LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Dewe\
MA:?i;. r:s:.
FEB
26
1
;?M.
1976
DtvV£Y LIBRARY
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ON SUMS OF LOGNORMAL RANDOM VARIABLES*
by
E.
Barouch and Gordon M. Kaufman
WP 831-76
February 1976
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
MASS.i;iST.T£CH.
FEB
26
^.976
nJAiVEY LIBl^f.RY
ON SUMS OF LOGNORMAL RANDOM VARIABLES*
by
E.
Barouch and Gordon M. Kaufman
WP 831-76
February 1976
ABSTRACT
Approximations to the characteristic function of the lognormal
distribution are computed and used to calculate approximations
to theden/sity of sums of lognormal random variables.
* This research was supported by NSF grant no.
SIA 74-22773.
The authors thank their colleague Hung Cheng for a very fruitful
discussion.
fgm
On Sums of Lognormal Random Variables *
by
E.
1
.
Barouch and Gordon M. Kaufman
Introduction
The lognormal distribution has been used as a model for empirical
Aitcheson
data generating processes in a wide variety of disciplines.
and Brown
(Lintner
[
[
1
]
3 ])
cite over 100 applications.
In portfolio analysis
and in statistical studies of the deposition of mineral
resources (Barouch and Kaufman
[
2
]
and Uhler
[
4 ])
= K
sums X +...+X
of lognormal random variables (rvs) are of central interest.
Here we compute approximations to the density of a sum of N
mutually independent and identically distributed lognoirmal rvs.
As is
well known, the density of a sum of independent, identically distributed
rvs is given by the inverse Fourier (or LaPlace) transform of the Nth
power of the characteristic function, so we begin by studying the
characteristic function of a lognormal density.
We perform an asymptotic
analysis of it in its various regions for N and Var(X.) = a
2
large and
compute approximations to the density of the sum by transforming back.
We find that (a) for values of K larger than its mean, the
density of K is approximately a three parameter lognormal density
(cf.
(44));
(b)
for values of K near its mean, the densitv contains
both lognormal-like and normal-like components,
(cf.(45))-
(c)
for
values of K larger than order one but smaller than its mean, the density
is approximately a three parameter lognormal density,
(cf.(46)) and (d) for
values of K smaller than order one, the density is approximately lognormal
(cf.(47)).
072G74I
Tf~TK^authors thank their colleague Hung Cheng for a very fruitful discussion.
2.
2. 1
Lognormal Characteristic Function
Properties of the Characteristic Function and Approximations
The characteristic function G(y) of the lognormal distribution
is defined by
G(y) =
where y is complex with Imy
loss of generality.
/
exp{-iyx
<_
^^^S x}-^
(1)
has been set equal to
and y
without
The function G(y) is analytic everywhere in the
lower half of the complex y plane, and continuous from below for real y.
However, G(y) is not analytic near y = 0, and this greatly enhances the
difficulty of approximating G(y).
""ivx
An obvious way to attempt computation of G(y) is to expand e
in a Taylor series around zero and integrate term by term;
i.e.
to
.
In so doing, we are expanding G(y)
express G(y) in a moment series.
around a point at which G(y) is non-analytic.
Hence it is not at. all
surprising that this expansion fails in every respect.
The first few
terms of such an expansion cannot be looked upon as a "small y" approximation
to G(y) as the resulting polynomial is analytic while G(y)
T,,
Furthermore,
.
T/.n\
m
.
since the (n+l)st term
2 2,„
./.\nnian/<i
—
1
.
the series is (-i) y
this moment series is divergent for all y ^ 0.
This fact is not very useful, since for a
series is well approximated by e
—iy
density concentrated on the point
,
1.
j-e
,
Trying hard, the moment
series can be viewed as an asymptotic series provided that
small.
is not.
2
o
2
is very
small enough, the
the characteristic function of a
5-in,ri>0
Consider a point y =
G(y)
=
——
and rewrite G(y) as
oo
/ exp{-nx - i^x -
a/li^
-^log^x}^
As long as n remains positive we may expand e
(2)
^
2a
-i£x
in a power series,
since we are expanding G(y) around a point in its domain of analyticity.
Hence we may write
00
G(y) =
The limit n ^
I
j-=0
oo
j
-^^TT
^
—
/
exp{-nx - ^log^xlx^'^dx
a/27
(3)
2a
is not allowed after expanding since these two operations
do not commute and a study of G(y) based on (3) must be done with extreme
caution.
Each term in (3) can be derived from the first term, by
differentiating with respect to
n-
This is allowed since the series
is uniformly convergent.
Consequently, to construct a good approximation to G(y) we need
only study its behavior for y lying on the negative imaginary axis.
This
of course is not surprising, since any operation on G(y) can be performed
by deforming the contour of integration through the axis Imy < 0.
We now focus Our attention on
00
G(y) =
/
exp{-yx
a/27
rlog x]
for real positive y (namely y = -in, n
We further assume that a
2
^
2a
is large.
>
0,
and so henceforth
n
=
y)
Immediate properties of G(y) are:
(i)
G(0) = 1
lim G(y) =
(ii)
y->oo
|G(y)|
(iii)
1
<_
2
2
(iv)
-—
dy
2
,
,
3
= -e
G(ye
)
2
G
2
2a „.
2a
,
8y
From properties (iv) and (v)
2
G(y) rescales y by e
it is apparent that differentiating
,
2
If e
.
is large,
even if y is small, sufficient
differentiation moves G(y) to its asymptotic region.
For example if y = 0(1),
2
-r— is
constant times G(ye
a
).
Thus, when we wish to perform an- integration
°y
involving G(y), asjonptotic evaluation of G(y) may be necessary.
Let y ^
°°,
2
and log y/a
Change variables in G(y):
be large.
yx =
z
to obtain
G(y) =
exp{-
^log^y}
a/lt
Since
—
^-r-^
/
2a
exp{-z - -iylog^z}z^
2a
to (4) cannot come from z
~
0.
Thus,
—12
r-
log
z
the major contribution
is assumed large and positive,
a
1°8 y dz
z
is small compared with
2a
z,
and can be dropped.
This is of course a crude approximation.
improve it we expand exp{log y/a
—j log
1
2
z} in a power series.
To
As long as
2a'
2
>
the resulting term by term integration is uniformly convergent.
Equation (4) takes the form
^
^
1
_
j^ 2
j=0
a/27r
2a
j!
^
-"q
z
(5)
The integrals in (5) can be expressed in terms of derivatives of
the
r
function:
/V^log
,^"^logydz
z)2j
= r^'-^^a-^log y)
(6)
z
with
i^
> 0.
a
Then G(y) takes the form
—i— exp{- ^Wy}
G(y) =
c.^
I
(-
j=0
2a
-^) ^-^^ ^^^
^
(o~^log y)
(7)
^•
2a
This expansion is exact but not really useful for computation since
higher derivatives of the
r
function are very complicated.
use of the assumption that log y/a
2"
o
r
where
4^
'^(^
(a'^log y)- ^
-2
2
We now make
is large, and approximate
log y)r(a
-2
log y)
is the logarithmic derivative of the
r
(8)
function.
To justify
this approximation we write a table:
r(a)
r' (a)
= ilia)
r"(a) =
r'"(a) =
r(a)
{/(a) +
{ij/^Ca)
+
i|;'(a)}r(a)
3i;;(a)i|j'(a)
+
r""(a) = {/(a) + 6i>^ia)i>'(a) +
<J;"(a)}r(a)
44^(a)i|-"(o)
+
3ii;'^(a)
+ (p'"(a)}r(a)
with a = log y/a
series of
\J^(a)
2
If we substitute the leading term in the asymptotic
.
which is log a,
~
il/'Ca)
—
we can drop all derivatives
,
a
of
ij)
compared with
i(;,
and
"*
T
-
(a)
Substitution of
-^rCa).
[i|;(a)]
(8)
into (7) yields a summable series for G(y)
G(y)
1
exp{
:=:
o/StT
12-''
12-2
~
log
ylog y}r(a 'log y)exp{
2a^
(a
tJ;
y)
(9)
}
2o
For instance, in
We can compute corrections to any order desired.
order to obtain the next term we write
r^^^^a)
/ha)
=
r(a) +
j
(2j-l)i|;
'
(a)^^^"^(a)r(a)
(10)
which yields
1
G(y) i
.
—exp{
o-ZIt;
* (1
12-2
2l°8 y}'^(°
ij;(a)
(a
y)}
-^M>\^)] +o([r(^)]h
a
a
This expansion breaks down when
relative to
^
2a
2a
--i-^.(i£Y)[i
2a
12-2 log
log y)exp{
i^'
(a)
a
(11)
a
can no longer be neglected
and is particularly bad for a
(y
•''
~
1)
2
Also,
(9)
is valid for y > e°
y close to expi-ro
;
surprisingly,
works well for
}.
We have already argued why y ^
(ii) and
(9)
is not legitimate here.
(iii) are immediate for (9)^ and it satisfies
of approximation computed.
are immediate].
[If
(iv)
is obeyed,
To see this, we neglect ^'
,
(iv)
Properties
to the order
the rest of the derivatives
differentiate (9), and show
1
that the result equals -exp{-ra
2
2
times (9) with argument ye°
}
in place
Property (iv) takes the form
of y.
1
jlog y}r(a log
—12-2
exp{-
^
2a
a/lTi
2-1
(ya
X
12-2 log
y)exp{
log y + (yo
)
(a
y)}
2a'
2-1
-2
i|;(a
)
log y)
(12)
- (yo
= -
-1
4
)
^(a
-2
log y)4''(a
-2
log y)}
^^ Ho
-jn
exp{- -ijlog^y)
a/27
2a
yo^
~2,
.0.2,
+
2
^log y)exp{- -^^[^'(o
^og
y)
+
t-^— ]^
^°S y
2a
Approximating
1
r
exp{
r
/
2^^(o
1
- expi/
,
2
log y)
|2^ -2^
2^ (a log
.
]
-2
.,_
y)}[l -
\p(a
^=^-=^
4*'
log
y),
&-J-1-]
log y
2a
and neglecting
}
^
ii^
the LHS of (12) we obtain (after some simplification)
the identity
^
-
a
^ ^(a-^log
y)
=
l^Y
a
a
^1 -
liT'^(^>>
^
a
(1^)
Thus, equation (iv) is obeyed by (9) when we keep all large and
order
1
terms.
The complicated way in which the equation is obeyed is
due" to the rich structure of G(y)
integral representation
,
despite its "simple"
^
Next we study G(y) for small y.
It seems reasonable to expect
that for very small y, G(y) can be approximated by a polynomial composed
of the first few terms of its moment expansion.
done carefully:
However, it must be
since G(y) is not analytic in a neighborhood of y =
a meaningful small y approximation must possess this feature.
Furthermore,
since the asymptotic expansion of G(y) for large y does not exhibit an
additive polynomial, one must show how the polynomial disappears as y
increases.
3 2
We begin with y = 0(exp{-
-ja
})
since this is a relevant order
,
of y for sampling without replacement and proportional to random size
(cf.
[
Adding and subtracting
2]) and then generalize.
3
exp{-yx} write G(y) for y = 0(exp{-
2
-ra
})
- yx from
1
as
00
/ exp{-yx
G(y) =
2^°8 ^'^~^
2a
a/2T\
(15)
A
= 11 - y exp{-20 2,}
00
1
J
H
r
exp{
a/27
__1
2-,c,-z.
(e
rlog y}
-l+z)exp{
,
T
1
J
2a
2,
jlog z}z
-2
a
^
2a
Treating the above integral in the same way as (4), we obtain
CO
C(y) = 1 -
yexp{^
}+
—
^log y}
;:;:2sxp{
av2Tr
2o
I
-1
^)
{(
j=0
2a
-rj-'
(16)
X
[-4t
da^
for a = a
log y
/"(e-^-l+z)-!^]}
—
log^ y
dz
^
In (16), -2 < a < -1, and so the integral exists.
In fact,
this
integral is an integral representation of r(a) for a negative and
non-integer valued.
Given (16) the method used to compute an expansion
for large y can be used and we find that for y = 0(exp{-(in + -7)0 }) and
integer m, we can add and subtract a polynomial of the m
degree from
exp{-yx} to obtain
G(y) -
I
Szlll^^pih^ah +
j=0
J-
exp{- -i-log^ylp (l^)exp{- -^^ (l2M^)
-^
a/27
2a
a^
2o^
(17)
for
-m
>
°^^y
>
-(m+1)
a
This expansion possesses both of the features needed for it to be a
meaningful approximation to G(y) for small y; it is non- analytic at
y =
and has a polynomial piece.
However, it has two major
it is not defined at y = exp{-ma
defects:
2
}
for integer m,
and it appears as if G(y) is discontinuous at y = exp{-ma
2
}.
Hence
formula (17) can be regarded at best as an approximation of limited
Furthermore, it does not explain the relation between the
validity.
rising power of the polynomial term as y decreases and the lognormal
term.
Therefore further analysis is needed.
split the integral representation of G(y) into a sum I^ +
We
I
of two integrals defined by
I
(y)
=
^
expi-yx -
J
a/2?r
—^log xj-^
^
2a
and
(18)
T
^
/
19(7) =
^
If
—=1
av^
00
/
exp{-yx
J
1/y
^log xj—
—
2a
1
-,
2
idx
a^
10
Wh en y is sufficiently small, I^ is small compared with Land in the
= G(0) = 1.
Consequently, we first analyze
limit y
->
small.
Expanding exp{-yx} and integrating term by term we obtain a
0,
I^
I
for y
uniformly convergent series:
-4=
=
I, (y)
-^
^"0
=
V
J
2a
(-y)-*
rl-2
exp{yj a
.\
I
2,
}
.
f
erfc[
logCye^^
^'-^
^-
j=0
(19)
-2
•
CO
1
^
/^"^expf- -i^log'xlx^
I
a/27 j=0
),
]
^o
with
oo
erfc(z)
for
z
_>
0.
=
-^
/
2
},
with
erf (z)
(20)
terms for which
j
X
>
c(j - X)< 0,
(21a)
a(j - X)> 0,
(21b)
a(j - A)- 0.
(21c)
The argument in (19) is
term
Hi-
In (19) we must distinguish between three types of terms:
setting y = exp{-Xa
be one
exp{-t^}dt
—
a(j - A), hence there are a finite number of
^
- A < 0.
Since a
for which a(j-A)-
one such term).
2
is assumed large,
(when
j
- A = 0(a
Remaining terms are of type a(j-A)>
We replace erfc(*) in each term for which
j
),
there may
there is only
0.
- A >
by its asymp-
totic expansion, namely,
err>-(z) =
-^
^"^
exp{-z^}[l +
y
£=1
(_i)
^
il^ciUl]
(2z^)^
(22)
12
"
C-1)
-
Since
J
—r^—
[j
-2
+
^"
j=0
incomplete gamma function y
Ii (y)
-
-^
log y]
a
is a series representation of the
_2
(25) may be rewritten as
log y; 1),
(a
exp{-rj
I
I
J-
j=0
-1
a
+
}
exp{-
'^
—j^oS yH(^
lo8 YJ 1)
2a
o/ZtT
(26)
+
(-l)^"^-*-^^
1
,
exp{
(k+D!
-
-2
^
-2
/
—rlog
1
T
/2a
y))""*-]
-2
log y; 1) has a pole for a
-2
2
log y)
}
-1
(k+l)a^,
(k+l)a^-,
1 1
2,
^
^log ye
)}erfc(.
(ye^
)
2a
2a
when y = exp{-(k+l)a
I
,rl
(a/27(l + k + a"^log
Even though yia
of (1 + k + a
2
,
rlog y}[-exp{
the pole of y(a
so that
log y a negative integer,
log y; 1) is cancelled by that
is a legitimate representation of
(26)
(y) for all small y.
.
We now turn to y or
j
- A
>
for all
I^(y) =
1
j
and first consider y > 1.
1
except
1
'
f
exp(
>{
j
= 0,
--'-
2
1
p-log y},
2o
Then log (ye
^
(-1)^
^^^
I
.,
3=1
^'
—
1 T
log(y e^
log(ye^-)
ja^,,
,
.
2,
...-.JL_.,-2.-Ja\.,„.,.
^log (ye-J
)}erfc(, ^-^—
exp{
i/2a
2a
—
(27)
1
1
/
2
The approximation (26) to
>
)
so
.1
log
y
^
^-^)
-r€rfc(
+
2
,,
»^a
I
(y)
for small y was computed keeping only the
first term of the asymptotic series for erfc('); it is not clear a priori
that this leads to an accurate approximation to
/2a
z
> 0,
when y = 0(1), so
log y) with its series
we replace the asymptotic series for erfc(
expansion for small argument
I, (y)
13
_ 2
00
—
2j+l
r
erf(z) = e
J
erfc(z) =
,
- erf(z),
1
j=0 r(| + j)
and approximate
I
with
(y)
rlog y}
exp{
I. (y)
2a/2^
1
-
exp{- -^log^y}
77-
[j
+ a
log y]
^•
—
°°
9
1
+ i[l
—
)_
j=2
2a
I
1
2=0 r(| + j)
2a
2j+l
T
(12^^)
r^^
(28)
]
,^0
2
-
When a
2
-ry
exp{-;rti
}erfc(
ye
^log
is large the last term in
)
can be incorporated into the
(28)
first sum.
For y - 1, y < 1, and a
2
2
large enough so that log(ye^
00
I
(y)
-^log^y
= iexp{-
J
-
erfc(-
-^
>
0,
2
izil_ exp{
1
-
j=l
2a
+ |[2
j
)
log2(ygja
2
)
}erfc(-^log(yeJ^
2a
log y)
/2a
]
/2a
=
-^
exp{- -^log^y} f -^=if^"
2/2TTa
j=2
2a
1
+
hi
exp{-
°°
o
1
-
^log^y
^
1
,.1
2,
-
+ a'^log y]"^
1
1
j=0 r(| + j)
2a
- -p' expi^tJ
—
[j
.
1
}ei:fc(
/2a
2
.
log ye
a
.
)
1
(-
1°^)
(29)
2j+l
]
/2a
))
14
As y
-»
1
both (28) and (29) approach the same limit, namely.
,
1
lim I,(y) - i{l - exp{4j^}erfc(-2-) +
^
y-^1
a/27 j=2
/I
'^
Izili
y
(30)
=
i{l - exp{^^}erfc(-^) +
—^[1
/2
+ Ei(-l) - y]}
0/2^
where y = .57721 56649, Euler's constant and
Ei(-l) = - / e'^^t" dt - .21938 3934.
1
We conclude our discussion of G(y) with an asymptotic analysis
When y is large the principal contribution
of I^(y) as defined by (18).
to G(y)
is from I«(y)
and is of the form (11).
but when y - 1, I«(y) is of the same order as I (y)
is small,
Consider y -
T
and a
1
1
\
t
12^^) "
2
z
and a
1
^^P^
- 1 and so
i
z
^
2
Rewrite
large first.
1
1
r
1
2
2
,
^^ ^^
/•
J
-2
1-1
exp{-
1
2a
a/2Tr
When y -
When y is small, I„(y)
2
,
jlog z}z
-a
—
log y dz
^ "
,„t,
(31)
2a
is large the major contribution to I„(y)
comes from
we approximate
J_
. 1 _
z-z
(i_z) + (i_z)2
~
2.
(32)
If we replace I/2 with z in (31) when y - 1 and a
l2(y) -
,
Ij^(y~''')
and
I
- I
(y)
+
(y"""")
I
2
is large, we see that
where I^ is given by (28) and (29).
For y << 1 we write
„
lo^y) ^
2^°^ y^ / exp{-z
^^P''^
a/27
-2
"
2o
1
jios 2^^
2a
^
z
(^^^
15
When a
2-2
is large a
exp{o
}
<< z for 1
z
1
2a
small
2
>
2
—j log
we may expand exp{z
log
is
2
z};
£
for large a
o/27
£ exp{a
2
}
and in this region
the contribution from the region
-^ exp{- -^log^y}
In(y) =
z
2
and so we ignore it.
I
j=0
2a
i^(-
-^)
-i—
^*
2a
9a
Explicitly
r(a;l)
(34)
-
where for negative a, r(a;l) is the incomplete gamma function
'^^•^'^
-
lohr
^
^o'"'"
^''^
For a large and negative a standard steepest descent calculation yields
^^"'^^
with
9
vV
-e -a
"ITO^^ ^
= -(1 + a
—
H
)
~
1
a/2TT
+
"^/^
2
e
+|]
(36)
so that the leading term in an asjnnptotic
expansion of I„(y), y <<
12^7) -
f2
1
is
12
-2
,2^2
exp{
2
l°g y>[r(l - a
log y)
-1
'
]
2a
log y
^2
^2
1/2 - a"^log y
log y
(37)
2
2
-1/2
By differentiating (37) with respect to
computed if desired.
ot,
higher order terms may be
16
2. 2
Summary of Approximations to G(y)
The approximations to G(y) for a
(1)
«
For y
k
V
rl-2 2,
^
^^Pi^yJ
Z.
(-1)
,
*"
,
1
,
1
e^Pi-
a/27
k
[-^expi
—
,
-2,
,,
log y;l)
/Za
k+1
1
a/lTjlk+a
2
2a
)
+ io(y)
(38)
log y;l) is an incomplete gamma function whose singularities
at integer -k and -(k+1) are cancelled by the singularities of [k + a
+
1
]
log y]
/2a
]
[k
,
log y]
a/2TT'[k+l+a
and
—
?
{^li..p(-i5loS^ye"=*"° )}erfc(^los(ye "'«>''
+
-2
,
2a
2a
where y(a
2
-
olog y)Y(o
2
2
ka.,^,l^,ka-,
.1,2,
^log(ye
)}erfc(
))
2"1°S (y^
rl
^,
-k,
-f
^*
+
large computed in 2.1 are:
°
i
(-y)
—TT
j=0
~
r^/ N
G(y)
"
£ ^°^/
£
2
and -(k+1)
1
2
-2
+
a
-1
log y] respectively, and I^(y) is given by (37).
-2
-1
log y]
An
asymptotic evaluation of I„(y) when y is small (cf. formula [37]) shows
it to be small by comparison with I^ (y)
(2)
For y -
and y
1
>
1,
1
,1
2,
}
r
J.
,
erfc(
[y
I, (y)
given by (28),
I/y ^--J--^xp{- ^log^y}
G(y) ^ I,(y) +
- -rexp{:ra
with
1
^
log
T
1
- exp{-
2a
(3)
9
-^log^}
)
y
°°
I
y^li^-o'hog^Y
2
2
a,,l
-la,,
£/li
+
log ye )]
-erfc(
ye
a/2
+
I
(39)
a/2
-i— (1°^) 2j+l
1
T
j=0 r(|+j)
a/2
For y = 1.
G(l) - 1 - exp{|a^}erfc(— )
+^^—[1
+ Ei(-l) - y]
where y = .5772156649 and Ei(-l) = .219383934,
(40)
17
(4)
For y -
1
and y
<
1,
(39) with y replaced by y
and
I^ (y)
given
by (29).
(5)
For y >> 1,
1
G(y) -
exp{
2l°g y^r(a
log y)exp{
la
a-Zln
X [1
12-2
1-2
^
i|j'(a
12-2 log
2 ^
(^
v)
^
la
(41)
log y)(l - a
-2
-2
2
^
log y))]
{a
2a
Figure
1
displays the graph of G(y) for a
2
=3.0
and y =
an approximation to it using the above approximation formulae.
and
The solid
line is drawn through points computed to five digits accuarcy by numerical
integration of
(1)
18
lliJ,
19
3.
Inversion of [G(y)
N
The density h(K) of the sum of N lognormal random variables is
hm
=
^
I
exp{Ky}[G(y)]%.
(42)
The major contribution to this integral comes from the region Ky = 0(1).
Since G(y) has a different form for each of the regions y << 1, y - 1,
and y >>
the functional form of h(K) is different for differing orders
For each of the following cases N and a
of K.
K
(1)
1,
>
NM
.
1
a(y) + ^!^- exp{
of K.
= 1,
1
2
y^og y>b(y)
(43)
2a
a/l-n
with a(0)
are assumed large.
For this case we approximate G(y) by (38) and write
~'^'^1
G(y) - e
2
The number of terms in a(y) is determined by the magnitude
Approximating [G(y)]
N
by the first two terms of the binomial
expansion of G(y) expressed as (43),
M^y
[G(y)]^ - exp{-NM^y} [a^(y)
+—
av2TT
exp{-
^logS}b(y)a^"^(y)
]
2a
the integral (42) is approximated by a sum of two integrals.
the first, that with integrand exp{(K - NM )y}a (y)
,
If a(y)
vanishes; when
=
1
20
a(y) -
for y -
1
it is exponentially small.
Consequently we
approximate
h(K) ^
2^
-^—
.
exp{-
^log^(K
^
-
exp{(K - [N-l]M^)y -
/
-
-i-
K
(44)
[N-1]M^
-
b({K - [N-llM^r-'-)a^'-'-({K -
X
(y)dy
[N-1]M )}
2^
0^
-^log y}b(y)a"
[N-llM^}"""-)
Corrections to (44) can be computed in a straightforward fashion.
The approximation (44) to h(K) shows that for large K
h(K)
is lognonnal-like;
>
NM
,
the leading term of an asymptotic expansion
i.e.
of it is composed of a three parameter (y=0, a
2
,
(N-l)M
lognormal
)
density with argument K - (N-l)M^, times a correction.
(2)
K - NM
,
This case requires specification of the scales of N
In particular, for
and K - different scales give different answers.
5
2
2
N = 0(exp{2o }) and K = 0(exp{-^ })
exp{-yM
+
12
-jVy
}.
,
a(y) may be approximated by
Upon expanding [G(y)]
N
written as (43) and
integrating term by term we obtain
.In
f(K) -
2
(2Tra
2
)
N
b'\K)exp{-
N
2
^log
K}
o
N-1
+
M(-)(2ttV(N-j))
j=0
X
-\
„
{(2Tra
(K -
exp{
^
)
exp{-
— ^log
(N-j)M^)^
}
(45)
2(jj_.)v
^^^qy^^
}
b(|j^ _
(n:j)mJ^^
^
21
(3)
1
K
<
<
(N-l)M^
case (1) with (N-l)M
This case may be treated in the same way as
.
replacing K - (N-l)M
- K
-^^[(N-l)M^
h(K) ^
- K]~-^exp{-
:
^log^((N-l)M^
- K)
}
(46)
b([(N-l)M^ - K]"^ a^"^([(N-l)M^ - K]"^
(4)
K << 1.
h(K) -
Use the expansion (11) of G(y) in its asymptotic region:
Yj^
•
——
/
exp{Ky
2l°8 y
2 ^
^^
^°^ y)r(a
(-^)^exp{- -^log^K}^
0^/27
X
„
N
2a^
2.
(47)
^
;:r---(N-l)^N,
,
2,
^.
'r (-a log K)exp{
(a/2TT)
log y)dy
N
2a
r
,2,
i(j
log
K,
9")
(
a
,
^
References
[L]
Aitcheson, J. and J. A. C. Brown, The Lognormal Distribution
Cambridge University Press, 1957,
New York:
[2]
Barouch, E. and Kaufman, G., "A Probabilistic Model of Oil and
Gas Discovery", Proceedings of the IDASA Energy Conference
Laxenburg, Austria, May, 1975.
,
,
[3]
Lintner, J., "The Lognormality of Security Returns, Portfolio
Selection and Market Equilibrium", forthcoming, 1976.
[4]
Uhler, F.
"Production, Reserves, and Economic Return in
,
Petroleum Exploration and Reservoir Development Programs,
1975 (forthcoming)
^N
1
MAR'
-Jm
4
1983
T-J5
w
143
76
no.830-
(lescript
Manoh/A Parsi"iomouS
Kalwani.
D»BK^,,„,„,p,Qq,?,(^,64^
726382
71
1060 000 bSb
T-J5
w
143
no.831-
76
sums
Barouch, Evtan/On
726741
lognormal ra
of
D«BKS
POWflMi.,
-i||iii|i!iln|l|il!il|ll!|f|l|ll!
""l" ll'lll
'"I
III
000 hSh 022
TOfiO
H028.M414 no832- 76
Cathe/The retraining programs
Lindsav.
"'TiT'iliiiii'
TDflD
3
000 hSS ISa
H028.IVI414 no.833- 76
Piskor, Wladys/Bibliographic survey of
D»BKS'
726744
II
3
1
iiiiMiiii
1
!|iiiii
TDfiO
III
III
00020496
III
II!
I'll
1
1
if:
'II
II
000 bS4 3TT
H028.M414 no.834Alvin
Silk,
731050...
3
76
J./Pre-test market evaluat
.0*BKS....
TOflO
.
.000^542
000 AMD ETS