Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2005,20(4):407415
A COMPARI
SON BETWEEN TWO KI
NDSOFBMSBASED
ON DI
FFERENTCREDI
BI
LI
TY
Wa
ngYi
xua
n ZhouShuz
i
.Thepa
Abs
t
r
ac
t
pe
rde
ve
l
opsade
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i
gnofopt
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ma
lBonus
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usSys
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e
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orf
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he f
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om c
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omobi
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d:2003MR Subj
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Vol
.[7]de
pol
i
c
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s
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gne
danopt
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malBMSf
r
om al
i
ne
arc
r
e
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l
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ai
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om c
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t
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ome
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r
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ght
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e
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i
c
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ore
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e
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l
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e
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l
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t
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nt
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t
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r
e
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l
ai
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hes
e
ve
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i
t
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ac
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ons
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e
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ti
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ume
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i
c
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pe
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e
ve
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t
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l
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ode
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t
ht
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hes
e
ve
r
i
t
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nt
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e
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i
mat
or
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l
ai
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numbe
rand c
l
ai
m s
e
ve
r
i
t
y ar
ee
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tc
r
e
di
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l
i
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s
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onGammaast
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l
ai
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f
r
e
que
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y di
s
t
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on and Expone
nt
i
al
Gammaast
hec
l
ai
m s
e
ve
r
i
t
y di
s
t
r
i
but
i
on.The
.I
pur
pos
eofe
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t
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ec
r
e
di
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l
i
t
ypr
e
mi
um i
st
omi
ni
mi
z
et
hei
ne
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t
i
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st
opol
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c
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s
ti
s
di
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f
e
r
e
ntf
r
om t
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adi
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onalBMS,butbot
hoft
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et
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ndsofBMSr
e
war
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i
z
e
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i
c
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rbyi
nt
e
gr
at
i
ngapr
i
or
iandapos
t
e
r
i
or
ii
nf
or
mat
i
on.
I
n Y2,we pr
e
s
e
nt t
he mai
nr
e
s
ul
t
s of s
quar
e
de
r
r
or c
r
e
di
bi
l
i
t
y and e
qui
t
abl
e
c
r
e
di
bi
l
i
t
y.I
nYZ,wes
e
tnot
i
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s
umpt
i
onsont
het
woc
ompone
nt
sc
ons
i
de
r
e
di
n
,andwes
.I
BMS.I
nY[,aBMSi
sde
s
i
gne
dunde
rc
e
r
t
ai
nc
ondi
t
i
ons
how i
ti
sopt
i
mal
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Y\,wec
ompar
et
woBMSsbas
e
dondi
f
f
e
r
e
ntc
r
e
di
bi
l
i
t
yt
he
or
i
e
st
hr
ough anume
r
i
c
al
e
xampl
e.
Y] ^_‘
abc
defg
hs
s
umet
hatt
het
ot
alc
l
ai
msofapol
i
c
yhol
de
ri
ni
t
h pol
i
c
y pe
r
i
od ar
ear
andom
,i
= 1,2,...,whe
edi
s
t
r
i
but
i
onde
pe
ndsonk
r
ekmaybeve
c
t
or
val
ue
d.
var
i
abl
ejiwhos
,anddi
Eac
hpol
i
c
yhol
de
rhasc
ons
t
antbutunknownunde
r
l
yi
ngr
i
s
k,orunknownk
f
f
e
r
e
nt
.hs
(c
)and
pol
i
c
yhol
de
r
smayhavedi
f
f
e
r
e
ntk
s
umet
hatjii
si
nde
pe
nde
nt
ondi
t
i
onalonk
i
de
nt
i
c
al
l
ydi
s
t
r
i
but
e
d.le
tusas
s
umef
ort
hes
t
r
uc
t
ur
ef
unc
t
i
onofkt
hatkhasapr
obabi
l
i
t
y
(de
).
ns
i
t
y)f
unc
t
i
onoff
or
m m(k
p
,hi
.row
noragi
ve
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i
c
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r
sc
l
ai
me
xpe
r
i
e
nc
eoq=(o1,...,oq)i
nt
hepas
tqye
ar
s
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al
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ul
at
eac
r
e
di
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l
i
t
ye
s
t
i
mat
ors
jqt 1 f
orjqt 1 i
n pe
r
i
odqt 1.ubvi
ous
l
y,unde
rge
ne
r
al
.
WangYi
xuan,e
tal
A COMPARI
SON BETWEEN TWO KI
NDSOFBMSBASED
ON DI
FFERENT CREDI
BI
LI
TY
409
→
c
r
e
di
bi
l
i
t
yt
he
or
y,^
Z t+ 1 s
houl
dbear
e
al
val
ue
df
unc
t
i
ononzt.
,
I
foneknowst
heval
ueofτt
hatde
t
e
r
mi
ne
st
hec
l
ai
m di
s
t
r
i
but
i
onofapol
i
c
yhol
de
r
)woul
+ 1,andi
t
he
nE(Zt+ 1|τ
dbet
hemos
te
qui
t
abl
epr
e
mi
um f
orpe
r
i
odt
ti
st
het
r
ue
.Le
)de
),µde
pr
e
mi
um f
ort
hegi
ve
npol
i
c
yhol
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ri
nt
hi
spol
i
c
yye
ar
tµ(τ
not
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not
e
E(Zt+ 1)andPt+ 1 de
not
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ve
nc
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ge
dpr
e
mi
um.Thei
ne
qui
t
yofPt+ 1 i
sme
as
ur
e
d
).A ge
r
e
l
at
i
ve
l
yt
ot
hi
smos
te
qui
t
abl
eandt
r
uepr
e
mi
um µ(τ
ne
r
alpr
oc
e
dur
eofpr
i
c
i
ngi
s
t
os
e
l
e
c
t an appr
opr
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at
el
os
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unc
t
i
on U and t
he
n t
oc
hoos
e^
Zt+ 1 t
o mi
ni
mi
z
e
^ t+ 1 i
))].A c
E[U(Pt+ 1,µ(τ
r
e
di
bi
l
i
t
ye
s
t
i
mat
orZ
sc
al
l
e
de
xac
ti
fi
ti
sal
i
ne
arf
unc
t
i
onof
→
.
t
hec
l
ai
m dat
aztand,e
qual
st
hel
i
ne
arc
r
e
di
bi
l
i
t
ye
s
t
i
mat
or
,t
,
I
nt
hec
l
as
s
i
c
alt
he
or
y,t
hel
os
sf
unc
t
i
onU i
st
ake
nt
obes
quar
e
de
r
r
or
hati
s
))= (Pt+ 1 - µ(τ
))2.
U1(Pt+ 1,µ(τ
(2.1)
Ther
e
s
ul
t
i
ngc
r
e
di
bi
l
i
t
ypr
e
mi
um mi
ni
mi
z
i
ngE[U1]i
st
hepos
t
e
r
i
ore
xpe
c
t
e
dval
ueoft
he
c
ondi
t
i
onalme
an
→
^
)π(τ
|zt)dτ
.
Z t+ 1 = E(Zt+ 1|τ
∫
[4,5]
J
e
we
l
l
(2.2)
ve
r
i
f
i
e
d:Us
i
ngs
quar
e
de
r
r
orc
r
e
di
bi
l
i
t
y,byas
s
umi
ng t
hatZi i
sc
ondi
t
i
onal
l
y
,e
i
nde
pe
nde
nt and i
de
nt
i
c
al
l
y di
s
t
r
i
but
e
d,unde
rc
e
r
t
ai
n r
e
gul
ar
i
t
y c
ondi
t
i
ons
xac
t
c
r
e
di
bi
l
i
t
yoc
c
ur
sf
orpr
obabi
l
i
t
ydi
s
t
r
i
but
i
onsf
r
om t
hel
i
ne
are
xpone
nt
i
alf
ami
l
ywhe
none
,i
,wec
us
e
st
hepr
i
orc
onj
ugat
e.Thati
s
nc
e
r
t
ai
nc
as
e
s
anobt
ai
ne
xpl
i
c
i
tl
i
ne
arf
or
mul
asf
or
t
heopt
i
malpr
e
mi
um.Theopt
i
male
s
t
i
mat
orhast
hef
or
m
-,
^
Z = (1- w )µ+ w z
t
+1
1
1
(2.3)
t
-= 1 z,
-i
,z
whe
r
ez
st
hes
ampl
eme
an,t
hati
s
i
tΣ
i
=1
w1 =
k=
t
,
t+ k
E[Var
(Zt+ 1|τ
)]
.
Var
[µ(τ
)]
(2.4)
(2.5)
(2.6)
Thef
ol
l
owi
ngt
he
or
ybe
l
ongst
o[9].
[9]us
e
dt
hef
ami
l
yofe
nt
r
opyl
os
sf
unc
t
i
onsi
nt
hepl
ac
eoft
r
adi
t
i
onals
quar
e
de
r
r
or
.Theobj
.The
l
os
sf
unc
t
i
ons
e
c
t
i
vei
sme
as
ur
i
ngandmi
ni
mi
z
i
ngunf
ai
r
ne
s
s
ys
howe
dt
he
Pt+ 1
,andt
he
µ(τ
)
,l
f
unc
t
i
ongs
houl
dbec
onve
xands
at
i
s
f
yg(1)=0.Thati
s
os
si
se
xpr
e
s
s
e
dasaf
unc
t
i
onof
))=µ(τ
)g(r
),whe
appr
opr
i
at
el
os
sf
unc
t
i
onsoft
hef
or
m U2(Pt+ 1,µ(τ
r
er=
t
he r
e
l
at
i
ve di
f
f
e
r
e
nc
e we
i
ght
e
d by t
he t
r
ue pr
e
mi
um.The pr
e
mi
um de
r
i
ve
df
r
om
mi
ni
mi
z
i
ngE[U2]i
sc
al
l
e
de
qui
t
abl
ec
r
e
di
bi
l
i
t
ypr
e
mi
um.
2
)=r-1,t
I
fg(r
he
nt
heopt
i
male
s
t
i
mat
ormi
ni
mi
z
i
ngE[U2]i
s
^
Z t+ 1 =
µ
→ [
(Eτ|z
µ(τ
)- 1]- 1).
t
→ [
E(Eτ|z
µ(τ
)- 1]- 1)
t
(2.7)
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
410
.20,No.4
Vol
Suppos
eZ|τi
sdi
s
t
r
i
but
e
dac
c
or
di
ngt
oadi
s
t
r
i
but
i
onf
r
om al
i
ne
are
xpone
nt
i
alf
ami
l
y.
Spe
c
i
f
i
c
al
l
y,t
hepdfofZ|τi
soft
hef
or
m
f(z|τ
)=
- zτ
p(z)e
,z≥ 0,τ∈ (τ
τ
,- ∞ ≤ τ
0,
1)
0< τ
1 ≤+ ∞.
q(τ
)
)= Thec
ondi
t
i
onalme
anofZ|τi
sgi
ve
nbyµ(τ
q'
(τ
)
,t
henat
ur
alpr
i
orc
onj
ugat
eofτ
q(τ
)
hast
hef
or
m
π(τ
)=
- µkτ
{q(τ
)}- ke
,τ
0 < τ< τ
1.
c
(µ,k)
(2.8)
(µ,k)i
Fors
omeµandk>0,t
heval
uec
sanor
mal
i
z
i
ngc
ons
t
antf
orgi
ve
nval
ue
sofµand
k.
,as
=π(τ
Toobt
ai
ne
xac
tc
r
e
di
bi
l
i
t
yf
ort
hee
qui
t
abl
ee
s
t
i
mat
or
s
umet
hatπ(τ
and
0)
1),
=π(τ
,whe
)= (µ(τ
))- 1.
π(τ
v(τ
v(τ
r
ev(τ
0)
0)
1)
1)
[5]
)}i
The
or
e
m 2.1 .Suppos
et
hat{f(z|τ
sal
i
ne
are
xpone
nt
i
alf
ami
l
yandt
henat
ur
alpr
i
or
c
onj
ugat
es
at
i
s
f
i
e
st
her
e
gul
ar
i
t
yc
ondi
t
i
onsoni
t
sboundar
ygi
ve
nabove.I
fvs
at
i
s
f
i
e
st
he
= av'f
di
f
f
e
r
e
nt
i
ale
quat
i
onv"
ors
omec
ons
t
anta,t
he
nt
hee
qui
t
abl
ec
r
e
di
bi
l
i
t
ye
s
t
i
mat
or
.Spe
mi
ni
mi
z
i
ngE[U2]i
se
xac
t
c
i
f
i
c
al
l
y,^
Z t+ 1=(1-w2)µ+w2z,whe
r
e
w2 =
t
.
t+ k- a/µ
st uov
wv
x
oywyzw{
{
|m}v
x
oy{
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ve
l
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c
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hef
r
e
que
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l
ai
msands
e
ve
r
i
t
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ac
h
c
l
ai
m ar
ec
ons
i
de
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e
d.Fur
t
he
r mor
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s
umpt
i
on a):The numbe
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l
ai
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h
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c
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pe
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r
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oge
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c
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t
antbut
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une
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nt
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umet
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ype
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nt
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d,i= 1,
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umpt
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s
umpt
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onc):Thec
l
ai
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ve
nt
hepar
ame
t
e
rθ
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t
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i
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or
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),andt
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Poi
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ur
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i
on'
sas
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umpt
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oni
sθ
o,
p(Xi= m)=
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)=
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θ
e
m!
m = 0,1,...,
βα α- 1 - βθ
θ e ,θ> 0,α> 0,β> 0.
Г(α)
(3.1)
(3.2)
)=α/β.
andwi
t
hme
anE(θ
+ 1)t
Toobt
ai
nc
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e
di
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l
i
t
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e
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um f
oragi
ve
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c
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ar
l
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l
ai
m numbe
rdat
axt= (x1,...,xt)dur
i
ngpas
ttye
ar
s
not
eby
^ t+ 1 t
.Le
)=E(Xt+ 1|θ
)whi
X
heopt
i
malc
l
ai
mf
r
e
que
nc
ye
s
t
i
mat
or
tµ(θ
c
hi
st
het
r
uec
l
ai
m
[6]
,and l
numbe
ri
n ne
xtye
ar
e
tµ= E(Xt+ 1).I
tc
an bepr
ove
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t
hatt
heunc
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t
i
onal
t
di
s
t
r
i
but
i
onofc
l
ai
m numbe
rwi
l
lbeNe
gat
i
veBi
nomi
al(α,β).Le
tM =
Σ x bethetotal
i
i
=1
.Thepos
numbe
rofc
l
ai
mst
hatapol
i
c
yhol
de
rhadi
ntye
ar
s
t
e
r
i
ors
t
r
uc
t
ur
eofθwi
t
hc
l
ai
m
→
).Soi
hi
s
t
or
yxti
sGamma(α+ M,β+ t
ti
sc
l
e
art
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ur
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nt
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nt
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us
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e
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i
t
at
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ame
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e
r
sofGamma,f
r
om αandβt
oα+M andβ
+tr
e
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pe
c
t
i
ve
l
y,andGammai
sac
onj
ugat
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ami
l
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orPoi
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i
ke
l
i
hood.
3.2 s
e
ve
r
i
t
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ac
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l
ai
m
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As
s
umewec
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ai
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ve
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t
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or
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c
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ve
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e
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os
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andom var
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t
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i
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umpt
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ei
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pe
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nt(c
ondi
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i
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i
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t
r
i
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i
on.
,i
ve
nt
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ame
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e
rλ
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s
t
r
i
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dac
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or
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o
As
s
umpt
i
on e):Thec
l
ai
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rYj,gi
(λ
),andt
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Expone
nt
i
al
hes
t
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uc
t
ur
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i
on'
sas
s
umpt
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oni
sλ
o
-λ
y
fY(y)= λ
e
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(3.3)
ξ
)=
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η ξ- 1 - ηy
λ e ,λ> 0,ξ> 0,η> 0,
Г(ξ)
(3.4)
)=ξ/η.
andt
heme
anE(λ
→
.De
i
ngpas
ttye
ar
s
not
eby
As
s
umewehaveal
lhi
ss
e
ve
r
i
t
ydat
ayM =(y1,...,yM )dur
^
.Le
(λ
)=E(Y|λ
)whi
YM+ 1 t
heopt
i
malc
l
ai
ms
e
ve
r
i
t
ye
s
t
i
mat
or
tφ
c
hi
st
het
r
uec
l
ai
ml
e
ve
l
[11]
,l
ofe
ac
hl
os
si
n ne
xtye
ar
e
tφ= E(Y).I
tc
an be pr
ove
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t
hatt
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t
i
onal
→
di
s
t
r
i
but
i
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l
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o(α,β).Thepos
t
e
r
i
ors
t
r
uc
t
ur
eofλwi
t
hc
l
ai
m hi
s
t
or
yyM i
s
M
r
eW =
Gamma(α+ M,β+ W ),whe
Σ y . So Gamma isa conjugate family for
j
j
=1
e
xpone
nt
i
all
i
ke
l
i
hood.
%& ’()*as
e
+o,e
-.i
t
a*l
ec
r
e
+i
*i
l
i
t
y
2
)=1-1,ands
/ewi
l
lde
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t
ht
hec
as
ei
nwhi
c
h0(1
ot
hel
os
sf
unc
t
i
oni
s
2(3,µ(4
))=
32
- µ(4
).
µ(4
)
(4.1)
5r
om The
or
e
m 2.1 wec
an s
e
et
hatunde
rc
e
r
t
ai
nc
ondi
t
i
onst
hes
quar
e
d6
e
r
r
or
c
r
e
di
bi
l
i
t
ye
s
t
i
mat
orande
qui
t
abl
ec
r
e
di
bi
l
i
t
ye
s
t
i
mat
orhavet
hes
amef
or
m,t
he
yar
eal
lan
af
f
i
nef
unc
t
i
on oft
hes
ampl
eme
an.Thedi
f
f
e
r
e
nc
ebe
t
we
e
nt
woe
s
t
i
mat
or
si
st
hatt
he
.
we
i
ght
smaybeune
qual
&.7 f
r
e
-.e
,c
ye
s
t
i
mat
or.,+e
re
-.i
t
a*l
ec
r
e
+i
*i
l
i
t
y
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
412
.20,No.4
Vol
),f
The
or
e
m 4.1.Unde
rAs
s
umpt
i
onsb)andc
orapol
i
c
yhol
de
rwhohasac
l
ai
mf
r
e
que
nc
y
→
,i
))=
dat
axt=(x1,...,xt)i
npas
ttye
ar
s
ft
hel
os
sf
unc
t
i
oni
st
ake
nasU(X,µ(θ
X2
-µ
µ(θ
)
(θ
),t
+1)t
he
nhi
sc
l
ai
m numbe
re
qui
t
abl
ec
r
e
di
bi
l
i
t
ye
s
t
i
mat
orf
or(t
hye
ari
s
^ t+ 1 =
X
α+ M
.
β+ t- β/α
(4.2)
.Not
～Poi
),i
=1,2,...,θ
～Gamma(α,β),i
Pr
oof
et
hatXi|θ
s
s
on(θ
ti
sanat
ur
alc
onj
ugat
e
δ
.µ(θ
)=θ
.Le
),t
=v'
,a=1.Thec
f
ami
l
yf
orXi|θ
tδ=-l
n(θ
he
nv(δ)=e andv"
ondi
t
i
onsi
n
^
The
or
e
m 2.1ar
es
at
i
s
f
i
e
d.Sot
hec
r
e
di
bi
l
i
t
ye
s
t
i
mat
ori
sXt+ 1 = (1- w2)µ+ w2x,whe
r
e
µ=
α- M
t
,x= ,k= β,w2 =
,
β
t
t+ k- a/µ
.
andt
her
e
s
ul
tf
ol
l
ows
.
Re
mar
ks
(1)I
,t
ft
hewe
i
ghti
ne
qui
t
abl
ec
r
e
di
bi
l
i
t
yt
os
ampl
eme
anw2 e
xi
s
t
s
he
ni
ti
sgr
e
at
e
r
t
hant
hec
or
r
e
s
pondi
ngwe
i
ghtw1 i
nt
r
adi
t
i
onalc
r
e
di
bi
l
i
t
yof(2.5).
:I
(2)I
l
ongt
o(0,1),andhavet
hef
ol
l
owi
ngc
onc
l
us
i
ons
f
fα>1,t
he
nw2 and1-w2 be
M/f> α/β,t
hemal
ust
of
r
e
que
nc
y unde
re
qui
t
abl
ec
r
e
di
bi
l
i
t
yi
smor
et
han i
tunde
r
s
quar
e
de
r
r
orc
r
e
di
bi
l
i
t
y;
=α/β,t
;
I
fM/t
hemal
ust
of
r
e
que
nc
yunde
rbot
hc
r
e
di
bi
l
i
t
yt
he
or
i
e
sar
ee
qual
< α/β,t
I
fM/t
hebonust
of
r
e
que
nc
yunde
re
qui
t
abl
ec
r
e
di
bi
l
i
t
yi
smor
et
hani
tunde
r
s
quar
e
de
r
r
orc
r
e
di
bi
l
i
t
y.
(3)I
fα=1,t
he
nw2=1.Thec
r
e
di
bi
l
i
t
ye
s
t
i
mat
ori
st
hes
ampl
eme
an,andi
ti
se
qualt
o
pr
i
oras
s
umpt
i
on.
(4)I
fα< 1,t
he
nt
he
r
emus
tbeoneofw2 and1- w2 whi
c
hi
sne
gat
i
ve,s
ow2 c
an
ne
i
t
he
rac
tasac
r
e
di
bi
l
i
t
yf
ac
t
ornorde
ve
l
opaBMS.
4.2 s
e
ve
r
i
t
ye
s
t
i
mat
orunde
re
qui
t
abl
ec
r
e
di
bi
l
i
t
y
The
or
e
m 4.2.Unde
rAs
s
umpt
i
onsd)ande),f
orapol
i
c
yhol
de
rwhohasac
l
ai
ms
e
ve
r
i
t
y
Y2
-φ
(λ
),t
he
n
φ
(λ
)
+1ye
hi
se
ac
hc
l
ai
ml
e
ve
le
s
t
i
mat
orunde
rbot
hc
r
e
di
bi
l
i
t
yt
he
or
i
e
sf
ort
arar
e
→
,i
(λ
))=
dat
ayM =(y1,...,yM )i
npas
ttye
ar
s
ft
hel
os
sf
unc
t
i
oni
sU(Y,φ
^
Y t+ 1 =
η+ W
.
M + ξ- 1
(4.3)
4.3 BMSunde
re
qui
t
abl
ec
r
e
di
bi
l
i
t
yt
hoe
r
y
:1.f
,t
,t
A BMSi
sc
al
l
e
dopt
i
mali
fi
ti
s
i
nanc
i
al
l
ybal
anc
e
df
ort
hei
ns
ur
e
r
hati
s
het
ot
al
;and2.f
,
amountofbonus
e
si
se
qualt
ot
het
ot
alamountofmal
us
e
s
ai
rf
ort
hepol
i
c
yhol
de
r
t
hati
se
ac
hpol
i
c
yhol
de
rpaysapr
e
mi
um pr
opor
t
i
onalt
ot
her
i
s
kt
hathei
mpos
e
st
ot
he
.
pool
The
or
e
m 4.3.As
s
umeAs
s
umpt
i
onsa)b)c)d)ande)hol
dandi
ft
oe
ve
r
yne
we
nt
e
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A COMPARz
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^ t+ 1^
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he
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e
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ul
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om (4.2)and(4.3)i
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um.
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Pr
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umpt
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he s
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or
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an
βη
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i
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ai
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1{6|}4pol
i
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e
eTabl
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bl
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~umbe
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a
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r
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s
{
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1
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2
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3
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Unde
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sar
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.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
414
.20,No.4
Vol
1.6049,β=15.8778.
(1)Be
c
aus
eα= 1.6049> 1,t
hec
ondi
t
i
onsofThe
or
e
m 4.1ar
es
at
i
s
f
i
e
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ai
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e
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i
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os
sf
unc
t
i
on (4.1)
(Tabl
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bl
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^
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t
+1
Ye
a
r
t
0
0
101.1
1
1
2
3
4
5
6
86.61
229.78
372.95
516.13
659.30
802.48
945.65
2
75.76
201.00
326.24
451.49
576.73
701.97
827.22
3
67.33
178.63
289.93
401.24
512.54
623.84
735.14
4
60.58
160.74
260.89
361.05
461.20
561.36
661.52
5
55.17
146.11
237.14
328.18
419.22
510.26
601.29
6
50.47
133.91
217.36
300.80
384.24
467.68
551.12
7
46.59
123.60
200.62
277.63
354.65
431.66
508.68
(2)I
fus
i
ngt
r
adi
t
i
onals
quar
e
de
r
r
orl
os
sf
unc
t
i
on(2.1),wec
anobt
ai
nanot
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rs
e
r
i
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s
ofe
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t
i
mat
or
s(Tabl
e3).
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bl
e3
^
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t
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a
r
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0
0
101.1
1
1
2
3
4
5
6
95.09
154.34
213.59
272.84
332.09
391.34
450.59
2
89.77
145.71
201.64
257.58
313.51
369.45
425.38
3
85.02
137.99
190.96
243.93
296.90
349.88
402.85
4
80.74
131.05
181.35
231.66
281.97
332.28
382.58
5
75.87
124.77
172.67
220.56
268.46
316.36
364.26
6
73.36
119.07
164.77
210.48
256.19
301.90
347.61
7
70.15
113.86
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201.28
244.99
288.70
332.41
.(1)Eas
Re
mar
ks
i
l
ys
e
e
nf
r
om Tabl
e2andTabl
e3,unde
rc
ondi
t
i
onsofThe
or
e
m 3t
he
e
qui
t
abl
ec
r
e
di
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l
i
t
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ampl
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or
mat
i
on.
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nt
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han t
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obonus
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bl
ei
nt
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.Thoughbot
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or
me
r
hTabl
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e3ar
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r
i
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r
om Tabl
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fal
l
owanc
ei
sf
or
.
WangYi
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tal
A COMPARI
SON BETWEEN TWO KI
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FFERENT CREDI
BI
LI
TY
415
,wec
bonus
hunge
r
anhar
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yobt
ai
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amei
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Re
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e
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s
1 Buhl
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l
si
nRi
s
kThe
or
y,Ne
w Yor
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but
i
onwi
t
har
e
gr
e
s
s
i
onc
ompone
nt
N Bul
l
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i
n,1989,19:1993 Fr
a
ngosN E,Vr
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ur
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N Bul
l
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t
i
n,2001,31:14 J
341.
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s
N Bul
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t
i
n,1974,8:776 Le
.Bonus
:Kl
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us Sys
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s
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348.
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