APPROXIMA nON OF T ~ PROBAB~ITIES USING THE
G~)-TRANSFORM
by
J.McWilliams.,J.Balusek.,andH L. Gray+
.Stephen F. Austin University, NacogdochesTexas
+Southem MedIodist University, Dallas Texas
Abstract: In Gray and Wang (1991) the detemtinistic version of the generalizedjackknife,
referredto as the G~") transfonn, was shown to be a powerful tool for obtaining simple
approximationsfunctions for tail probabilities of mostpdfs. Theseapproximationfunctions are
highly accuratein the tails of the distribution and are all of the form.f{x )R(x), wheref is thepdf and
R is a rational function. Gray and Wang were only able to give R(x) for G~I) for n = 1,2,3 due to
the extensivealgebrarequired. Even so theseapproximationsyielded relative errors typically in
the I 0-5range. In this paper we review the generalizedjackknife theory for this application and
make usethe computer algebraprogramsMaple and Madtematicato obtain approximation
functions.f{x)R(x) for n = 1,2...7 (up to n = lOin dIe normal case).The resulting approximation
functions have relative errors typically in the 10-10range and in somecases10-20or better. Thus,
for most practical purposesone can considertheseapproximationsas good as closed form
solutionsin the tails of the distributions.
Keywords: Tail probabilities, Generalizedjackknife, Rational approximations,Gmtransfonn.
Introduction
Gray and Wang (1991) introducedthe G~")-transfonnationas a generalmethodfor obtaining
approximatingfimctions for tail probabilities. The G~") approximationfunctions have the Wlusual
property that they actually improve in the tails. The needfor approximationfimctions that are
highly accuratein the extremetails arisesin a number of areasof statistics. For example,in
clustering problems, it is often necessmyto computeextremely small probabilities becauseof the
large number of ways, (2n+l -I),
to separaten + 2 points into two groups. h1reliability the need
for such probabilities can arise from the desirefor a highly reliable componentthat dependson
other componentsin series~seeGood (1986).
Whell viewed in the more gelleral setting as introducedin Gray (1988), the G~m)-transform
is
actually a "generalizedjackknife." Moreover, in Gray and Wang (1991), it was demonstratedthat
these')ackknife approximations" are more accuratethan other existing approximationfunctions.
Additionally, the approximationsare all of the form j{x )R(x), wheref is the pdf and R is a rational
function. Unfortunately, the function R(x) is not easily obtainedfor large valuesof m or n due to
extensivealgebrarequired In Gray and Wang (1991), thesefunctions were only given for m = 1
and n S 3. Even so, they achievedrelative errors that are typically in the range 10-'.
In this paper, we review the necessarygeneralizedjackknife theory for this application and
make use of the computer algebraprogram in Mathematica to obtain j{x )R(x) for valuesof n S 7
(up to lOin the normal case). The resulting approximationswith relative errors generallyin the
range 10-10and in someinstances10-20.
Finally, in the appendix,we include tables giVing the actual approximationfunctions for n S 6
(10 in the normal case). Additionally, we include the code that can be run in Mathematica to give
more extensiveapproximations.Such approximationsare also availableby writing the authors.
TheGeneralized
Jackknife
Let 91, 92, . ",
9k+l
be a collectionof estimatorsandlet Cjbe a setof constants
such d1at
..,
1+1
(1)
E[9j -c~ = Latfbi(9),j = 1, 2,
1-1
where c} = 1. the constantsaij are given and the bi(O) are unknown functions
orB. The kth order generalizedjackknife is defined as follows:
where
H1-tol
(zJ;a,) =
%1
%2
Zfil
all
Q12
Ql,k+l
Qt.1 Qi,2 ..
ak,io+l
In (2), the gmeralized jackknife is defined as an extensionof the original jackknife definition
given in Schucany,Gray and Owm (1971). The extensiondefined in (2) has beendemonstrated
as having significant value in numerical approximationin Gray (1988). If atjb;(O)
= 0 for i
> kin
equatioo(1). then taking the expectedvalue of both sidesofequati on (2) revealsthat
G(81. 82. "'. 8k+l;atj) is an unbiasedestimatorforO. Ifatjbl(O) * 0 fori > k. the generalized
2
jackknife is not unbiased,however, under generalconditionsit is of lower order bias.
The generalizedjackknife is well-known as a methodfor reducing bias in estimators. Not as
well-known is the fact that the generalizedjackknife can provide a way of reducing error in
numerical approximations;seeGray (1991). To elaborate,let OJbe an estimatorwhich has its
probability massconcentratedat a single point. This kind of point estimatoris more commonly
known as an approximationand E[9j - eJ=Oj - e, so the bias in this caseis the numerical error.
When the generalizedjackknife is appliedto a collection of approximations,it can be used as an
approachfor reducing error on thesenumerical approximations. This method of using the
generalizedjackknife as an approachto finding numerical approximationsis discussedin Gray
(1988). In addition, Gray pointed out that such well known numerical methodsas Simpson's rule,
trapezoidal rule, Romberg integration,LagrangeInterpolation, etc. can be viewed as generalized
jackknives.
Let f be a pM, and let
(4:
J{t) > 0 for t:$ Q, as well as t > Q. Now let e(x) = S-F(x)
and
00
(5)
Uk(X) =
x1k~~
;-ox'
where Ik is an integer suchthat Ik ::5k and ak,O* O. Supposefurther that m is
the smallestpossibleinteger such that the differential equation
Um(x)£(m) + Um-l(X)£(m-l)+
+ Ul(X)e' -e = 0,
is satisfiedby E(x) for some set of Uk'S. The a k,i in (6) neednotbe known.
For example,suppose
al.i
= a2,i = ... = am,i = Ofori?:
, nm, respectively,the sum in (5)
I:: 7- = I:: ak,ixlj-i, and the differential equation(6) becomes
nj-I
becomes Xli
nl,n2,
PO
nj-1
;..()
3
&(x)= 0
Then
(7)
and
(8)
(k = 0, 1,2,. ,N -l;N
WI
= }:ni).
tel
Equations(7) and (8) defIne a systemof equationsof the sameform as (1) with
Cl = 1,
and Cj = 0 for 2 ~ j ~ N + 1, and with the aj,/ correspondingto the b,(9).
Clearly, this systemcan be solved for S and, therefore,the generalizedjackknife
defInedby equations(7) and (8) is exact~that is, it gives an exacttail probability
when applied to the N+l functionsFk)(X) (k = 0, 1,2, ..., N) ife(x) satisfies(6)
Thus ife(x) satisfies (6) for x?: a, then for any x?: a
..,
(9)
G[F(x),F(x),...,pOO(x)~a'J(x)]
= S. = fJ(x)dx
a
ifilie aiJ areproperlydefinedby (7) and(8). Now noteiliat (9) holdsfor x ?: a.
But F(a) = 0,sowe cantakex = a andno integrationis requiredin (9).
To be morespecific,supposethat
m = 1, nl = n, andx = a. ThenN = n anddenotingG(O,
F{x),...,F(">{x);aij(x» by
G~m)(f(x);aij(x», we have
4
G~l)Utx);aij(x»=
= [rl-t+l.f{x)](;-l)
Qtj(X)
Now assumethatJ{x) satisfies(6) for someset of Uk(X) definedby (5) and
,'j = 1" 2 ...) ,
Levin and Sidi (1981) showedthat there exists a~,i suchthat
(13)
where
"...1
(14)
Uk(x) = 1:a~,xI.-t.
~
should converge"super fasf' to S. This is the caseand it also is true for
m ~ 1. SeeLevin and Sidi (1981). Equation (13) leadsus to define the generalized
jackknife tail probability function as follows:
5
Definition 1: Let./{x) be a pdf, with infinite support and supposethat./{x) satisfies
dIe differential equationin (6) for somemand somecollection of Uk(X). Then we
define the G~m)-transformation
of..f{x) as dIe generalizedjackknife approximation
~
of f./{t)dt correspondingto (13); dIat is,
x
(15)
G~.)[I(x);a,(x)] = 6(0, .l(x), .
"
f--l)(x);a.-(x)],
where
Qtj(X)
= (rl-i+JJ(x»(;-I),; = 1, " n; j = I,
= (r%-l+ff+l.f(x}}(;-l), i = n + 1,
, mn + 1,
" 211; j =
= (X'_I+(_l)t+lf-l)(X}}C;-l), ; = (m - l}n + I,
..., mn + 1,
m
,
n
"
,
j = I, ..., mn + 1
It was demonstratedin Levin and Sidi (1981) that the foregoing assumprionscover a wide
classof integrands. In fact, it is difficult to think of a differ~tiable pdf that doesnot satisfy (or
approximatelysatisfy) (6) for someset of Uk(X) and somem. For instance,in the simple casein
which all of the Uk(X) are constants,that is, Uk = Ck,then for any m there exists an mth order
homogeneousdifferential equationwith coefficientsCk whose solution is givm by some linear
combinationof the elementsof S = {efJ.x,k = 1,2, ..', m}, where.8k = ak + ibk, which would
imply that iff can be representedby a Fourier seriesthe method should be effective. In fact, WIder
generaldifferentiability conditions,it has beenshown in Gray and Lewis (1971) that
wi'dl only non-positivepower terms.
6
Applications
In Gray and Wang (1991), the G~")-transformationwas usedto approximatetail probabilities
for the standardnormal, Studentt, Inverse Gaussian,PearsonType N and, the ratio of a X2 and a
log nonnal. In Gray and Lewis (1971), the Gammadistribution was usedto demonstratethe
power of anotherless generaljackknife method called the B n-transformationin getting
approximationsto tail probability. The B n-transformationonly assumesthat the pdf satisfiesan
mth order constantcoefficient homogeneouslinear differential equation.
In this sectionwe extendthe resultsof Gray and Wang for the G~l)-transfonnationfor larger
order transforms and introduce somenew examples. Hereafter we will use Gn to mean G~l).
ExampleI: TheNormalDistribution
The standardDOnna!pdfis./{x) = -!ii-e-t.iZ. Clearly,
x-If(x) - ./{x) = 0, and
therefore,./{x)satisfiesEquation6 with m = I. Then
U.(x)
=rl
~
1:::
7
= X'I
~ '.
= -1, andal,; = 0, for i > O,sincealO :#:0.
j8O
Then.for example
G. (x) = J{x)~,
G.(X)
~(,xI+21zA+1Mz2'" )
=J(X) (.zI+~+l~+lfk2+2A)
~+ltr+lI)
03 (X) =j(X) %6+1~+27x2+6'
,
G,
(x)
= .f(X)
.I(zi+~+~+l~.+fOO)
%lo+3Sz'+~~+127~+1~+120
Seethe Appendix for a completetable of G1 through Glo
The relative errors in estimatingthe tail probabilities for various valuesof x are
listed in Table 1 for G1(x) through G.(x).
8
TableI Relativeerrorsfor theG~I )-transformsof the 00Ima1
true
l!;«il)
-2 1.1507(1)
- 1.7(1)*
...6 I 5.47993(2) 19.~
2 12.27501(2) I 5.1(2)
3
1.3499(3) 11. S(2)
6 9.86588(10)
10 7.61985(24)I L ,}
z
E(G7.)
2.3(2)
5.1(3)
I 17.5{4)
,
3.2(4)
2.4(5)
11.6(6)
E(G3) E(6.)
E(G,)
3.0(4) 1.6(3) 9.3(4)
1.0(3) 5.9(4) 1.8(4)
6.0{4) 1.7(4) 3.2{5)
8.2(5) 7.7(6) 2.7(7)
5.0(7) 3.4{10) 3.5(10)
5.1(9) 2.6(11) 4.3(13)
E(G,)
3.9(4)
4.4(5)
3.7{6)
6.8{1)
1.3(11)
5.8(16)
£{6,)
1.4(4)
7.0(6)
2.7(7)
2.1{1)
5.7{15)
6.0(17)
£{G.)
4.0(5)
3.1(7)
3.5(7)
3.7{9)
3.1(14)
8.5(19)'
12 3.67097(51)
- 13.8(5) 11.6(7)
- 1.1(10) 3.6(13) 1.1(15) 2.0{11) 2.3{20) 9.9(21)
(8
1.1507(1)~
1.1507xl0-1)
Example 2; The GammaDistribution: .f(x) = ~;xII-.e'f"
Cleariy.!{x)satisfies(6) with m = 1;thatis. (ax-I - t)-lf
thusagain,m= 1 andU1(x) = (ax-l_t)-I.
(x) - .f(x) = O. and
In orderto identifythe/l parameter
in equation(15) we find theLaurmt seriesofU.(x) expandedaboutzero.
UI(x) = ~1
1
= -b[l + -f- + ~
+
.]. Hence the parameter 11 = O.
Now, for example,a = 7 andb = 2; thatis, a X2 distributionwidtl4 degreesof
freedom In this case,thepdfbecomes}tx) = ~x6e=t
, and}tx) is the solution
to somelinearconstantcoefficient7thorderhomogeneous
differentialequation.
(X)
Therefore,G~l)f.f(x);af(X)] . f(t)dt. SeeGray, Atchison and McWilliams
x
(1971). For the given values of the parametersa and b,
G7(X) = 2e-xI2(X6 + 12X"s+ 12Ox. + 96Qx3 + 5760x2 + 23040x + 46080)
as could have beenobtainedby repeatedintegrationby parts. In fact, the Gamma
pdf./{x) is a solution of a homog~eous ODE with constantcoeffici~ts for any
integer a ?: I, so that G~l)[/{x);aij(x)] will be exact for n
= a.
However, if a is not an integer G" (/{x» is still highly accuratein the tails. As an
examplewe include a table of relative errors for parametervaluesa = t
and b = 2, and approximationsG1 through G.
9
Table2 Relativeerrorsfor the G~l)-transfonns
of theGammafor a = t and
b = 2.
trIIe
K=+ 8.3265(2)*
6 1.4306(2)
12
~
35
50
75
120
(.
5.3200(4)
5.7330(7)
3.2971(9)
1.5375(12)
1.7071(18)
6.3261(28)
8.3265(2)~
E(Gl) E(G2) E(G3) E(G.)
E(G,)
E(G,)
E(G,)- E(G.)
7.4(2)
2.8(2)
9.4(3)
2.6(3)
1.4{3)
7.2(4)
3.3(4)
1.3(4)
2.8(4)
1.2(')
2.4(7)
1.5(9)
1.1(10)
5.9(12)
1.8(13)
2.5(15)
1.0(4)
3.0(6)
3.3(8)
9.7(11)
4.7(12)
1.5(13)
2.5(15)
1.6(11)
4.0(5)
8.0(7)
5.4(9)
7.6(12)
2.5(13)
5.0(25)
4.5(17)
1.3(19)
1.2(2)
2.4(3)
3.4(4)
3.1(5)
9.7(6)
2.7(6)
5.9(7)
9.9{8)
2.9(3)
3.3(4)
2.2(5)
7.7(7)
1.4(7)
2.2(8)
2.4(9)
1.7(10)
8.5(4)
5.8(5)
2.0(6)
2.9(8)
3.3(9)
3.0(10)
1.7(11)
5.4(13)
1.7(5)
2.4{7)
9.8(10)
6.9(13)
1.5(14)
2.0{16)
1.0(18)
1.5(21)
8.3265x 10-2)
are listed
TheG~I)(/{X» for n e {1.2. ., 6} and parametersa and b \D1Specified
in the appendix.
Example3: The Studentt Distribution
The pdf of the t distribution with parameter k degrees of freedom is
./(.%)
= ~lIL
r(
t
)JIi:
( 1 z2
)
+
T
_.!:t!.
2
The Studentt pdf satisfiesthe differential equation(6) with m
= . ascanbe s~
below:
Hence U1(x) = Xl (--t;r - "(.!i);'), yielding /1 = I. Once againDefinition I
gives us the medlOdfor generatingthe Gn(x)-transforms for the Studentt which
10
are tabulatedCorn E {1,2,3,
} in theappendix.For example,if k = 3
Gz(f(x»
G4(f(X»
=f(x)~
~7+~+~+7S6%
= f(x) 9o.K6+'76x4+1~+324
Table 3 Relative «fOIS for 1110
G ,,-tr81Sformsof the t distrIbution for various ~
of
freeOOmk.
k.l'
3
3
4
4
5
S
8
8
11
11
20
20
10.214'
22.2037
7.1731
13.0336
5.8934
9.677S
4.S007
6.4420
4.0147
S.4527
3. SS18
4.S38S
* 1.0000(3)*
TIW
AlGi]
1.0000(3). 1.'(2)
1.0000(4) 3.3(3)
1.0000(3) 3.3(2)
1.0000(4) 9.9(3)
1.<XX>O(3)S.0(2)
1.<XX>O(4)1.8(2)
1.0000(3) 9.1(2)
1.0000(4) 4.4(2)
1.0000(3) 0.1(0)
1.0(xx)(4) 6.3(2)
1.0000(3) 0.2(0)
1.0000(4) 9.4(2)
AlG2]
3.8(3)
8.1(4)
6.6(3)
2.0(3)
8.S(3)
3.1(3)
1.1(2)
S.2(3)
1.2(2)
6.1(3)
1.3(2)
7.0(3)
EtG)]
3.7(S)
1.7(6)
1.2(4)
1.1(')
2.2(4)
3.1(5)
4.3(4)
9.9(5)
S.3(4)
1.4(4)
6.3(4)
1.9(4)
AlG4]
S.6(6)
2.8(7)
1.4(')
1.'(6)
1.8(5)
3.4(6)
I.S(S)
6.1(6)
6.4(6)
S.S(6)
6. S(6)
2.S(6)
~G5]
2.4(7)
2.4(9)
1.3(6)
4.1(8)
2.8(6)
1.7(7)
S.3(6)
7.4(7)
'.4(6)
1.0(6)
4.2(6)
1.0(6)
AlG,]
2.2(8)
2.8(10)
6.1(8)
3.8(9)
1.2(8)
1.1(8)
4.1(7)
1.2(9)
6. '(7)
3.9(8)
7.2(7)
8.3(8)
~G7]
2.2(9)
S.I(12)
2.0(8)
2.3(10)
4.8(8)
1.5(9)
'.3(8)
8.1(9)
8.2(9)
8.4(9)
S.0(8)
1.6(9)
AIG.]
1.1(10)
4.6(13)
3.4(10)
1.4(11)
4.5(9)
3.1(11)
2.0(8)
7.4(10)
2.0(8)
1.6(9)
7.6(9)
1.'(9)
means1.0000x ]0-3
Example 4: The Inverse GaussianDistribution
TheinverseGaussian
pdfis./{x)
= f5:" e~,
differentialequation :-2"3P~~Jl2
UI
(X )
whichsatisfiesthefirst order
)f(X) - ./{x) = o. The coefficient
,4
p4...oL
JJ
= -2- Jl2
+ ...
l +6---2~
l2%
.a,.r2
from which we can see1hat/1 = O. Since UI (x) has the parameter/1 = 0, then
G £I(x); a ij (x)] hM the fonn of equatioo(19), from which we can generatethe
ft
G~I)-transformation. Table 4 showsthe relative errors for Jl = ). = 1 and
11
= 1,..., 8. The approximationis given for anyA and Jl in dIe Appendix for
II
n~5
For example,by the Appendix. for J.l= A = 1
G1(f(x» = l(x);;1:/;:"
G3 (/;(
V' x»
Gz(f(x» = I(x) .z4+~~~~~~1'
~+~'+110%4_.,+2.z2
-- J (,x ) .z'+2Ix'+1~4+75.z'-4~+9.z-1
/;
Table4 Relativeemxs for theGft-u.-lif{jIu~ of ~ inverseG81ssi8l.A ~ Il
-x
1hIe
E(G1) E(G2) E(G3) E(G.) E(G,) E(G,)
1.' 1.8923(1)* 1.7(1) 7.1(2) 2.8(2) 1.3(2)
2 1.14'2(1) 1.'(1) 4.1(2) 1.7(2) 6.6(3)
3 4.6812(2) 1.1(1) 2.5(2) 7.0(3) 2.2(3)
4.5 1.4301{2) 7.4(2) 1.2(2) 2.5(3) 6.0(4)
6 4.8499(3) 5.3{2) 6.5(3) 1.1.(3) 2.1(4)
~
6.2(3)
2.1(3)
7.9(4)
1.7(4)
4.9(5)
3.2(3)
1.3(3)
3.0(4)
5.3(5)
1.3(5)
10 9.4392(4) 2.I{2) 1.9(3) 1.9(4) 2.4(S) 3.7(6) 6.6(7)
16 9.4392(6) 1.4(2) 5.1(4) 2.9(5) 2.3(6) 2.3(7) 2.6(1)
- 1.
E(G,)
E(G.)
1.7(3)
6.3(4)
1.2(4)
1.8(5)
3.6(6)
9.4(4)
3.2(4)
5.4(5)
6.4(6)
1.1(6)
1.3(7) 2.9(1)
3.5(9) 5.3(10)
132 1.2201(9) 4.4(3) 5.7(5) 1.3(6) 4.2(8) 1.8(9) 9.6(11)
6.2(12) 4.6(13) I
* 1.8923(1)means1.8923)( 10-1
~
~
Example 5: The F distribution
The pdf of the F distribution is
Ax)
whichs~es
r(.,..) (.L)
= r(f)r(t)
b f
!~
(I+(t)%).
thefirst orderpdf ( -2%
-a~~+axb
)f
(x) - J{x)
= O.Hence
U 1(x) = -2% -ab+~2a¥'toarb
' which can be rewritten in a Laursrt series of the form
aloX+all+~+...,soll
= 1.
SinceU1(x) has the parameter/1 = I, G,,[ltx);alj(x)] hastheform of equation
(20), from whichwe cangeneratetheG~l)-transformation
of/(x). Table6 shows
therelativeerrorfor n = 1,2,...8 anda = 3,b = 4. Theapproximation
is given
in theappendixfor all a andb whenn = 1,2,...5.
12
a=3b=4
.
When
G1(f(X) =f(x)~
G2(f(X» = f(x)
z:~~~~
Table5 Re1ative
errorsfor 1heG"-u-rUi~
z
TrIIe
E(G1) E(G2) E(G)
offle F distributiG1:
E(G.)
E(Gs)
4.19 1.000296(1) 1.2(2) 6.2(3) 1.3(4) 7.0(6) S.4(7)
6.S9 S.OOI69(2) 6.3(3) 2.3(3) 3.3(5) 1.2(6) 6.2(1)
E(G,)
E(G,)
E(G.)
S.2(8) S.7(9) 7.0(10)
..1(9) 3.1(10) 2.6(11)
9.98 2.4996S(2) 3.7(3) 9.7(4) 9.2(6) 2.3(7) 1.2(9) 3.7(10) 20(11) 1.2(12)
16.7 9.99383(3) 2.0(3) 3.3(4) 1.9(6) 29(8) 6.6(10) 1.9(11) 6.1(13) 2.2(14)
Other Methods
There are, of course,other methodsfor obtaining approximationfimctions for tail probabilities.
However, the only onesto datethat are effective are of the form f(x )R(x) whereR(x) is a rational
fimction andfis the pdf. Besidesthe G~") - transform, the only generalmethodthat is competitive
with G~m)is the methodof continuedfractioos. However, the only result availablefrom that
approachcurrently is for f(x) a normal (0,1) pdf. In that particular case,G~l) and the continued
fraction results are very similar in form and accuracy. Other results in the normal caseof this
rational function have been given by Hawkes (1982) and Lew (1981). Their 8th order
approximationsare comparablein complexity and accuracywith G.(f(x».
Their methodsdo not
extendto other distributions.
A Mathemati~ Program for GeneratingThe G~1) - transfonn
ThesimpleMathematica
programlistedbelowwasusedto generateeachof the G~I)transformsin this paper.Thereareonlythreeparameters
needed:thepdfAx]. theorderof the
transfoInl n. and the parameter II from equation (5) with k
= 1.
Theseare defined on the first line
of theprogram
Theoutputconsistsof therationalexpression
R[x] followedby theG~I)- transfoInl
G[x] = Ax]R[x]. BothR[x] and G[x] are then defined functions and can be evaluatedat any value
ofx.
An examplefor the G~l) - transformof the standardnormal
13
X2
/fi;
L=-}-n=S-Kx
,
vl:- I = -1-e-T.
,
Num = Table[O,{i, I,n + I}, {i, I,n + I}];
Num[[l, I]] = 0;
Forfj = 2J <= n + I,j + +,Num[[IJ]] = D[,1x],{x,j
- 2}]]
For[i = 2,i <= n + I,n ++,
Forfj = IJ <= n + IJ + +,Num[[iJ]]
= D[x"'{L
-
i + 2) * J[x], {xJ
-
I}]]]
Den = Table[O,{i, I,n + I}, {i, I,n + I} ];
Den[[I, I]] = I;
For{; = 2J <= n + IJ + +,Den[[IJ]] = 0]
For[i = 2,i <= n + I,n + +,
Forfj = I,j <= n + I,j + +,Den[[i,j}] = D[x"(L
- i + 2) * J[x], {xJ -I}]]]
R[x- ] -- S.Implify[ Ax]Det[Num
]
* Det[Den]
G[.%_]= Ax] * R[x]
The exact form of G~l) can be found in the appendix. Evaluationsin Mathematicayielded
G[I.645] = 0.0499849, G[I. 96] = 0.0249979, and G[2.326] = 0.0100093.
Concluding Remarks
In this paper we have presentedthe theory of the G~m)-transformation
and the general
methodologyfor finding easily evaluatedfunctions that accuratelyapproximatetail probabilities.
Unlike most approximationfunctions which are distribution specific, the G~m)-transformation
is
very generalin application and can be applied to nearly any distribution that is differentiable. The
examplespresentedin this paper are an extensionof the results in Gray and Wang (1991) and were
madepossibleusing a computer algebrasystemthat was not availablein 1991. The examples
presentedshouldprovide the readerwith amplenumerical evidencethat when
~
has a
convergentLaurent expansionabout zero with only non-positivepower terms G~) convergesquite
00
rapidly to f./(t)dt. In fact, the approximationgiven here should be sufficiently accuratefor virtually
or
any applicationrequiring highly accuratetail probabilities.
14
In our examples,we have usedm = 1 and determined/1 for eachdistribution, however G~I) is
robust and in most casesaccurateapproximationscan be obtainedby simply letting /1
= 1 for any
relevantdistribution and thm increasingthe order n \D1tilthe desiredapproximationis reached. An
exampleof this feature is presentedfor the ratio of a Z2 and a lognormal in Gray and Wang (1991).
Finally, it should be mentionedthat valuesofm > 1 are also ofinterest. For example,Gray
and Wang (1993) have used G~;~to obtain tail probabilities approximationfunctions for the
standardnon-cmtral distributions.
IS
~
APPENDIX
-~<%<~
Coefficients of the xl tenn
, Tsm-
~ Nt
~Dt
N2 I D2 ~ N3 'D,
1
1
2
4
16
27
l'
1
1
600
168
104
12
D,
12C
~24
96
18
s
1
D4 ~N,
N.
1200
1<XX>
123
21
127S
333
22
1
365
34
35
1
I
.
1
An empy cell ~
thatthecoefficientof that teID1is zero.
Example:
G~l)[/(X)]
= Ax]
+ IIx3+ 27x2
+ I8x + 6
x6 x'+ Ilx4
16
-«><x«<>
Coefficients of the xk teI:m
Example
G(I)[/{X)]
6
--JI;'-X ] Xl2 +XIISlxlO
+ SQx9+ 807x7 + 493&x' + 10200x3 + 4320x
+ 8SSx'+ S6SSX6
+ 139S0x4
+ 9720X2
+ 720
17
n
r(¥-)
Student
t ./{x) = r(t)~
Example:
¥)-AJl.
G~l)[f(X)] =JtX] (k+2~
+i(k+7~
+S1lx
1(k + 2)x4 + Q2X2 + 31l
18
n
Studentt
~x) =
r(-AfL)
r(t)1i';E
(~
Example: G~l)(/{X)]
)
-1fL
k
=JtX]
(k+2)%' +i(k+Z~
i(k+2)%4+6I2r+3k2
+ Sk2x
19
n
Student
t
./{x)=
r(..tsL)
r(t)5':i:
(
¥ ) -~
(continued)
Coefficients of the Xk tem1.
20
m
Example:
-br + 2bZ(a- 4)il- b3(r - 6a+ 11)x
a~l)[/(x)] = Ax] -r + 3b(a - 3)il- 2bZ(a-2)(a - 3)x+b3(a -1)(a -2)(a - 3)
21
IV
In~
GaussianDistributioo:.f(x) = J~~;~:~~~f~
Example
G~l)[.f{X)]
=Ax]
2p2u4 + 14p.x3_2p~2
12X. + 1~2x3-J&2(2l2_15p2)x2-6A1&.X
+ ).2p2
21
REFERENCES
Abramowitz. M and Stegun.I. A. (1964). Handbook of Mathematical Functions, Dover
Publications, pg. 932
Bamdorff-Nielson. O. E. and Cox. D. R (1989), Asymptotic Techniquesfor Use in Statistics,
Chapmanand Hall Publishers,pp. 55-57
Good, I. J. (1986). Very Small Tails of the t Distribution and SignificanceTests for
Clustering. JounKll of Statistical Computing and Simulation, 1.3,pp.243-248
Gray, H. L. and Lewis, T. O. (1971). Approximation of Tail Probabilities by Means of the
B,.-Transformation. JoumaI of the American Statistical Association, 66, No. 336, pp.897-899
Gray, H. L., Atchison, T. A., and McWilliams, G. V. (1971) On Higher Order
G-Transformations,SIAM Journal on Numerical Analysis, 8, No.2, pp.
Gray, H. L. (1988). On a Unification of Bias Reductionand Numerical Approximation.
Probabi/ityand Statistics, ed J. N. Srivastava,Amsterdam,Eisivier SciencePublishersB. V
(North-Holland)
Gray, H. L. and Wang, Suojin (1991). A Generalmedlod for Approximating Tail
Probabilities. JoumaJ of theAmerican Statistical Association, 86, No. 413, pp.159-166
Gray, H.L. and Wang, Suojin (1993). Approximating Tail Probabilities ofNoo-Central
Distributions. Computational Statisticsand Data Analysis, V 01.15,No.3, March 1993,pp
343-352.
Hamaker,Hugo C. (1978). Approximating the CumulativeNormal Distributioo and its
Inverse. Applied Statistics, 2.7,No.1, pp.76-77
22