defined in (1) above, the result of Martin (2006) can be
used whenever the moment generating function of X exists. For certain problems however, X
does not possess an mgf, but may permit a stochastic representation in terms of random variables
that do. The present paper derives a saddlepoint approximation that is applicable in this setting.
At first glance, it might perhaps appear as if the most natural route to such a result were via
a routine application of asymptotic expansions. This is certainly possible (if a bit tedious), and
would lead to a result which generalizes that of Martin (2006) in much the same the way Daniels
and Young’s result generalizes that of Lugannani and Rice. However, careful scrutiny of the
results available in the univariate setting reveals that an alternative strategy may lead to an
approximation that is both more accurate and simpler to compute. This is the approach taken
here.
The plan is as follows. Section 2 reviews some useful results from the theory of saddlepoint
1
approximations. Section 3 demonstrates an alternative method of proof for some known saddlepoint approximations to expected shortfall, and motivates the new approximation given in
Section 4. Section 5 applies the approximation to the noncentral t distribution and concludes.
2
Saddlepoint Approximations for Densities and Tail Probabilities
Consider the mean of n independent and identically distributed variables, X̄ = n−1
Pn
i=1 Xi ,
and suppose that each Xi possesses a density, and that its cumulant generating function KX(t)
converges in a nonvanishing interval containing the origin. The cumulant generating function is
the natural logarithm of the mgf, which is, in turn, defined as
MX (t) ≡ E etX
whenever the expectation exists. It generates the moments of a distribution, in the sense that its
jth derivative evaluated at zero equals the jth moment of X. The usefulness of the mgf in the
context of risk management derives from the fact that the mgf corresponding to the convolution
of two distributions (viz., the mgf of a sum of two independent random variables) is given by the
product of the individual mgfs. Hence, the mgf possesses a tractable form in many cases in which
the distribution function does not. If the distribution of X̄ is absolutely continuous, as we shall
assume throughout, then its density fn (x̄) can be recovered from the mgf via the Fourier-Mellin
integral
n
fn (x̄) =
2πi
Z
i∞
en(KX(t)−tx) dt.
(2)
−i∞
Similar expressions exist for the distribution function and the expected shortfall. Typically,
the integral in (2) permits no analytical solution, and numerical quadrature must be used. An
alternative is to approximate the integral by means of a truncated asymptotic series: Daniels
(1954) derived the asymptotic expansion
#
m
X
ak
+ O n−(m+1) ,
fn (x̄) = gn (x̄)
nk
"
where
gn (x̄) =
s
(3)
k=0
n
2
e−nŵ /2 ,
′′
2π KX t̂
ŵ = sgn t̂
q
2t̂x̄ − 2KX t̂ ,
and the saddlepoint t̂ = t̂ (x̄) solves
′
x̄ = KX
t̂ .
In most applications this has to be evaluated numerically for each x̄ at which the density is to be
approximated. The first two coefficients in the expansion (3) are
5
1
a0 = 1 and a1 = λ̂4 − λ̂23 ,
8
24
2
′′)j/2 , and the truncated series g is referred to as the saddlepoint approxiwhere λ̂j = KX(j) / (KX
n
mation to fn .
A uniform asymptotic expansion for the distribution function of X̄ has been derived in Lugannani and Rice (1980). It is given by
Fn (x̄) = Φ ŵn
1/2
+
r
#
"m
1 −nŵ2 /2 X bk
−(m+1)
,
e
+O n
2πn
nk
(4)
k=0
where Φ is the cdf of the standard normal distribution. The first two coefficients in the expansion
(4) are
b0 =
1
1
−
ŵ û
and b1 =
where
û = t̂
and a1 is as in (3).
λ̂3
1
1
a1
− 3+ 2− ,
3
û
ŵ
2û
û
q
′′ t̂
KX
Lugannani and Rice derived expansion (4) using complex variable methods. It can, however,
also be obtained from (3) by means of the following result of Temme (1982), which we shall use
frequently.
Theorem 1 (Temme, 1982). Suppose Ψn (ζ) is analytic in a strip containing the real axis in its
interior and such that, for some fixed λk and ω,
dk
λk ωζ 2
ζ
e
Ψ
(ζ)
=
O
n
dζ k
as Re ζ → ∞.
If Ψn (ζ) permits the asymptotic expansion
Ψn (ζ) =
m
X
ψk (ζ)
k=0
nk
+ O Ψn (ζ)n−(m+1) ,
where the ψk (ζ) are analytic in ζ and do not depend on n, then
Z η
n
2
e−nζ /2 Ψn (ζ)dζ =
Hn (η) =
2π −∞
"m
#
# r
"m
X
Ak
1 −nη2 /2 X Bk (η)
1/2
−(m+1)
−(m+1)
= Φ ηn
,
+ O(n
) +
e
+O n
nk
2πn
nk
r
k=0
k=0
where Ak are the coefficients in the asymptotic expansion
Hn (∞) =
∞
X
Ak
k=0
B−1 (η) = 0,
nk
and ηBk (η) = Ak − ψk (η) +
3
,
d
Bk−1 (η),
dη
k ∈ {0, 1, . . .}.
3
Saddlepoint Approximations for Expected Shortfall
The first step to approximating the expected shortfall of X̄ is to find an approximation to its
quantiles x̄q . This can be achieved by truncating the series for Fn (x̄) after k terms, yielding the
approximation F̂nk (x̄), say, and numerically solving
ˆkq ) = q,
F̂nk (x̄
q ∈ (0, 1).
(5)
It then remains to find an approximation to integrals of the form
In (c) =
Z
c
x̄fn (x̄)dx̄.
(6)
−∞
Martin (2006) shows that
In (c) = Φ(ŵc n
1/2
)µ +
r
1 −nŵc2 /2
e
2πn
µ
c
−1
−
+O n
,
ŵc ûc
(7)
′ (0), and hatted quantities with a subscript c correspond to those in (3) and (4),
where µ = KX
but evaluated at c rather than x̄. Martin derived his approximation using the same technique
employed by Lugannani and Rice in obtaining their expansion of the distribution function. The
same result is obtained by applying Theorem 1 to the integral in (6) and retaining only the leading
term. Incorporating the second term, which is derived in Appendix A, one has
In (c) = Φ(ŵc n1/2 )µ+
r
1 −nŵc2 /2
+
e
2πn
!
"
#
cλ̂3,c ca1
c
1 c
1
µ
µ
−
+
−
+ O n−2 , (8)
− 3+
−
ŵc ûc n û3c
ŵc
2û2c
ûc
t̂c ûc
where a1 is as in (3).
10
0
−0.1
5
% relative error of I(c)
−0.2
I(c)
−0.3
−0.4
−0.5
−0.6
Martin
2nd order SPA
Butler and Wood
Exact
−0.7
−0.8
−15
−10
0
−5
−10
−15
−5
c
0
−20
−15
5
−10
−5
c
0
Figure 1: Approximations to In (c) (left panel) and relative error in percent (right panel).
4
5
20
30
Martin
2nd order SPA
Butler and Wood
Exact
18
20
% relative error of ES(q)
16
14
ES(q)
12
10
8
6
4
10
0
−10
−20
2
0
0
0.2
0.4
0.6
0.8
−30
1
0.1
0.2
0.3
q
0.4
0.5
q
0.6
0.7
0.8
0.9
Figure 2: Approximations (left panel) and relative error in percent (right panel) for the expected shortfall at level
q.
Butler and Wood (2004), via a completely different method of proof, obtained an approximation for the mgf, and its logarithmic derivatives, of the truncated random variable a X̄b with
density fn (x̄)I(a,b) (x̄)/(Fn (b)−Fn (a)), where as before, fn and Fn are the density and distribution
function of X̄, respectively. Setting a = −∞, b = c, and evaluating their approximation for the
logarithmic derivative at zero produces the following approximation:
In (c) = Φ(ŵc n
1/2
)µ +
r
1 −nŵc2 /2
e
2πn
µ
c
1 c
µ
1
−1
+O n
.
−
+
−
−
ŵc ûc n ŵc3 ŵc3 t̂ûc
(9)
We illustrate the relative merits of approximations (7), (8) and (9) by considering one copy
(i.e., n = 1) of a Normal Inverse Gaussian (NIG) random variable with parameters (α, β, δ, µ) =
(1, −1/2, 2, 0). The NIG belongs to the family of Generalized Hyperbolic (GHyp) distributions.
These, and the NIG in particular, have been very successfully applied to financial returns data,
see, e.g., Eberlein and Keller (1995), Barndorff-Nielsen (1997) and Broda and Paolella (2009).
As shown in the latter paper, the saddlepoint approximation for the NIG is explicit, and the
√
′′(t̂ ) = ȳ 3 δ 3 α, ŵ =
relevant quantities are t̂c = zα/ȳ − β, z = (c − µ)/δ, ȳ = 1 + z 2 , KX
c
c
sgn(t̂c )[2δ(ȳα − zβ − (α2 − β 2 )1/2 ]1/2 , λ̂3,c = 3z(ȳαδ)−1/2 and λ̂4,c = 3(1 + 5z 2 )(ȳαδ)−1 .
The results are shown in Figure 1. As expected, the second order approximation (8) clearly
dominates the first order approximation (7), with under 10% error across the entire support. It is
also observed that despite being formally accurate to the same order, approximation (9) cannot
compete even with (7). This picture changes dramatically if instead of the tail integral In (c), one
considers its normalized version, viz., the expected shortfall: the approximation is
ES(q) (X) ∼ −
ˆkq )
Inℓ (x̄
,
ˆkq )
F̂ k (x̄
where ℓ = 1 refers to approximation (7), ℓ = 2 to approximation (8), and ℓ = 3 to approximation
ˆkq is determined according to (5) with k = 1 if ℓ ∈ {1, 3} and k = 2 if ℓ = 2. In
(9), and x̄
5
other words, the order of the approximation to the distribution function used in determining the
quantile is chosen to agree with that of the approximation to In .
The approximations are illustrated in Figure 2. Despite the fact that (9) is only an expansion
to order n−3/2 , the resulting approximation to the expected shortfall is comparable to that based
on (8), which is O n−5/2 . The reason for this is the following: In order to benefit from error
cancellation, one would ideally approximate the expected shortfall by
ES(q) (X) ∼ −
Z
ˆkq
x̄
x̄fˆnℓ (x̄)dx̄
−∞
"Z
ˆkq
x̄
fˆnℓ (x̄)dx̄
−∞
#−1
,
that is, use the same approximation fˆnℓ (x̄) in both the numerator and denominator. If the
ˆkq ), is used as the denominator, this
first order approximation to the distribution function, F̂ 1 (x̄
amounts to approximating the numerator integral by applying Theorem 1 to
ˆkq
x̄
d 1
−
F̂ (x̄) dx̄.
x̄
dx̄ n
−∞
Z
Now, differentiating the distribution function approximation yields
c1 d 1
,
F̂n (x̄) = gn (x̄) 1 +
dx̄
n
where
c1 =
1
λ̂3
û
+
− 3.
2
û
2û ŵ
A simple way of seeing that this approach leads to approximation (9) is to replace a1 with
c1 + O n−1 in the second-order expansion (8).
4
Expected Shortfall for Transformed Means
The approximations given in the foregoing section are limited to random variables which possess
an mgf. This section derives an approximation for the case where the random variable of interest
does not possess an mgf, but can be represented as a smooth function of two underlying random
variables that do. The derivation is motivated by the following observation: as demonstrated
above for the NIG distribution, the expected shortfall approximation based on (9) provides similar
accuracy to that based on (8), despite the fact that the former only involves quantities which
are anyway needed for the saddlepoint approximation to the distribution function (4). Note that
prior to computing the expected shortfall, one needs to find the relevant quantile of the loss
distribution, and thus evaluate the approximate distribution function. Therefore computing the
expected shortfall via approximation (9), unlike using (8), entails no extra effort.
To fix ideas, let X = (X1 , X2 ) be a bivariate random vector having a density and a joint
′ (t) and K ′′(t) its
cumulant generating function, and denote the latter as KX(t). Denote by KX
X
gradient and hessian, respectively, and by X̄ = (X̄1 , X̄2 ) the mean of X from a sample of n
6
observations. Consider a smooth bijection g mapping the support of Y onto the support of X,
such that X̄ = g(Y ) = (g1 (Y ), g2 (Y ))′ , and denote its inverse by h so that Y = (Y1 , Y2 ) = h(X̄) =
(h1 (X̄), h2 (X̄))′ . We wish to obtain an approximation for the expected shortfall of Y1 which is
analogous to (9); that is, we will approximate
In (c) =
Z
c
x̄fn (y1 )dx̄
(10)
−∞
by replacing fn (y1 ) with the first derivative of the appropriate distribution approximation and
then applying Theorem 1.
Our starting point is the following result from Daniels and Young (1991). Let ∇yi g(y) =
(∂g1 /∂yi , ∂g2 /∂yi )′ , i ∈ {1, 2}, and let Jg (y) = ∂g/∂y denote the Jacobian of g. The density
and distribution function of Y1 are
r
fn (y1 ) =
n −nw̃2 /2 e
1 + O n−1
2π K̃
and
Fn (y1 ) = Φ w̃n
1/2
respectively, where w̃ = sgn(y1 − α)
q
+
r
1
1 −nw̃2 /2 1
−1
e
− +O n
,
2πn
w̃ ũ
(11)
′ (0),
2 t̃′ g(ỹ) − KX(t̃) , ỹ = (y1 , ỹ2 ), α = h1 (µ), µ = KX
i
′′(t̃) h
−1
det KX
′
′′
′ 2
∇
g(ỹ)
K
(
t̃)
∇
g(ỹ)
+
t̃
∇
g(ỹ)
,
K̃ =
y
y
X
y2
2
2
det2 Jg (ỹ)
ũ = t̃′ ∇y1 g(ỹ) K̃ 1/2 , and, for each value of y1 , t̃ and ỹ2 solve the system
′
KX
(t̃) = g(ỹ)
t̃′ ∇y2 g(ỹ) = 0.
The next step is to differentiate the expansion for Fn . This yields
fn (y1 ) =
r
n −nw̃2 /2
1 ũ
1/2 d 1
−1
+ K̃
+O n
e
.
1−
n w̃3
dy1 ũ
2π K̃
As shown in Appendix B, substituting this into (10) and applying Theorem 1 yields
1/2
1
) A0 + A1 +
n
In (c) = Φ(w̃c n
r
1 −nw̃c2 /2
e
+
2πn
!
#
"
A0
c
1 A1
c
A0
1
−1
, (12)
+O n
−
+
+ 3− 3−
′
w̃c
ũc n w̃c
w̃c
w̃c
t˜c ∇y1 g(ỹc ) ũc
where as before, tilded quantities with a subscript c are evaluated at c rather than x̄, and A0
and A1 are coefficients in the asymptotic expansion of E[y1 ]. Denoting the elements of the
′′(0) as κ , these are A = α
Hessian of h1 (X̄) evaluated at µ as hij and the elements of KX
ij
0
7
and A1 = hij κij /2, see Hurt (1976). Here we have used the summation convention, i.e., indices
appearing as both a subscript and a superscript are to be summed over. For practical purposes,
saddlepoint approximations are often applied for n = 1, incorporating the convolution into the
mgf itself. Following that practice, we have the following result.
Theorem 2. Let X = (X1 , X2 ) be a bivariate random vector possessing a density and joint cumulant generating function KX(t). Let g be a smooth bijection such that X = g(Y ) = (g1 (Y ), g2 (Y ))′ ,
with inverse Y = (Y1 , Y2 ) = h(X) = (h1 (X), h2 (X))′ . Then
φ(w̃c )
E Y1 Y1 < c ≈ (α + A1 ) +
F̂ 1 (c)
where w̃c = sgn(c − α)
q
(c − α)
1
1
−
3
w̃c
ũc
A1
−
+
′
ũc
t˜c ∇y1 g(ỹc ) ũc
1
!
,
(13)
′
′ (0), A =
2 t˜c g(ỹc ) − KX(t˜c ) , ỹc = (c, ỹ2,c ), α = h1 (µ), µ = KX
1
′′(0), respectively,
hij κij /2, hij and κij are the elements of the Hessian of h1 (X̄) at µ and KX
i
′′(t̃ ) h
−1
det KX
c
′
′′
′ 2
K̃c =
∇y2 g(ỹc ) KX(t̃c )
∇y2 g(ỹc ) + t̃c ∇y2 g(ỹc ) ,
det2 J(ỹc )
ũc = t̃′c ∇y1 g(ỹc ) K̃c1/2 ,
, and t̃c and ỹ2,c solve the system
F̂ 1 (c) = Φ (w̃c ) + φ(w̃c ) w̃c−1 − ũ−1
c
′
KX
(t̃c ) = g(ỹc )
t̃′c ∇y2 g(ỹc ) = 0.
5
Application
In this section, we apply our approximation to the doubly noncentral t distribution, which has
recently found application as a model for financial returns, see Harvey and Siddique (1999),
Broda and Paolella (2007), Paolella (2010), and the references therein. Despite the excellent fit
it offers, the widespread adoption of this distribution has been hampered by the computational
difficulties associated with it: neither the density nor the distribution function can be expressed
in closed form. This limitation is overcome by the saddlepoint approximations for the density
and distribution functions derived in Broda and Paolella (2007). The resulting approximations
are explicit and hence trivially computed, while offering extraordinary accuracy. This benefit is
inherited by the expected shortfall calculation as well.
Apart from location and scale parameters (which, without loss of generality, can be taken
to be zero and one, respectively), the doubly noncentral t distribution has 3 parameters: the
numerator noncentrality µ, denominator noncentrality θ, and the degrees of freedom, n. The
relevant quantities entering the approximations are
t̃′c ∇y1 g(ỹc ) = (t̃1,c ỹ2,c ),
w̃c =
q
2
.
2
2
,
K̃c = (c2 + 2nt̃2,c )(2nν 2 + 4θν 3 ) + 4n2 ỹ2,c
2nỹ2,c
−µt̃1,c − n log ν − 2θν t̃2,c sgn c − α ,
8
p
α = µ/ 1 + θ/n,
where ν = (1 − 2t̃2,c )−1 , t̃1,c = −µ + cỹ2,c , t̃2,c = −(ct̃1c )/(2nỹ2,c ), and, with a3 = c4 + 2nc2 +
n2 ,
a2 = −2c3 µ − 2cnµ, a1 = c2 µ2 − nc2 − n2 − θn, a0 = cnµ,
c2 =
a2
,
a3
c1 =
a1
,
a3
c0 =
a0
,
a3
1
1
q = c1 − c22 ,
3
9
√
1
1
r = (c1 c2 − 3c0 ) − c32 , m = q 3 + r2 , and s1,2 = (r ± m)1/3 ,
6
27
p
p
c2
ỹ2,c = −4q cos cos−1 r/ −q 3 /3 − ,
3
and it is easily seen that A1 = 3α(n + 2θ)/(4(n + θ)2 ). With these, approximation (13) can be
computed. For illustration, we choose the set of parameters (µ, θ) = (−10, 10), which corresponds
to a heavily left-skewed distribution. For n ∈ {2, 4, 8, 16}, Figure 3 displays the accuracy of (13).
The exact values (the solid line in the graph) are obtained from the relationship
In (c) = cFn (c) −
Z
c
Fn (x̄)dx̄,
c < 0,
(14)
−∞
which follows from (6) upon integration by parts, and where the exact distribution function Fn
is computed via the algorithm described in Reeve (1986). The accuracy of the approximation
clearly improves with larger n, as expected, but is already quite good for n as low as 2 (note
that for n = 1, the doubly noncentral t does not possess a mean, and thus the expected shortfall
is not defined). When the t distribution is fitted to actual financial returns data, the degrees
of freedom parameter is typically estimated between 4 and 8. The approximation is certainly
accurate enough in this range for all practical purposes.
9
n=2
n=4
−4
−4
−6
−6
−8
−8
−12
E[T|T<tq]
E[T|T<tq]
−10
−14
−16
−18
−10
−12
−14
−20
−24
0
−16
Exact
SPA
−22
0.2
0.4
0.6
0.8
−18
0
1
0.2
0.4
q
0.6
0.8
1
0.6
0.8
1
q
n=8
n=16
−7
−8
−8
−9
−9
−10
−11
−11
E[T|T<tq]
E[T|T<tq]
−10
−12
−13
−12
−13
−14
−14
−15
−15
−16
−17
0
0.2
0.4
0.6
0.8
−16
0
1
q
0.2
0.4
q
Figure 3: Approximations to expected shortfall of a doubly noncentral t with µ = −10, θ = 10, and n = 2, 4, 8, 16,
from top left to bottom right.
10
References
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Value at Risk,” Economic Notes, 31, 379–388. 1
——— (2002b): “On the coherence of Expected Shortfall,” Journal of Banking and Finance, 26,
1487–1503. 1
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Mathematical Finance, 9, 203–228. 1
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Volatility Modelling,” Scandinavian Journal of Statistics, 24, 1–13. 5
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Springer, 2nd ed. 1
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631–650. 1, 2
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11
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A
Proof of (8)
Replacing fn (x̄) in (6) with its second order saddlepoint approximation and applying Theorem 1
yields
In (c) =
=
=
=
Z
c
x̄fn (x̄)dx̄
−∞
r
n
2π
Z
r
n
2π
Z
r
n
2π
Z
ŵc
−∞
ŵc
−∞
ŵc
h
i
x̄
a1
2
q
+ O n−2 dx̄
e−nŵ /2 1 +
n
Kx′′(t̂)
h
i
x̄
dx̄
a1
2
q
+ O n−2 dŵ
e−nŵ /2 1 +
dŵ
n
Kx′′(t̂)
e−nŵ
2 /2
Ψn (ŵ)dŵ,
−∞
where
h
i
a1
x̄
dx̄
q
1+
+ O n−2
dŵ
n
Kx′′(t̂)
i
ŵx̄ h
a1
=
+ O n−2
1+
û
n
ψ1 (ŵ)
= ψ0 (ŵ) +
+ O n−2 ,
n
Ψn (ŵ) ≡
with
ŵx̄
and
ψ0 (ŵ) ≡
û
5 2
ŵx̄ 1
λ̂4 − λ̂3 .
ψ1 (ŵ) ≡
û 8
24
12
(15)
Equation (15) is in the form of Theorem 1. The coefficients in the expansion evaluate to
B0 (ŵc ) =
µ
c
−
ŵc ûc
and
d
B0 (ŵc )
ŵc B1 (ŵc ) = A1 − ψ1 (ŵc ) +
dŵc
µ
5
1 dc
c d
ŵc c 1
λ̂4 − λ̂23 − 2 −
+ 2
ûc
=−
ûc 8
24
ŵc
ûc dŵc ûc dŵc
ŵc c 1
5 2
µ
ŵc
c ŵc 1
=−
λ̂4 − λ̂3 − 2 −
+ ŵc λ̂3
+
ûc 8
24
ŵc
2
t̂c ûc û2c ûc
5
µ
c 1
1
c
cλ̂3
λ̂4 − λ̂23 − 3 −
⇔ B1 (ŵc ) = −
+ 3 + 2.
ûc 8
24
ŵc
û
2û
t̂c ûc
c
c
B
Proof of (12)
Replacing fn (y1 ) in (10) with the first derivative of the appropriate tail probability approximation
and applying Theorem 1 yields
In (c) =
Z
c
y1 fn (y1 )dy1
−∞
=
r
n
2π
=
r
Z
n
2π
=
r
Z
n
2π
Z
c
−∞
w̃c
−∞
w̃c
1 ũ
y1 −nw̃2 /2
1/2 d 1
−1
p e
1−
+ K̃
+O n
dy1
n w̃3
dy1 ũ
K̃
1 ũ
dy1 y1 −nw̃2 /2
1/2 d 1
−1
p e
1−
+ K̃
+O n
dw̃
dw̃ K̃
n w̃3
dy1 ũ
e−nw̃
2 /2
Ψn (w̃)dw̃,
−∞
where
with
dy1 y1
1 ũ
1/2 d 1
−1
p
+ K̃
+O n
Ψn (w̃) ≡
1−
dw̃ K̃
n w̃3
dy1 ũ
w̃y1
1 ũ
1/2 d 1
−1
=
+ K̃
1−
+O n
ũ
n w̃3
dy1 ũ
ψ1 (w̃)
+ O n−1 ,
= ψ0 (w̃) +
n
ψ0 (w̃) ≡
w̃y1
and
ũ
− ψ1 (w̃) ≡
13
y1
w̃y1 K̃ 1/2 d 1
+
.
w̃2
ũ
dy1 ũ
(16)
Again, Equation (16) is in the form of Theorem 1, with the coefficients in the expansion given by
B0 (w̃c ) =
c
A0
−
w̃c
ũc
and
w̃c B1 (w̃c ) = A1 − ψ1 (w̃c ) +
d
B0 (w̃c )
dw̃c
1/2
= A1 +
c
w̃c cK̃c
+
w̃c2
ũc
1/2
c
w̃c cK̃c
= A1 + 2 +
w̃c
ũc
d 1
d A0
c
+
−
dc ũc dw̃c w̃c
ũc
d 1
A0
d c
− 2−
dc ũc w̃c
dw̃c ũc
1/2
d 1
A0
dc d c
− 2−
dc ũc w̃c
dw̃c dc ũc
1/2
1/2 A0 w̃c K̃c
d 1
w̃c cK̃c d 1
1
c
− 2−
+c
= A1 + 2 +
w̃c
ũc
dc ũc w̃c
ũc
ũc
dc ũc
= A1 +
w̃c cK̃c
c
+
2
w̃c
ũc
1/2
A0 w̃c K̃c
c
= A1 + 2 − 2 −
w̃c
w̃c
ũ2c
.
14