FRACTIONAL MOIvlENTS OF A QUADRATIC FORN
IN NOIWENTBAL NORNAL BANDON VARIABLES
by
James R. Harvey
Institute of Statistics
Mimeograph Series No. 433
April} 1965
v
TABLE OF CONTENTS
Page
LIST OF TABLES. . .
1.
· · · · vii
···· 1
INTRODUCTION.
1.1
1.2
1,3
1.4
2.
.
,
Statement of the Problem .
Literature Search. . . .
Distance Problems, in General . . .
Organization of the Paper..
. ..•.
1
2
3
5
NONCENTRAL CHI-SQUARE .
6
2.1 Introquction.
2.2 Probability Density Function, Moment Generating
Function, and Cumulants. •.
. ....
2.3 Moments and Moment Recursion . . .
2 . 4 Reznarks. . . . . . . . . . . . . . .
3.
6
····
TWO SPECIAL CASES OF THE GENERAL PROBLEM. .
14
3·1 Introduction .
3·2 Special Case 1 [k and! General, D = Ia ].
3·3 Special Case 2 [k = 2, ! = Q, D = diag(ai,a~)]
3.4 Remarks . . . . .
" .
..
•
•
•
•
•
•
•
(I
..
..
.. . . . . . .
4.
•
~
14
14
•
......
THE GENERAL PROBLEM, .APPROXD1ATE SOLUTIONS.
········
··
····
·····
····
4.4 Patnaik-extension Approximation [Y = ~X2(V3'Sl)]
3
4·5 Pearson-extension Approximation [Y4=a4X2 (v4,S2)+b2 ] .
,
4.6 Sunnnary.
··
5·
···
THE GENERAL PROBLEM, EXACT SOLUTIONS.
5·1
5·2
5·3
5.4
5·5
5.6
22
24
25
27
33
33
34
· · · · · · · · . . · · ·, · · · · · · ·
· ·
·
····
..·····
·,·
·
·
····
,
Introduction .
F(q) as Given by Shah and Khatri
F(q) as Given by Imhof · ·
· .
The Choice
·
·
·
·
Reduction to a Single Integral (I) .
Reduction to a Single Integral (II) .
17
20
22
,.
4.1 Introduction .
· r: vI)]
4.2 Patnaik's Approximation [Y = a
.
1
l
4.3 Pearson I s Approximation [Y2 = a2 ~ ( v2 )+b ]
l
. .····
6
10
12
!
34
34
36
37
37
39
vi
TABLE OF CONTENTS (continued)
Page
6.
47
NUMERICAL COMPARISONS . . . . . • .
6.1
6.2
6.3
6.4
Introduction . . . . . . . . • . .
Distribution Function of Q . .
Fractional Moments of Q. .
Conclusions.
.., .
47
47
48
51
7~
SUMMARY. • • • • • •
8.
LIST OF REFERENCES.
60
9.
APPENDICES. . . . .
62
9·1 The Confluent Hypergeometric Function (Kummer's
Function). . . . . . . . • . . . . . • . . • .
9.1.1 Properties and Relationships. . . • . . . . . .
55
62
62
9.1.2 Representation of F{~ ~;~;-x) in Tenms of
Bessel or Error Functions, General . . . .
64
9.1.3 Representation of F(-~J~;-~) in Terms of
Bessel or Error Functions, Specific Cases
9·2 A Convenient Approximation •
9·3 Proof of Equation (5.6.19) . . . . .
9·3·1 Introduction. . • • . . . . . .
. ....
9.3.2 Proof, Part I . . . • .
. . . .
9.3.3 Proof, Part II. . . . . . . . • .
9.3.3.1 Proof of (9·3.8.1) . . . . .
9.3.3.2 Proof of (9·3.2.2) .
9.3.4 Properties of Functions . . • . . . • . . • . .
9.3.4.1 The Functions A(t) and p(t). .
9.3.4.2 The Functions fl(t) and t 2(t).
9.3.4.3 The Function~ h(t,a,r) and H(t,a,r).
9.3.4.4 The Function I 2(v,o) • . . . .
9.3.4.5 The Functions t~p)(t), p ~ 1,
j = 1,2. . . . . . . . . . .
67
69
72
72
74
77
78
79
84
84
85
85
86
87
vii
LIST OF TABLES
Page
6.1. Values (approximate and exact) of F(q)
6.2.
Values (approximate and exact) of
E(Qt)
= p(Q ~
=
q)
E(Z).
52
APPENDIX TABLES
Values of x, A, and B such that .lA-BL
A
$ €, where
r(r-l)
A = r x and B = xr e 2x . . • . • . . . . . . . . . .
f(fj)
71
1.
INTRODUCTION
1.1 Statement of the Problem
Let the kxl vector random variable (r.v.)
! be normally distributed
with vector mean! and variance matrix D, where D is a nonsingular diagonal matrix with diagonal elements rr~, i
~
quadratic form XIX.
= l, •.. ,k.
Let Q be the
In compact notation, we write
The problem is to determine the fractional moments of Q, !.~., E(Qr),
r
> O.
More generally, suppose!l is N[!,V], and Ql
symmetric nonnegative-definite matrix.
= !i.A:!l'
where A is a
If V is nonsingular, it is well
known that we may, by means of a nonsingular linear transformation,
expres s
in the form
~
where the u
i
are independent noncentral chi-square r.v.'sj and the a
are the nonzero characteristic roots of AV.
i
When so expressed, Q is
l
said to be in canonical form.
In our problem,
where the
(~~
~
r
Q
= Xl X =
- -
~ rr~
i=l
( Xi ) 2 ,
rri
~
are independent noncentral chi-square r. v.' s, each
2
1fith one degree of freedom, and the rr
characteristic roots of ID.
i
are, of course, the nonzero
Thus Q is seen to be in canonical form,
and hence any result that we might obtain pertaining to Q would pertain
to
~
as well.
2
1
Let Z
= Q2.
Clearly, Z may be interpreted as the distance from an
appropriate origin to a k-dimensional normal random point.
We should
like, in particular, to determine the integer moments of Z.
The even
moments of Z are, of course, the integer moments of Q, while the odd
moments of Z are the "half-fractional " moments of Q,
r is an odd multiple of~.
!. ~., E( Qr) ,where
Now the characteristic function for Q is
known [see, ~.~., Patnaik (1949)] and hence the integer moments of Q
are essentially known.
In effect, then, we are primarily interested in
the half-fractional moments of Q, although the treatment will usually
be more general.
1.2
Literature Search
A search of the literature suggests an apparent lack of interest
in fractional moments of Q; yet, in the past year, the author was twice
asked to consider special cases of the problem.
In only one paper, an
internal report of' Space Technology Laboratories by Scheuer (1961), was
related work found.
Scheuer treats the case!
= Q and
k
= 2;
pleteness, a brief look at his work is included in Section 3.
almost trivial case, !
many times.
= Q and
D
= Icl,
for com-
An
has undoubtedly been solved
But aside from these two special cases, the problem
appears not to have been treated.
Perhaps the apparent lack of interest in fractional moments stems,
in reality, from the complicated nature of the distribution of Q.
Helpful to this study, however, is the fact that explicit expressions
for the distribution function of Q have been recently given by Shah and
Khatri (1961) and Imhof (1961).
3
1.3
Distance Problems, in General
The problem stated in the first paragraph of Section 1.1 is quite
general in that practically all distance problems in normal r.v.'s can
be reduced to that form.
To see this, suppose that we are interested
in the distance between two points, Xl and
12,
in k-dimensional space,
where
(1.3.1)
Clearly, Q ;:: <X - X ) r <X - X ), and we have the most general dista.nce
2
1
2
l
problem in normal r. v. 's that can be formulated.
To reduce this problem to the desired form, we first make the
linear transformation X
where
~
= ~1
-
~2
and V
=
Xl - X2 ·
= V1
+ V2
Then
c-
C'.
nonsingular, and, hence, positive definite.
We require only that V be
This requirement is cer-
tainly met if the variance matrix in (1.3.1) is nonsingular (hence,
positive definite).
u'Vu
=
For suppose there exists a vector ~
f
0 such that
o. Then
which is not possible.
Next we make an orthogonal transformation.
It is well known that
for every real symmetric matrix A there exists an orthogonal matrix P
such that P'AP
= D,
where D is a diagonal matrix whose diagonal elements
4
are the characteristic roots of A.
Therefore, let P be such a matrix
for V, and let us make the orthogonal transformation!
where!
= pl~,
D
= plvp,
= ply.
Then
and the reduction is complete.
In theory, at least, finding such an orthogonal matrix P is simple
enough.
One determines the characteristic roots of V, and a correspond-
ing set of orthonormal characteristic vectors.
vectors constitute the columns of P.
The characteristic
In practice, the computation of P
may be tedious if k is large; various computing techniques eXist, but
we shall not consider the matter further here.
In (1.3.2), we required that V be nonsingular.
The reason for
this requirement becomes clear if we suppose that V is singular (hence,
positive semidefinite).
In this case, there exists an orthogonal matrix
P such that
l
where D is a diagonal matrix whose diagonal elements are the nonzero
1
characteristic roots of V.
If we make the transformation X
= pil,
we
see that
and
Q
= -XiX- = -1-1
XiX
+
X~x_
-~2
= -1-1
XIX
+
/;~/;
.iiUe--2
.
Now, because of the presence of the constant term !~2' investigation
5
of the fractional moments of this r.v. is an altogether different
problem from that proposed.
Furthermore, it appears to be of less
practical interest, and, accordingly, we shall not consider it further.
1.4
Organization of the Paper
Because of recently given explicit expressions for the distribution function of Q, mentioned in Section 1.2, it is not difficult to
write explicit expressions for fractional moments of Q.
Rather, it is
in the numerical evaluation of such expressions that one encounters
difficulty.
After developing, in Section 2, some properties of non-
central chi-square required in later parts of the work, we turn in
Section 3 to consider two special cases of the more general problem for
which exact solutions are not particularly difficult to evaluate
numerically.
In Section 4, we develop several approximate solutions to
the general problem, while, in Section 5, exact solutions to the general
problem are considered.
In Section 6, comparisons of exact and approxi-
mate solutions for a set of numerical examples are given.
contains a concluding
three appendices:
SUlmary
of the paper.
Section
7
The work also contains
Appendix 9.1 contains material relating to the con-
fluent hypergeometric function; Appendix 9.2 gives a convenient approximation for the ratio of two gamma functions; Appendix 9.3 contains a
proof of an important result obtained formally in Section 5.
6
2.
NONCENTRAL CHI-SQUARE
2.1
Introduction
In general, we shall be dealing with a quadratic form which is a
linear combination of independent noncentral chi-square random variables.
Because of this fact, we shall pause briefly to establish a few
properties of the noncentral chi-square distribution.
Let the kxl vector X be N[!,I].
It is well known that X'X is a
noncentral chi-square r.v. having k degrees of freedom and noncentrality
parameter \, where, consistent with recent practice, we define
More generally, let U be a noncentral chi-square r.v. having v
degrees of freedom (v not necessarily an integer) and noncentrality
parameter \.
In compact notation, we write
U
is
l-
(v,\).
If \=0, then U is the well-known central chi-square r.v. [designated
here by
l-(v,o)
2.2
or simply
l-(v)].
Probability Density Function, Moment
Generating Function, and Cumulants
The probability density function (pdf) of U is variously derived
and presented in the literature.
Fisher (1928), p. 663, equation (B),
obtained it as a particular case of the distribution of the multiple
correlation coefficient.
feu)
=1.(1i VIu)
2).
2
He gives (in the present notation)
v-2
2
e-~(u0f'2\)
J _
v 2
2
(i~)
(v > OJ\,u ~ 0),
(2.2.1)
7
where J (x) is the Bessel function of the first kind of order T).
T)
(In
Fisher's expression, there is a slight but obvious sign error in the
exponent of the exponential.)
given in terms of I
TJ
Equation (2.2.1) is now more popularly
(x), the modified Bessel function of the first kind
of order TJ,
v-2
4
f( u)
= t ( 2u,.)
(v > O;A.,U ~ 0).
T'ne pili' of U has long been viewed as being somewhat other than
attractive.
One recent author, Lindgren (1960), derives
v-2
2
u
(A.~)
f(u) :::; _u__e
_
-
00
E
i=o
(2A.ulr(i+k )
(v > O;A.,U
(2i)~r(fri)
~
0),
and then states,
The expression finally obtained for the noncentral
chi-square density is still not especially pleasant to
behold or to work with. However, various tables are
available; . . . .
Had Lindgren made use of the duplication formula for the gamma function,
his expression for f(u) could have been reduced to a more appealing one,
viz.
f(u) =
~[~]
~.
(v
i=o
>
O;A.,U ~ 0)
(2.2.4)
(v > O;A.,U
~
0).
8
In this form, feu) is not particularly unpleasant to behold.
function g
.J.rI' (
V·c;l.
The
u) is the pdf of a central chi- square r. v . with v+'2i
degrees of freedom, while its coefficient PA(i) is the i-th probability
term of a Poisson r.v. with parameter A.
Thus the pdf for a noncentral
chi-square r.v. is the sum of an infinite number of central chi·wsquare
pdf's, each weighted by a term from the Poisson distribution.
appears that Robbins and Pitman
It
(1949) were the first to view the non-
central chi-square in this light.
Equ.ation (2.2.4.) may be quickly obtained from (2.2.2), and vice
versa, with the aid of the following relationship [see Sneddon
(1961),
p. 127, equation (33.4)]~
I
Tj
(x)
= (2"x )TJ
Moreover, for some purposes,
work With.
(2.2.4) is not at all unpleasant to
Consider, for example, the ease with which we derive the
moment generating function (mgf) for U.
Formally,
9
ex)
~(t)
=J
e
ex)
ut
~
i=o
0
00
ex)
=
~
PA(i)gv~i(u)du
PA,(i)
i=o
J
e
ut
gV~i(u)du
0
00
=
~
i=o
PA(i) M~
(v~i)
-
00
::=
~
i=o
PA(i)(1-2t)
e -A
=
(1_2t)V/2
ex)
~
i=o
(t)
-2
v~i
(t <-k)
[A!(1-2t)]i
i~
e2At !(1-2t)
=
(1-2t)v /2
(t<-k).
(2.2.5)
The change in order of integration and summation is justified, of
course, because the integrand is nonnegative.
An elegant derivation of
(2.2.4) is afforded by direct derivation of ~(t) from knOWledge of the
distribution of ~ (!,~., Mu(t) = E[exp(~'~t)]), followed by straightforward application of the customary inversion formula to ~u(t) = Mu(it).
Readily derived, using the mgf, is the reproductive property of
noncentral chi-square,
!.~.,
a sum of independent noncentral chi-square
r,v,'s is itself a noncentral chi-square r.v.: specifically,
10
We shall have need of the cumulants of U.
By definition, the
cumulant generating function is the natural logarithm of
2At
2At
~
1=0
=
~
Thus
~ 10g(1-2t)
Cu(t) = 1-2t
=
~(t).
(2t)i + Yo ~ (2tl
2 i=l
i
(It I
<!)
~i
2i-1 (v+2iA)
~
i=l
Hence, the r-th cumulant of U is seen to be
(r=1,2, ... ).
2.3
(2.2.6)
Moments and Moment Recursion
It is clear that we may obtain integer moments of U from (2.2.5)
or (2.2.6),
We shall, however, also require fractional moments of U.
In general,
00
E(lf)
=J
0
00
u
r
00
!:
i=o
PA(i)gv+2i(u)du
J
00
= I: PA(i)
i=o
0
r
u gv+2i (u)du
00
= !: PA(i) E[X2(v+2i)]r
i=o
00
= I: PA(i) { 2"
i=o
r
(V~i
+
r (V~i)
"l}
11
(At this point, it is interesting to note that for r
part in braces is a polynomial in i of degree r.
= 0,1,2, ""
the
Accordingly, the r-th
integer moment of a noncentral chi-square r.v. is expressible as a
linear combination of the moments of orders 0,1, ... ,r of a Poisson r,v.)
=
(~ + r) (~ + r+l)
r -r.. r(~ + r)
2 e
= 2 r e -r..
(~) (~ + 1)
r(~)
2
2
r(~ + r)
r
(~)
2
F
2
(oY
+ r'' ~.
'\)
2
2' f\o ,
where F(a;c;x) is the confluent hypergeometric functi.on (Kummer's
function), some of whose properties are given in Section 9.1.1 of
Appendix 9.1.
Using now equation (9.1.1.3) of that section, we have,
finally,
r(~ + r)
V
--~--- F(-r; -2 ; -t..)
r(~)
2
(r > - ~) .
We shall now derive a useful second-order recursion for the
moments of U.
In equation (9.1.1.9), let a
= -(r+l), c =~, and replace
x by -r.. to obtain
In this recursion equation, substitute for F(.;~;-A.) from (2.3.1) to
obtain
12
which upon simplification reduces to
(r>-~),
the desired recursive relationship.
The results (2.3.1) and (2.3,2) are
believed to be new.
2.4 Remarks
Of
course, for
~=O,
all numbered relations in this chapter
specialize to the corresponding expressions for central chi-square.
However, although (2.3,2) specializes to a valid central chi-square
moment recursion of order two, there does exist an obvious first-order
recursion, viz.
(2.4,1)
Possibly the most troublesome aspect of the noncentral chi-square
distribution has been the numerical evaluation of F(u)
= p(U
~
u) (as,
for instance, when one wishes to investigate the power function for
some statistical test based upon chi-square theory).
Various more
tractable r.v.'s, approximating U in some sense, have been offered with
the hope that their distribution functions would serve as acceptably
good approximations, for practical purposes, to that of U,
Without
doubt, Patnaik's (1949) approximation has become the most popular of
these (it is also one of the earliest and very likely the simplest).
13
More recently, an approximation by Pearson (1959) has been gaining
favor.
In fact, the proposal, justification, and comparison of such
approximations have occupied a fair amount of space in statistical
journals in the past decade.
At last, however, in November 1962,
extensive tables of the cumulative noncentral chi-square distribution
were published by Haynam, Govindarajulu, and Leone (1962).
The tables
referenced by Lindgren [cf. the quotation following (2.2.3) of this
chapter] are those by Fix (1954), a short and limited set.
Fortified by this brief consideration of the properties of noncentral chi-square, let us now consider two special cases of the
problem at hand.
14
3.
'!WO SPECIAL CASES OF THE GENERAL PROBLEM
3.1
Introduction
For certain choices of the values of the parameters k,
1,
and D,
the general problem may be reduced to one for which the exact solution
n~y
be expressed in terms of well-tabled special functions.
Therefore,
before proceeding to an attack on the general problem, we shall, in
this section, consider two special cases; the first is believed to be
new, while the second is the work of Scheuer (1961), and its inclusion
here is for the purpose of completeness.
3.2 Special Case 1 [k and 1 General, D = I~2 ]
We have that X is N[1,I~2], and
2 kL: (Xi)
Q = X'X = ~
-
-
k
where A..
~
and A.
= i:l A.i
i=l
S'S
= 2~2
2
~
By equation (2.3.1),
k
(r>-'2)'
and by (2.3.2), we have the following moment recursion:
k
(r > - '2),
15
which together with the zero-th and first moments of Q, will generate
all integer moments of Q [note that (3.2.1) and (3.2.2) are not confined
to integer moments].
Although it is clear that the integer moments of
Q may be obtained from (2.2.5) or (2.2.6), it is worth noting that
(3.2.2) generates them more conveniently.
1
For moments of Z (=Q2), we have
~'!'-A.)
2'2'
(n > - k),
and
(n > - k).
(3.2.4)
Thus (3.2,4), together with the zero-th and first three integer moments
of Z, will generate all integer moments of Z.
It is worthwhile, per-
haps, to remark that, although the variance of Q tends to infinity with
k, the variance of Z remains finite; in fact, it is not difficult to
show that
as
k
-00.
Tables of the confluent hypergeometric function are available [see
Rushton and Lang (1954) and Slater (1960)].
Alternatively, one may,
for k a positive integer and n a positive odd integer, express
F(- ~'!'-A.) in terms of more extensively tabled special functions
2'2'
(recall, again, that the case of n even is not of interest because the
even moments of Z are the integer moments of Q and these are essentially
known).
In particular, when k and n are each positive odd integers,
16
F(- ~;~;-A) may be expressed in terms of the error function, erf(~),
defined by (9,1,1,6), or in terms of the incomplete gamma function,
'Y (~, A ), defined by (9.1, 1, 7 ) ,
When k and n are positive even and odd
integers, respectively, F(- ~;~;-A) may be expressed in terms of modified Bessel functions of the first kind of orders zero and one, alone.
Proofs of the claims of the last two sentences constitute Section 9.1.2
of Appendix 9,1,
The first three integer moments of Z are readily seen to be
In Section 9.1.3 of Appendix 9.1, the two confluent hypergeometric
functions appearing in the expressions for E(Zl) and E(z3) are given
Bessel or error function representations for k = 1,2,3
univariate, circular, or spherical normal).
(!.~.
for!
These choices for k will
serve for distance problems that one is most apt to encounter of the
present type.
For k = 1, see (9.1.3,1) and (9.1.3.2); for k=2,
(9.1,3.3) and (9.1,3.4); for k=3, (9.1.3.5) and (9·1.3.6).
Finally, if
1
=
Q,
then Q = ~2~(k,O), and special case 1 is
reduced to an almost trivial problem.
izes to
In particular, (3.2.3) special-
17
(n > - k),
and, rather than (3.2.2), we have, by
(204.1),
(n > - k),
In fact, specializing further to the physically mea.nin.gful
d:iU1ETJSl,)!lS,
we have that
Var(Z)
(2-
1
\j\ -1l'
2
~- CT = 1.2533CT
CT =:
( 1 - g)
1l'
0,7979CT
3.3 Special Case 2 [k
=:
2,
!
=:
2
CT
~) CT2
(2
-
(3
- ~)
rC
2
1kr2
"" 0 0:l\67:
...
j
o,,'+2
j~92 IT 2
-
.;:>
.;l
0.45350"- .
CT- :::
D ::: diag(oi,a~)]
2,
We have that
and without loss of generality, suppose
Q :::
(X1)2
i!:1 + i:.2 :::
r
2
CT
~
2 (X1)2 +
1
CT2
CT
CT
1
l
2
CT22 , We have
(X2)2
,
0"2
and (:2
are independent i' (1) r. v .' s , Ull:ier a trans2
X
X
formati.on to polar coordinates, say -l: ::: R cos e.1 2:::; R sin e, it is
where
CT
l
CT
1
CT
2
18
2
well known that R is
a
I
(2) and a is uniform on (0,2:rr), and that Rand
are independent r.v.'s.
Thus
( 2r) [ I-h2.
2_]r
= 0"12rERE
s~n~ ,
where
°~
2
2
h
2
0"1-0"2
=
2
<1.
0"1
Clearly,
(r> -1).
Now
2
2
E[1+h sin S]r
2:rr
J
= ~:rr
2
2
(1-h sin elde
o
=~
:rr/2
J
2
2
(1_h sin e lde
o
(t
= sin2e)
1
=
*J
2
t-! (l-tr! (1_h t)rdt
o
2
= F(-r '2"
.l.·1·h ) ,
where F(a,b;c;x) is the well-known hypergeometric :function [see Erdelyi,
(1953), p. 59, equation (10), for this integral representation].
We have, then
(r > -1) ,
19
and for moments of Z,
(n>-2).
In the recursion [see Erdelyi (1953), p. 103, equation (31)]
(c-a)F(a-l,b;c;x)
= [c-2a-(b-a)x]F(a,b;c;x) + a(l-x)F(a+l,b;c;x),
2
let a = - (~ + 1), b =~, c = 1, replace x by h , and substitute for
F(.,~;1;h2) from (3.3.1) to obtain the moment recursion
(n>-2).
(3.3.3)
The recursion (3.3.3), together with the zero-th and first three
integer moments of Z, will yield all integer moments of Z.
hypergeometric function is not extensively tabled.
Now the
Happily, however,
it turns out that the odd moments of Z may be expressed in terms of the
well-tabled complete elliptic integrals of the first and second kinds,
K(h) and E(h), respectively, since
K(h)
1
2) ,
= 2:If F (12'2;1;h
E(h)
= ~2 F(-~2'2"
~'1'h2)
[see ~.~., Sneddon (1961), p. 42, examples (Viii) and (ix)].
first three integer moments of Z are
Thus the
20
::: (J
1
E(' Z2) :::
~ E(h);
.
-
1C
'
2 + cr2 '
1
2'
(J
This last expression, for E(Z3), is developed in detail here because it
does not agree with that given by Scheuer, the error apparently bej.ng
his.
3.4 Remarks
These two special cases should serve to answer moment questions for
a fair share of the distance problems that one will encounter.
Attempts
were made to generalize Scheuer's work to higher dimensions (!,~",
k '" 3,4,
o.o)~
but with little real success.
Scheuer's work does not
generalize in the two-dimensional case for noncentral rov.'s because
then, of course, the polar transformation variates R and
independent.
a
are not
21
Finally, in thi,s section and elsewhere in the present 'work, we
often encounter expressions of the type
r(fj)
r x
0
When r is not an integer, the numerical evaluation of
nuisance
tion for
Q
()0401) can be a
Presented as Appendix 9.2 is an exceptionally good approxima-
(304.1)0
We next turn to consider several approximate solutions for the
more difficult general problem.
22
40
THE GENERAL PROBLEM, APPROXIMATE SOLillIONS
401
Introduction
In general, the quadratic form Q is a linear combination of
independent noncentral chi-square r.v.'s, in contrast to the simpler
Bpecial cases dealt with in the previous section.
In this section we
sh3.11 consi,der several approximate solutions for the positive fractional
momcc,ni:.s of Q, while in the following section exact solutions will be
investigated,
In Section 6, the quality of the approximate solutions
will be judged on the basis of comparisons of the approximate and exact
eOLutlons for a set of numerical exampleso
As in Section 1, we have that! is N[!,D], where D is diag(CT~),
i·- 1 1
0
"
0'
k, and
Q ::::: X'X
k
22
L: CT. X-(l,A. ),
. 1 ~
~
~:::::
wheYe A, :;
~
obtained,
is
t( CT~i)2
Again, the integer moments of Q are readily
i
In particular, by relation (2.2.6), the r-th cumulant of Q
given by
(r : : : 1,2,3, "0)'
SpeCifically,
(401.1)
23
(4.1.2 )
However, there seems to be no easy way to obtain fractional moments of
Q.
Following Patnaik (1949) and Pearson (1959), we might try approximating Q b~,r more tractable central chi-square related r.v.' s, using the
method of matching moments.
On the other hand, because of the new
"machinery" developed in Section 2 [specifically, (2.3.1) and (2.3.2)],
we might try to extend the Patnaik and Pearson approximations by fitting
more general, noncentral chi-square related, r.v.'s.
The approximating
r.v.'s would be of the following forms:
l(
Y' : :;: a
v )---------2-moment approx. (Patnaik)
11.1
Y2 : :;: a x2(v )+b ------3-moment approx. (Pearson)
2
2
1
Y : :;: a3x2(v3'~1)------3-moment
3
approx. (Patnaik extension)
We should like to determine the parameters of each of the above approximating r.v.'s, and an explicit expression for a general moment of each.
24
4.2
Patnaik's Approximation [Y
1
= al,t( 'VI)]
and 'VI' Patnaik equates the first
l
two cumu1ants of Q, given by (4.1.2), to the first two cumulants of Y ,
l
and solves to obtain
To determine the
paramet~rs
a
Then, for the moments of Y , we have by (2.3.1),
l
(4.2.1)
and by (2. 4 .1 ),
(r>-f) .
(4.2.2)
Patnaik (1949) offered his two-moment central Cf~i-squarEl approximation primarily for the purpose of evaluating the distribution
function of a single noncentral chi-square r.v.
At the same time, how-
ever, he pointed out that Y might also be used to approximate a linear
l
combination of independent noncentral chi-square r.v.'s.
mation is well known and
w~dely
used d"\le to its
His approxi-
simpli~ity;
however,
the recent publication of extensive tables of the cumulative noncentral
chi-square distribution by Haynam, ~~. (1962) makes the primary intended use of Y passe.
l
More recently, Pearson (1959) offered his
three-moment central chi-square approximation Y , again for the same
2
primary purpose as Patnaik's approximation, and again, for this purpose,
25
it too is passe.
Imhof (1961) used Pearson's approximation to evaluate
pfEaix2(vi''''i) ~uJ and found, as did Johnson (1959) for
F(u)
=
F( u)
= p{ x2( v,,,,) ~ u}, that in general Pearson's approximation is
superior to Patnaik's.
Pearson considers the r.v.
Clearly, the first two cumulants of Y2 agree with the first two cumu1ants of Q.
Equating the third cumu1ants of Y and Q, he obtains
2
~
8v 2
(2::r
=
"3
yielding
(4·3·1)
It follows that
Now this approximation is fully applicable for the approximate
evaluation of F(q)
= p(Q
~
q)j however, we shall soon see that b 1 must
26
be restricted to nonnegative values for the purpose of approximating
fractional moments of Q.
Moreover, because of the presence of b , it
l
is clear that our prior knowledge of the moments of central chi-square
will be of little value in determining E(~) when
We have Y2 = a2 1( v2 )+b l .
The pdf
- -
1
l'
is not an integer.
for Y2 is readily shown to be
(y-b )
~a2
1
Hence,
J
E(~) =
b
and now it is clear that b
l
00
yrg(Y)dY,
l
must be restricted to nonnegative values in
order to assure that E(~) be real when
r(~+r)
r (
Less trivially, suppose b
y-b
t
l
= -- ,
b
l
V; )
> 0.
l'
is fractional.
When b
l
= 0,
(I' > _ :2)
Making the transformation
we obtain
l
00
J
o
V
-2+ r
= b
1
2
l'
+ V22 + 1; ~)
2a
2
(11'1
< (0),
(4.3.4)
27
where the function U(~;c;~) is also known as a confluent hypergeqmetric
function [see Slater
representation].
(1960), p. 38, equation (3.1.19), for its
inte~ral
This function is expressible in terms of the simpler
I\:ummer's function [see
(9.1.1.15)]. Although (4.3.4) exists for all real
r, this fact does not, of course, mean that
(4.3.4) may be regarded as
a suitable approximation for all moments of Q.
v
In the recursion
(9.1.1.15), l€lt a ::;
by b1/2":1' and substitute f'or U
v
-f ' c ::; r + i + 2,
(:2 ; .; ~~)
replace x
f'rom (4.3. 4) to obtain,
after simplification, the moment recursion
(Irl
<(0).
(4·3.5)
Consider now the implication of the imposed restriction that
bl ~
0, which by (4.3·3) is equivalent to
Thus, only when
(4.3.6) holds will the Pearson approximation be
applicable for approximating fractional moments of Q.
We next consider a logical extension of Patnaik's approximation.
4.4 Patnaik-extension Ap~roximation [Y3 ::; a x2(v ,Sl)]
3
3
,
Rather than solving for the parameter a , we shall solve for
3
Equating the first t;hree cumulants of KQ and x2( v , ~l) we
3
3
obtain the system
K ::;
-l:.-.
a
28
= v.3
+ 2~1'
(a)
~K2 = 2v.3
+ 8~1'
(b)
~K.3 = 8".3
+
48~1'
(0)
KK l
(4.4.1)
Obse:rve that, with respect to the right-hand sides of (4.4.1),
8[(b)-(a)]
t~e
= (c), yielding a quadratic equation in
K with respect to
left-hand sides, viz.
Solving this system of equations, we get
(4.4.2)
(4.4.4)
Clearly, we must require that v.3 > 0, Sl
(4.4.2) be nonnegative.
~
0, and that the radicand in
Note, with some anticipation, that this last
requirement is a happy complementation to (4 ..3.6),
!.~.,
here we re-
quire that
With the inequalities just imposed on v.3 and Sl' we obtain from (4.4 ..3)
and (4.4.4) the following inequalities on K:
(4.4.6 )
29
Substituting for K, from
(4.4.2),
in
(4.4.6)
we obtain
We shall now show that the smaller solution for K is inadmissible,
being always below the lower bound.
Suppose
Multiplying through by -2 and adding 4K~ to each side we obtain
or
which is not possible because, by' (4.1.1), the
K
r
are all strictly
po~itive.
For the larger solution for K, it is easily shown that the lefthand inequality of
(4.4.7)
is always satisfied.
the right-hand inequality of
(4.4.7)
We shall now show that
is also satisfied.
We have to show
that
(4.4.8)
To establish
Then
(4.4.8),
suppose (as will be proved shortly) that
30
Hence
Then
Thus (4.4.8) is established, given (4.4.9).
To prove (4.4.9), we substitute for
K ,K
l
2
, and
K
3
from (4.1.2),
obtaining
k
f(i,j) > 0,
L:
(4.4.10)
i,j=o
where
The truth of (4.4.10) is readily established.
f(i,i) =
~~(1+8~.)
>
J.
J.
First, for i=j, we have
0,
and hence
k
L:
f(i,i) > 0.
i=l
To sweep through the remaining k(k-l) terms, consider jointly the
index pairs (i,j) and (j,i), irj.
f(i,j) + f(j,i)
=
We have
~:(j~
f4(1+2~.+6~.+12~.~.)~~
J. J
J.
J
J. J
J
4(1+2~.+6~.+12~'~')~i4
J
J.
J.
J
+
2 2} .
- 6(1+4~.+4~.+16~.~.)~.~.
J.
J
J. J
J. J
Now it is not difficult to show that the part in braces is a positivedefinite quadratic form in ~: and ~~.
J.
J
Hence,
31
k
~
f(i,j)
>
O.
i,j=l
irj
Thus the truth of inequality
(4.4.10)
is established, and with it the
admissibility of the larger solution for K.
By way of a summary for approximation Y3' we have
where
(4.4.11)
We recall that approximation Y3 is applicable for approximating
moments of Q only when
(4.4.5)
is satisfied.
Then, by
(2.3.1),
v
F(-r . ..2._,. )
, 2' ~l
(4.4.12 )
and by
(2.3.2),
(4.4.13 )
32
Looking again at the complementary nature of the inequalities
(4.3.6)
and
(4.4.5),
it is easily verified that, under equality in each,
Y and Y specialize to the same r. v. (in fact, to Y ). Thus, combining
2
l
3
Y and Y , with their respective regions of applicability, we obtain a
2
3
single three-moment approximation of full applicability for approximating moments of Q.
Letting Y be the resulting r.v., we have
if t:. < 0,
if t:.
~
(4.4.14)
0,
where
(4.4.15 )
the discriminant of applicability.
At this point, the following conjecture regarding the distribution
function of Q is suggested.
Conjecture:
For the purpose of approximating F(q) ~ p(Q~q), the r.v.
Y is, in general, superior to Y2 [recall that Y is fully
2
applicable for the approximate evaluation of F(q)].
In
particular,
F(q) == p(y~q)
=
This conjecture will be tested by numerical examples in Section
(4.4.16 )
6.
33
An approximation of this nature, as it turns out, is much too
involved to warrant serious consideration.
Moreover, it would neces-
sarily have restricted regions of applicability for approximating
either fractional moments or the distribution function of Q.
Further-
more, we could hardly expect to find a counterpart of Y to serve in
3
complementary fashion to Y as Y3 does to Y .
2
4
Accordingly, approxima-
tion Y4 will not be considered further.
4.6
Summary
In summary, then, for purposes of approximating fractional moments
of Q, we have two approximations of full applicability.
The first, a
two-moment approximation based upon Patnaik's r.v., Y , viz.
l
(4.6.1)
where E(~) is given by
(4.2.1).
The second, a three-moment approxima-
tion based in part upon Pearson's r.v., Y , and in part upon the newly
2
proposed r.v. Y ,
3
i.~.
E(~),
E(Qr) ~ E(~)
~ < 0,
=
(4.6.2)
E(~),
where E(~), E(~), and ~ are given by
~ ~ 0,
(4.3.4), (4.4.12),
and
(4.4.15),
respectively.
In order to judge the quality of the approximations
(4.6.2),
(4.6.1)
and
we now turn to consider exact solutions for E(Qr), even though
their numerical evaluation may be difficult.
5.
THE GENERAL PROBLEM, EXACT SOLUTIONS
5.1
Introduction
In the search for exact solutions to the general problem, we are
aided considerably by the work presented in two recent papers.
Shah
and Khatri (1961) and Imhof (1961) give the distribution function for a
nonnegative-definite quadratic form in noncentral normal r.v.'s.
Thus,
by definition,
=
E(Qr)
J
00
qrdF(q)
o
where F(q)
~ q!.
= P {Q
=
E(Qr)
Alternatively,
J
00
[l_F(ql/r)]dq
(r > 0)
o
where (5.1.2) is a straightforward generalization of a corresponding
relation for r
=1
given by Parzen (1960), p. 211, equation (2.39).
The choice between (5.1.1) and (5.1.2) would be based upon efficiency
of numerical evaluation.
5.2
F(q) as Given by Shah and Khatri
In the notation used here, Shah and Khatri (1961) give F(q) as an
infinite double series:
k
F( q) = ( n
i=l
where A.
=t
k ( S.
L: 2:.
1 CTi
2
CT.
)
~
)2
,
e
Q*
(_1)i2j-iqi+jE[Q*iL*2j]
-A.
i~(2j)~r(i+j~+1)
=t
k
L:
1
(5·2.1)
35
They show that the series in (5.2.1) is absolutely convergent, and
state that, in practice, one would compute a f'inite number of' terms and
then give upper and lower bounds on F(q).
tion of' F(q),
!.~.,
We
•
requ~re
an
II
exact
tI
evalua-
for any preassigned small quantity 8 > 0, we must
be able to evaluate enough terms so that the absolute value of' the
error committed is not greater than 8.
Success in this regard would
depend sharply upon our being able to supply the joint moments of' Q*
and L*.
Shah and Khatri state, "The joint moments of' Q* and L* are easy to
obtain from the joint cumulants of' Q* and L*. "
Now the joint cumulants
of' Q* and L* are readily obtained, and, hence, theoretically at least,
the joint moments may be obtained.
Hm.,ever, bivariate moments expressed
in terms of' bivariate cLUuulants have been tabled only for
[see Cook (1951)],
our purposes.
j.l ~ .,
~J
i +j
~
6
Such a limited tabulation would hardly suff'ice f'or
Moreover, in view of the present techniques discussed in
the literature [see,
~.g"
Kendall and Stuart (1958)], we would have to
regard the prospect of generating further moment-cumulant relationships
as quite unattractive.
Hence, it would seem that we must either abandon
further consideration of' (5.2.1) in conjunction with either (5.1.1) or
(5.1.2), or we must f'ind a simpler device f'or generating moments in
terms of' cumulants.
A simpler device was discovered, and it was to
have been presented here as an appendix, but, as it turns out, the
discovery is essentially a "rediscovery" of a device published by David
(1952) .
36
5.3 F(q) as Given by Imhof
Again, in the notation used here, Imhof (1961) gives F(q) as an
integral,
00
F(q)
=~ _ 1
:rr
J
sin[A( t) - ~]
t pet)
dt,
o
where
2
2
A(t) = ~ ~k [ v.tan- 1 (~.t)
+ Si t4 2 ] '
i=l
~
~
l~ t
i
and
exp(~ i=l~
(note that p(O) = 1, and p(t) is a monotonic increasing function), where
the v. are the degrees of freedom associated with chi-square r.v.'s in
~
the quadratic form
Q*
=
k
~
. 1
~=
22
~ . X- ( v . , A.. )
~
~
(v. > 0,
~
~
i
= l, ... ,k).
Clearly, Q* = Q when the v. are set equal to unity, and for greater
~
generality in work to follow, the vi are retained.
Actually, Imhof's
expression for the distribution function, serves for a wider class of
quadratic forms than we consider here [see Imhof's paper].
Imhof further gives
1:..J
f( q) = dF( q) =
dq
2:rr
o
00
cos[A(t) - ~ tq]
pet)
dt,
37
although, possibly due to a printing error, the factor ~ before the
integral sign was not given.
factor
Primarily because of the convergence
t1 present in the integrand of (5.3.1), but absent in (5.3.4),
the moment expression (5.1.2) would be preferable to (5.1.1), should we
choose to use Imhof's expression for F(q) rather than that given by
Shah and Khatri.
5.4
The Choice
Thus, to evaluate E(Qr), at least at the present stage of deve10pment, we have a choice of the following nature:
(i) we may write a
program for the numerical evaluation of the integral of an infinite
double series, with a side program for evaluating the joint moments of
Q* and L*, or (ii) we may write a program for evaluating what is essentia11y a double integral.
Before choosing between (i) and (ii), we
note that the absolutely convergent double series (5.2.1) may, of
course, be expressed as a single series.
Even so, it seems clear that
the advantage rests with (ii) rather than (i).
As it turns out, the choice is well made because, as we shall now
see, the double integral may be reduced to a single integral.
5.5
Reduction to a Single Integral (I)
upon substituting (5.3.1) into (5.1.2), we obtain
sin[A( t)- ~ ql/r]
]
dt dq.
t pet)
(5·5.1)
Now, because q appears in the integrand of (5.5.1) in a particularly
simple way, we might hope that by changing the order of integration, we
38
would be able to perform the q integration analytically and thus reduce
(5.5.1) to a single integral.
With this idea in mind, we should like to replace the ~ in (5.5.1)
by an integral with respect to t of some function of q and t.
It turns
out that
J
00
o
1!
which is equal to '2 for all q > 0, is a good choice.
Making the
substitution and employing the rule for the sine of a difference between two angles, we obtain
co
=J J
1!E(Qr)
o
00
[fl(t)cos(~ ql/r)+f2(t)sin(~
ql/r)]dtdq,
0
where
f
1
(t)
= Si«)
tp t
and
f
()
2 t
= -1t
( 1 _ cOps(t() ) .
Proceeding now in a formal way, we have that
00
d{Qr) =
J [f J
1
00
(t)
cos(~
00
ql/r)dq+f2 (t)
J
Sin(~
0 0 0
t
cos('2 x)dx
+ f (t)
2
J
o
00
r 1
x -
Sin(~ x)ax }dt.
ql/r)dq ] dt
39
However, the two integrals inside the braces exist only for 0 < r < 1
[see ~.~., Grooner and Hofreiter (1961), p. 118, (lOa) and (lOb)].
particular interest in this study, though, is the case r =
For r
= t,
Of
t.
we obtain the result
00
=J
(5.5.6)
o
=
~
J
00
o
p(t)+sinA(t)-cosA(t)
t 3 / 2 p( t)
It can be proved that (5.5.1), with r
= t,
however, we shall not pause now to do so.
dt.
is in fact equal to (5.5.6),
Rather, we shall proceed to
a more general result and prove that result rigorously.
5.6 Reduction to a Single Integral (II)
The results presented in this section follow in elegant fashion as
a direct consequence of two key suggestions offered by Yrr. A. A. Fojt.
Again, we shall proceed in a formal way, without regard for the
validity of steps taken, deferring a rigorous proof of the end result
until later.
We have seen that the integrals inside the braces in (5.5.5) do
not eXist, in particular, for r
~
1.
Mr. Fojt suggested that the diffi-
culty might be circumvented by temporarily introducing some convergence
factor, say e
-a:x,with a > O.
Then
40
J
co
x r-l e -axsin (t
'2 x )dx =
0
r(r)
2
( a2 + 4'
t 2
)!
sin(rcl»,
(5.6.1)
J
00
x
t
r-l e -axcos(x)dx =
2
0
where
cl>
r(r)
r
(a2+
t)
= tan-l( 2a'
(5a) and (5b).]
= lim
cos(r4»,
~)'2
[See, ~.~., Grabner and Hofreiter (1961), p. 139,
Now, with thes e results, we might hope that
J
co
a-a o
If passage to the limit under the integral sign were made at this point,
we would get
o
and it is readily seen that, for r =~, this result agrees with (5.5.6).
However, for r ~ 1, we still have problems near the origin (t=O).
is not difficult to see that, for t small enough,
sin A(t) = ct + O(t3 ), c a constant,
cos A(t) = 1 + O(t2 ),
It follows that
It
41
Clearly, the integral in (5.6.3) would be well behaved, in the neighbor-
i-a'
hood of t = 0, if the integrand acts like
a > 0. For example,
3
t
-3/2
suppose r = 2' Then the integrand behaves like t
,the trouble
stemming from the first term a10nej for t =
22'
we have trouble stemming
from the second term as well.
Mr. Fojt now suggested that the difficulty might be avoid.ed by
subtracting from f (t) and f (t) the leading terms in their power
1
2
series expansions for t small, these terms to be treated separately.
To establish the nature of the f (t) and f (t) power series
1
2
expansions for t small, we shall find it helpful to obtain first the
characteristic function for Q*/2.
exp [
~ iA.j(J~t)
_ _ _ j=l
_ _1-i(J~t
_....J<--
~
k
n (l-i(J~t) 2
j=l
J
is~t
exp
J
(~ j=l
~ 1-i(J~t
)
J
=
v.
,
(5.6.6)
.-.J.
k
n (l-i(J~t) 2
j=l
J
where M 2
(t) is given by (2.2.5), and
X (v,A.)
that
s~ = 2(J~A.j'
It turns out
42
~Q~(t)
1
= Pm '
2
and
th~
Although Imhof did not point out explicitly
is clear that he was aware of them.
relations (5.6.7), it
He does, in fact, give other
relations that enable us to verify readily (5.6.7).
~
(t)
Q*
= e iA(t) = cosA(t)+isinA(t )
p(t)
"2
By
It follows from
p(t)
definition,
where
!J.
j
= E( Q*j ) .
It follows then, from (5.6.8) and (5.6.9), that
and
~
cosA t
P t
=
en
L:
7=0
!J.'
(_1)7 ~ (1)
2~. 2
'
and, for t small, these expansions exist.
27
'
The cor;responding expansions
for fl. (t) and f 2 ( t) follow immediately from (5.6.10), $.,
f l (t
_ 1
) - t
00 (
~
-1
)7
~;l+l
t 2r+l
(2 +lH (i)
,
r=07
G
and
f
2
(t)
= 1t
00
~
1
r=
~'
(_1)7- 1 ~
2~.
'
(!)
2
2r
.
Let us define
and
where the symbol
Cal,
as used for the upper ~imits of summation in
(5.6.l2), means "the largest integer ~ arr (the smns being zero if the
l~mit
lower
exceeds the upper limit).
It further follows that, for t
small,
Let us now designate the integrand Of (5.6.2) by h(t,a,r), and
rewrite it as
(5.6.14)
where
44
K( t,a,r )
r (r+l)
i
r
=
(c?+ ~ )2
and we recall that
[k( t,r ) C013 ( rei> ) +k ( t,r ) sin(r4l)],
l
F
= tan- l (2~)'
<II
co
:rcE(Qr) = lim
J
[H(t,a,r) +
a~o 0
= lim
J':(
We know, by
(5.6.13),
Then (5~6.2) may be rewr+tten
K(~,a,r)]dt
t,a,r )dt + lim
a-o o
a-o
J
00
K( t,a,r )dt.
0
that if pass~ge to the limit under the integral
(5.6,17)
sign were now to Qe made, for t~e first term of
res'Ulting integral would be well behaved for t small.
nature of the second term?
J
00 K( t,a,r )dt
(5.6.16)
,,0
alone, the
But what is the
Happ1ly,:t,t turns out that
(for a > 0; r > 0, r
~ 1,2, ... ).
o
To see the truth of this claim, we note that
co
J
o
= 0,
for all
r-l
r <2
'
f
providing a > 0, r > 0, and r
[See,~.~.,
1,2, ....
Hofreiter (1961), p. 110, (15a).]
( )J
Grebner and
Similarly, we note that
1f./2
qos r-21.,1 4lsin(r4l) sin21-14l d4l
= 92 a
o
=
r
0, for all 1 < 2
f
providing a> 0, r > 0, r
[See, ~.~., Grabner and
1,2, ....
Hofreiter (1961), p. ~09, (lOb).]
'
T~e
vanishing of these two integrals
clearly implies the truth of (5.6.18).
Hence, (5.6.17) reduces to
1f.E(Qr) = lim
J
00
H(t,a,r)dt
a-o o
=
J
00
H( t, o,r )dt,
o
and we have now obtained a fairly simple single integral expression for
the fractional moments of Q*.
A formal proof that this single integral
does, in fact, represent the fractional moments of Q* is given as
Appendix 9.,3.
We conclude this section by giving a more detailed expression for
46
1fE(Qr)
=J
[ r-l]
00 r
2 r(r+l) { [
t r +l
Sine(,) _ 2
(-If'
L:
7=0
p t
o
[~]
- [ cost())
L:
P t
(_1)7
7=0
where A(t) and p(t) are given by
III
~
(1)
2~.
2
I
Il'
27+1
(27+l)~
27 ]
}
sin(r!£) dt
2'
(5.3.2) and (5.3.3) respectively.
6.
NUMERICAL COMPARISONS
6.1 Introduction
With this section, we come, at last, to the consideration of
numerical comparisons of approximate with exact solutions for the
fractional moments of Q, and, somewhat incidentally, for the distribution function, F(q), of Q.
.!.
E(Q2)
= E(Z),
Moment comparisons are confined to the case
and these, with three exceptions, for two-dimensional
(k=2) examples.
More elaborate comparisons were planned; however,
funds available for numerical purposes were exhausted before such
comparisons could be made.
Nevertheless, those comparisons judged to
be most stringent (in some sense) were accomplished first, and it is
believed that the conclusions drawn from these comparisons vould have
been supported by the other comparisons.
6.2 Distribution Function of Q
With regard to the conjecture of Section 4.4, consider the approximation (4.4.16) for F(q).
When 4 < 0, the approximating r.v.
Y,
(4.4.14), is given by Pearson's r.v. Y2 , the quality of Y2 for approximating F( q) having been considered by Imhof (1961). In particular, we
should like to consider some examples where 4 > 0 (then Y
= Y3 ) so that
comparisons between Y and Y might be made.
2
3
Of the examples considered by Imhof, three (his ~, Q6' ~$6)
were found to have 4 > O.
Moreover, for these three examples, Pearson's
approximation is seen (see Imhof) to give particularly close agreement
with the exact values determined by (5.3.1).
Thus these examples were
judged to be a fairly stiff test for the r.v.
Y , in that Y2 leaves
3
48
little latitude for improvement.
Exact values for F(q) were obtained
using (5.3.1), while the various approximate values, based upon the
Y (Patnaik's), Y , and Y , were determined by suitable
l
2
3
specializations (of the parameters) of (5.3.1). The results obtained
three r.v.'s
are presented as Table 6.1.
We note that, without exception, the approximate values based upon
Y are closest to the exact values.
3
While these results do support the
conjecture, we do not, of course, conclude that determinations of F(q)
based upon Y would always be closer to exact values than those based
3
upon Y2 (or even those based upon Yl).
In fact, it may well be, for
practical applications, that Y is not enough better than Y2 to warrant
3
serious consideration of the
combined
ll
Y over Y alone, if one
2
is merely interested in approximate values of F(q). In any case, we
II
r. v.
shall not pursue the matter further here, it being quite incidental to
this study.
6.3 Fractional Homents of Q
For moment approximations, as noted in Section
4,
the situation
is a bit different from that just considered in the previous section,
in that Y applies only when 6.
2
when 6.
~
0).
< 0 (and Y3' as before, applies only
The comparison here, then, is essentially Y versus Y ,
l
1:. •.!::" the three-moment approximation versus the two-moment approximation.
In an attempt to avoid comparisons that might be particularly
favorable toward Y, rather than Y , we deliberately attempted to bias
l
matters in favor of Y1' trusting that the superiority of Y would show
through in spite of such bias (we were not disappointed).
e
e
e
Table 6.1. Values (approximate and exact) of F(q) = p(Q
Quadratic Form
~
q)
Approximate
q
P(Yl~q)
Exact
P(Y2~q)
P(Y3~q)
p(~q)
Qa
=
(.7)X2(6,3) + (.3)X2(2,1)
2
10
20
.0046
·5955
·9770
.0072
·5911
·97793
.0061
·5913
·97792
.0061
·5913
·97792
~
=
(.7)X2(1,3) + (.3)X2(1,1)
1
6
15
.0281
.6052
.9754
.0485
.5916
·97771
.0448
.5924
.97766
.0452
.5924
.97761
3·5
8
13
.0395
.05899
·9526
.0440
·5847
·9539
.0437
·5848
·9538
.0437
.5848
·9538
Qc =lQ
2
a
+l~
2
\5
50
Of the odd moments of Z [!.~., E(Qr), r
clear from considerations of
interpolation
choice to consider the first (r
=
= ~,~,~, .. _],
~_
it seems
extrapolation
that a
~), or possibly the next (r = ~),
would be a choice favorable to Y - We chose the first (r
l
c
~)
_
Having
selected the moment to be calculated, we then faced the more difficult
problem of choosing the actual examples (specific quadratic forms).
To aid in this choice, we determined the 6 values for all twodimensional quadratic
fo!~~
formed by taking all possible combinations
(864) of the following values for the parameters:
2
0"1
=
1-,
2
0"2
=
1, 2,
A. ,A.
l 2
(v
l
4, 9, 16, 25;
= 0,
1 1 1 1 1
1, 2,
32' lb' '8" 4' 2'
=V
=1)
2
4, 8, 16, 32
This unwieldy set of examples was then cut in half by an arbitrary
1
decision to consider only every other value for A.l (starting with 32).
Also, the value 0"22
=
1 was judged to be relatively uninteresting
(special case 1 of Section 3 applying to such examples).
number of combinations was reduced to 360-
Thus the
Now the advantage, if any,
of Y over Y tends to vanish as 6 tends to zero (recall that Y = Y2 =
l
3
for 6
= 0)_
1-
Thus, for each (O"~,A.l) combination, it was decided to
select, from among the 12 possible values for A.2 , that value of A.2 such
that
16\
is smallest, or next to smallest, depending upon which choice
gave the "nicer to work with" values for the
s-
~
(recall
s- = 0". ~)
~
~
~
51
--this rule for selection was followed, but with the added provision
that, when possible, alternating selections should correspond to
values alternating in sign.
~
In this manner, then, the number of two-
dimensional examples to be treated was fixed at 30, approximately half
with
~
> 0 (implying that Y :::: Y ) and half with
3
Y :::: Y ).
2
~
< 0 (implying that
In one instance, the rule yielded an example for which
The example was retained, serving as a check.
~
::::
o.
In addition, the three
quadratic forms shown in Table 6.1 were included as examples.
1
Exact values of E(Q2) were determined by (5.5.7), while the approx1
1
imate values, values of E(Yf) and E(y2), were determined in accordance
with Section 4.6.
The results obtained are presented as Table 6.2.
On inspecting Table 6.2, one is immediately struck by the apparent
excellence of the values based upon the Patnaik (Y ) approximation, so
l
much so that it would seern foolish to strive for a more sophisticated
approximation.
In this respect the results presented are misleading,
the examples having been selected so as to
agreement of approximate and exact values.
ins~~e
reasonably close
What is important, though,
is that, in spite of our attempts to bias matters in favor of Y , in
l
only six cases are the approximate values based upon Y closer to the
l
exact values than those based upon Y.
6.4 Conclusions
In view of the results obtained, it does seem quite safe to
conclude (as one feels intuitively) that the moments of Y will, in most
cases, serve as better approximations to the corresponding moments of Q
(or Z) than those of Y .
l
We hasten, however, to add a rather obvious
52
1
Table 6.2. Values (approximate and exact) of E(Q2) = E(Z)
Approximate
Quadratic Form
1
E(yi)
Exact
1
E(Y2)
1
E(Gl)
Qa=(.7)X2(6,3)+(.3)X2(2,1)
~=(.7)X2(1,3)+(.3)X2(1,1)
3·021
3.020 (+)a
3·020
2.290
2.282 (+)
2.281
Qc~2 Qa + 21 Qc
2.728
2·727 (+)
2·727
~=X2(1,t2)+2X2(1,~)
1·765
1.778 (- )
1.772
~=X2(l,g)+2X2(l,l)
2.437
2.423 (+)
2.423
~=x2(l,~)+2X2(l,g)
1.904
1·904 (0)
1·901
Q4=X2(1,2)+2X2(1,~)
2·521
2·512 (+)
2·508
~=X2(1,8)+2X2(1,0)
4.249
4.244 (+)
4.243
Q6=X2(1,32)+2X2(1,4)
9.046
9.045 (+)
9.045
Q7=X2(1,f2)+4X2(1,~)
2.095
2.141 (- )
2.123
Q8=X2(1,g)+4X2(1,~)
2.648b
2.643 (+)
2.652
~=X2(1,~)+4X2(1,g)
2.318b
2.360 (- )
2·337
Q10=X2(1,2)+4X2(1,2)
4.706
4.694 (+)
4.693
Q11=X2(1,8)+4X2(1,2)
5.8898
5·8922 (-)
5.8914
~=X2(1,32)+4X2(1,~)
8.302
8.301 (+)
8.301
53
Table 6.2.
(continued)
Approximate
Quadratic Form
1
E(1'r)
2
1
y?-
1
(1'8)
Exact
1
E(1'2)
1
E(Q2)
2.908
2.963 (- )
2·953
Q14=Y?-(1,~)+9Y?-(1,t)
b
3.752
3·729 (+)
3·751
Q15=Y?-(1,t)+9Y?-(1,~)
3·081
3·157 ( - )
3·133
~6=Y?-(1,2)+9Y?-(1,2)
6,574
6.540 (+)
6.542
~7=Y?-(1,8)+9Y?-(1,2)
7.496
7.499 (- )
7·504
~8=Y?-(1,32)+9Y?-(1,8)
b
14.5489
14.5488 (+)
14.5493
~9=Y?-(1,t2)+16Y?-(1,~)
3.758
3·780 (- )
3.808
~0=Y?-(1,~)+16Y?-(1,t)
4.894b
4.846 (+)
4.876
~1=Y?-(1,t)+16Y?-(1,§)
3·895
3.976 (- )
3·963
~2=Y?-(1,2)+16Y?-(1,2)
8·521
8.454 (+)
8.467
~3=X2(1,8)+16Y?-(1,t)
6.449
6·533 (- )
6·530
~4=Y?-(1,32)+16Y?-(1,8)
18.015
18.011 (+)
18.011
~5=Y?-(1'~2)+25Y?-(1,~)
4.626
4.639 (- )
4.674
~6=Y?-(1,~)+25Y?-(1,t)
6.052
5.980 (+)
6.014
~7=X2(1,t)+25X2(1,§)
4.808 (- )
4.812
~8=Y?-(1,2)+25X2(1,t)
4.739
b
6.395
6.377 (+)
6.437
~9=Y?-(1,8)+25Y?-(1,t)
7·380
7.483 (- )
7.494
~0=Y?-(1,32)+25Y?-(1,8)
21.663
21.653 (+)
21.654
~3=X (1'32)+9
aThe symbols (+),(-~(o) mean that 1'=1'3'1'2,1'1' respectively.
bpatnaik value closer to exact value.
comment:
Even as extrapolation is usually risky, wlllle interpolation
is generally sOlmd, so it must also be in applications of the moment
approximations proposed in this study.
In particular, the early
fractional moments of Y [E(~), 0 < r <
4, say] should serve well for
approximating those same moments of Q, but we would place little reliance on the approximation for moments of higher order (r >
4, say).
However, interest usually centers on lower moments, the first four
integer moments (beyond the zero-th) being those most often investigated.
If one must evaluate many moments, or moments of orders much
beyond those upon which the approximating r.v.
Y is based, then he
would do well to enlist the aid of a modern electronic computer so as
to obtain exact evaluations based on
(5.6.20).
55
7.
SUMMARY
Fully stated, the problem treated in this thesis is that of
determining positive fractional moments of a nonnegative-definite
quadratic form in nonsingular noncentral normal random variables.
By
way of sunnnary, let us review the contents of the several sections and
appendices.
Section 1:
The stage is quickly set by an immediate statement of
the problem in its simplest symbolic formatj interest in fractional
moments of a quadratic form Q is motivated by an interest in integer
moments of random distance, a topic that has apparently received little
attention in the past, perhaps because of the complicated nature of the
distributions involved.
All distance problems in nonsingular normal
random variables are shown to be reducible, by well-known linear transformations, to the same format as that given to the statement of the
problem of the thesis.
Section 2:
The noncentral chi-square distribution may be regarded
as being the basic statistical distribution underlying quadratic forms
in normal r.v.'s.
The section is a brief resume of the nature and
properties of the noncentral chi-square r.v.j two properties are neWly
established, viz., a general expression for its moments, and a secondorder moment recursi.on.
Section 3:
Each of Sections 3, 4, and 5 contains a different
approach to the problem.
The approach of Section 3 might be labeled
"approach by specialization."
parameters k,
1,
By suitable specializations of the
and D (see first paragraph of Section 1), we hoped to
obtain cases admitting fairly simple exart solutions ror fractional
moments.
The two special cases that are treated in the section yield
exact solutions that are shown to be expressible in terms of welltabled special mathematical functions, viz., the error and modified
Bessel functions, and the complete elliptic integrals.
The first of
these cases is believed to be new, while the second is the work of
Scheuer (1961), being included for purposes of completeness.
·
S ec t ~on
4:
.
This sec t'~on .right be labeled tI approac11 'b y approxlma-
tion," in the sense that ,{e try to approximate Q by various more
tractable r.v.'s whose fractional moments are relatively easy to obtai.n.
This rather popular approach to dealing with noncentral chi-square
r.v.'~
has its foundation in the
wor~
of Patnaik (1949), who proposed a
central chi-square r.v. for the purpose of evaluating (approximately)
the distribution function of a noncentral chi-square r.v.
We have
considered three such approximating r.v.'s, Y (Patnaik's two-moment
l
approximation), Y (Pearson's three-moment approximation), and Y (a
2
3
pewly proposed three-moment approximation). The most interesting result obtained in the section is that,
~though
neither
Y2
nor Y ,
3
individually, provides a fully applicable APproxirration for fractional
moments of Q, Y and Y together yield a r.v. Y of full applicability.
2
3
:j:t is this fact that gave rise to the conjecture that Y is, in geperal,
superior to Y (alone) for the purpose of evaluating (approximately)
2
the distribution function of Q.
Section
5: At the very outset (before much of the research re-
ported in this thesis had been accomplished), it seemed quite Unlikely
that a significant contribution toward a tractable exact solution of
57
the general problem would be possible.
It was clear that an exact
expression for the fractional moments of Q could be written explicitly,
because explicit expressions for the distribution function of Q, F(q),
had been recently published.
Thus
E(Qr)
=
J
00
qrdF(q),
o
but the expressions did present a fOrmidable sight.
How~ver,
with the
decision to use F(~) as given by Iffi40f (1961), and with the observation
that E(Qr) could be e~ressed alternatively as
J
E(Qr~ =
00
[l_F(q:J./r)]dq,
o
matters did look better.
We had what was essentially a double integral,
and, for a modern high-speed electronic computer~ that shouldn't
present too much of a problem.
J;3ut it did~
After many long weeks of programming and
II
debugging,
II
it became
clear that the double-integral evaluations would consume too much
machine time.
The thought of possibly reducing the double integral to
a single integral
'Or
changing the order of integration returned to mind;
the idea had occurred earlier, but, at that time, it didn't look
promising and was quickly dismissed.
tion for taking a more critical look.
the reduction
~as
But now there was stronger motivaSome success followed, in that
apparently possible for 0
This didn' t seem right.
< r < 1, but not for r > 1.
The author's understanding of the nature of
Fubin1's theorem led him to believe that the requction should be
58
possible tor all
~
> O. The author put
analyst (and friend),
As noted in the
made two key sU&gestions; the
f~rst
t~e
te~t
matter tq a numerical
of the
ch~pter, t~is
seemed to accomplish little by it-
self, but, when it was considered together with the second
the
~edu9tio~ ultimat~ly
the exact
e~ression
followed in fine
fa5~on.
sug~estion,
This reduction of
for the fractional moments of Q to a single inte-
gral (along with the rigorous proof that the single integral
correct--Appendix9~)is consid~red
this thesis.
friend
~s
to be an important contribution of
This single integral is no worse than a great many
others that are routinely evaluated by electronic computers.
Section .6:
In this section, a few numerical oomparil3ons are
given which support the conjecture that the r.v.
Y is, in general,
superior to Y for eva1ua~ing (approximately) F(q). other comparisons
2
made in the section support the hypoth~sis that the r.v. Y is, in
general, superior to Y for evaluating (approximately) E(Qr), provided
1
r is not too large, of courser In fact, for either purpose, it is
believed that apProximations based upon Y will be
AppendiX
i
9.1~
~uite ~eliable.
This appendix gives various properties of the confluent
hypergeometric function.
The function arises in Sections 2, 3, and 4,
Th,e two theorems stated and proved in the appendix appear to be of a
fairly general and useful nature, and we would hqpe i;hat they may find
important applications in areas other than that treated here.
Appendix 9.2:This appendix contains a very useful approximation tor
the ratio of two gamma functions,
It would be presumptuous to imagine
that it has not peen previously discovered.
eluded because it has obvious application
Nevertheless, it
he~e,
~s
in-
and partly because
59
of the refinements given that enable one to obtain any required
accuracy.
Appendix 9.3: With this appendix, we conclude the
thes~s.
It
cOntains the rigorous proof that the single integral obtained in
Section 5 does, in fact, represent fractional moments of Q.
60
8.
LIST OF REFERENCES
Cook, M. B1 1951. Bi-variate k-statistios and cumulants of their
joint sampling distrib'\ltion. Biometrika 38:179-195.
DaVid, H. A. 1952. An operational method for the derivation of
tions between moments and cumulants. Metron 16 :3-9.
r~la­
Erdelyi, A., Editor. 1953. Higher Transcendental Functions, Vol. 1.
MoGraw-Hill Book Company, Inc., New York.
Fisher, R. A. 1928. The general sampling distribution of the multiple
correlation coefficient. Proc. of the Royal Soc., Series A, 121:
654-673.
Fix, E. 1949. Tables of noncentral x?-.
in Statistics, Vol. I, No. 2:15-19.
Un!v. of Calif. fublications
Grooner, W. and N. Hotreiter. 1961. Integraltafel, Zweiter Teil,
Bestim;rp.t~ Integrale.
Sp;ringer... Verlag, Wien.
HaytlI3.ID, G. E., Z. GOvindara~ulu, and F. C. Leone. 1962 . Tables of the
cumulative non.. central ehi-squ~re distribution. Report No. 104,
Nov. 1962. Statistical Laboratory, Case Inst. of Technology,
Cleveland.
Imhof, J. P. 1961. Computing the difltribution of quadratic forms in
normal variables. Biometrika 48:419-426.
Johnson, N. L. 1959. On an extension of the connexion between Poisson
and X2 distributions. Biometrika 46;352-363.
Kendall, M. G. and A. Stuart. 1958. The Advanced Theory of Statistics,
Vol. I. Charles Griffin and Company, London.
Kolmogorov, A. N. and S. V. Fomin. 1961. Elements of the Theory of
Functions and Functional Analysis, Vol. 2. Graylock Press, Albany,
N. X. [translated from the first (1960) Russian edition by Kamel
and Konnn].
Lindgren, B.
York.
w.
1960. Statistical Theory. The Macmillan Company, New
Munroe, M. E. 1953. Introduction to Measure and Integration.
Wesley Publishing Company, Inc., Reading, Massachusetts.
Addison-
National Bureau of Standards. 1964. Handbook of Mathematical functions. Applied Mathematics Series 55, June 1964. U. S. Government Printing Office, Wa$hington, D. C.
61
Parzen, E, 1960. l>1odern Probability Theory and Its Applications.
John Wiley and Sons, Inc., New York.
Patnaik, P. B.
1949. The non-central -1- and F distributions and their
application~.
Biometrika 36:202r232.
Pearson, E. s. 1959. Note on an approximation to the distribution of
non-central X2 . Biometrika 46:364.
Robbins, H. E. and E. J. G. Pitman. 1949. Application of the method
of mixtures to quadratic forms in normal variates. Ann. Math.
Stat. 20:552-560.
Rushton, S. and E. D. Lang. 1954. Tables of the conf'luent hypergeometric function. Sankhya 13:377-411.
Scheuer, E. M. 1961. tIDments of radial error. Report PA 3328-01
(revised), 10 Oct. 1961. Space Technology Laboratories, Inc., Los
Ar+geles.
Shah, B. K. and C. G. Khatri. 1961. Distribution of a definite
quadratic form for non-central normal variates. Ann. Math. Stat.
32:883-887.
Slater, L. J. 1960. Confluent Hypergeometric Functions.
Univ. Press, Cambridge.
Cambridge
Sneddon, I. N. 1961. Special Functions of Mathematical Physics and
Chemistry. Oliver and Boyd, Edinburgh.
Tricomi, F. G. 1954.
Cremonese, Rome.
Funzioni Ipergeometriche Confluenti.
Edizioni
62
9.
APPENDICES
9.1 The
Function
9.1.1 Properties and Relationships
The few properties and relationships of the confluent hypergeometric function presented in this first section are well known and
may be found, for the most part, in standard works on the subject.
In
particular, see Erde1yi (1953), Slater (1960), Tricomi (1954).
Definition:
<Xl
F(ajcjx)
where (a)
n
(a)o
(a) n
= E
n=o
(a)
~
(9.1.1.1)
\l;Jn
is Pochhammer's symbol:
= 1,
(9·1.1.2 )
= a(a+1)(aol2) ... (a+n-1)
The defining series is absolutely and uniformly convergent
for all values of a, c, and x, real or complex, excluding
c
= 0,-1,-2, ....
Kummer's transformation (Kummer's first theorem):
F(ajcjx)
= e~(c-ajcj-x).
Special cases:
X
F(ajajx) :::: e .
(9.1.1.4)
x
where erf(x)
=f
2
e- u du
o
=! r(!,x2 )
=:!f
x
2
t-!e-tdt
(x>o),
o
where r(·,·) is the incomplete gamma function.
where I v(x) is the modified Bessel function of tbe
first kind of order v.
Recursions:
=
(c-a)F(a-l;c;x)+(2a-c~)F(a;c;x)-aF(a+l;cjx) Q.
c( c-l )F(a; c-l;x)-c( c,.l+x)F( a; c;x)+( c-a)xF( aj c+ljx) = o.
= o.
(a-c+l)F(a;c;x)-aF(a+l;cjx)+(c-l)F(~;c-l;x)
cF(a;c;x)-cF(a-l;c;x)-xF(a;c+l;x) =
o.
(9.1.1.9)
(9.1.1.10)
(9.1.1.11)
(9.1.1.12)
c(a+x)F(ajc;x)-(c-a)xF(a;c+l;x)-acF(a+ljc;x)
=
O.
(a-l+x)F(ajc;x)+(c-a)F(a-l;cjx)-(c-l)F(ajc-l;x) = O.
(9.1.1.~~)
(9.1.1.14)
U-function:
The U-function is also known as a confluent
geometric function.
hype~-
It is related to F(ajcjx) as
folloW$ [see Slater (1960 ), p. 5]:
U(ajc;x) =
(
x1-cF~ 1+a-c'2-c'x ) }
F a'c'X
r(1+a2cjr
0)
r a)r(2:C)'
. (9·1.1.15)
s n1tC {
i 1t
f
64
The U-function
sa~isfies
the following recursion:
(c-a-1)U(a;c-1;x)+(1-c-x)U(a;c;x)+xU(a;c+1;x) ~ O.
9.1.2
(9·1.1.16)
Representation of F(-~;~;-x) in Terms of Bessel or Error
Functions, General
Note:
To facilitate the proofs in this section, the
symbol € is used to mean "may be expressed in
terms of'."
Theorem 9.1.2.1:
F(- ~;~j-x)
Io(~) and I1(~)' whenever nand k
€
are positive odd and. even integers, respectively, and x
Proof:
f o.
We first establish three preliminary results:
By (9.1.1.12), with a ~ i and c
= 1,
F(i·
+ xF(i·
2· x )
2' 1·, X ) = F(~'l'x)
2' ,
2'"
Also, by (9.1.1.10), with a
=i
and c
= 2,
Eliminating F(i;2;X) from these two equations, we obtain
2
F(~;l;X) = (x+1)F(~;1;x) + ~ F(~;3;X).
Eliminating F(~;l;X) from the same equations, we obtain
And, by (9.1.1.13), with a
=~
and c
= 1,
fo11owe~ by substitution f'rom
(9.1.2.1), we obtain
F(~;2;x) = F(~;l;x) - ~(i;3;X).
We neW proceed with the proof:
Let nand k take the values stated in the theorem.
k+n k
F(~;2;x)
€
k+n
k+n
F(~;2;X) and F(~;l;x),
either immediately (if k
(9.1.1.10).
Then
=2
or 4) or by repeated
x f 0,
app1iq~tion
of
Similarly, by repeated application of (9.1.1.9),
F(k;n;l;X)
€
F(~;l;X) and F(!;l;x)
€
F(~;l;x) and F(~;3;X), by (9.1.2.1).
Also, by repeated application of (9.1.1.9),
F(~'2'X)
2 ' ,
€
F(~'2'x)
and F(A'2'x)
2' ,
2' ,
€
F(!;l;x) and
F(~;3;X),
Hence, for n, k, and x as stated in the theorem,
F(-
~;~;-x)
Theorem 9.1.2.2:
€
F(!;l;~) and F(~;3;X)
F(- ~;~;-x)
€
erf(~), whenever nand k are
each positive odd integers and x > 0.
Proof:
Then
Let n, k, and x take the values stated in the theorem.
66
F(- ~'!'-x)
2'2'
€
=1
either immediately (if k
(9.1.1.10).
F(- ~.2·-x)
and F(- ~'~'-x)
2'2'
2'2'
,
or 3) or by repeated application of
Similarly, by repeated application of (9.1.1.9),
F(- ~'~'-x)
2'2'
€
and by (9.1.1.13), with a
F(2.l·_
2'2' X )
E
F(~'~'-x)
and F(2.l·2'2'
2'2" x )
= c = ~,
F(l.l·-x)
and F(l.2·- x )
2'2'
2'2"
Therefore
F(- ~.l·-x)
2'2'
€
F(l.l·-x)
and F(l.2·- x )
2'2'
2'2"
Also, by repeated application of (9.1.1.9),
F(- ~.2·-x)
2'2'
€
F(-21.2·-x) and F(2.2·- x )
~ '2'
2'2"
Thus for n, k, and x as specified in the theorem,
F(- ~'!'-x)
2'2'
E
F(l.2·-x
and F(~·2·-x)
2'2' ) and F(*·l·-x)
~'2'
2'2'
E
erf(I[:X).
Also, by (9.1.1.7), it is further c1e&r that
Representation of F(- ~;~;-~) in Terms of Bessel or Error
Functions, SRecific Cases
9.1.3
In this section, six specific cases (n
c
1,3; k
= ~,2,3) of
F(- ~;~;-~) are expressed in terms of Bessel or error functions,
consistent with Theorems 9.1.2.1 and 9.1.2.2, for use in conjunction
with special case 1 of Section 3.
(i)
= 1,
n
Proof:
= 1:
k
In (9.1.1.12), let a = c = ~ and replace x by -~to
obtain
F(-~'~'-~)
= F(~'~'-~)
+ 2~(~·L._~)
2,2J
2J2'
2'2"
The result follows from (9.1.1.4) and (9.1.1.5).
(ii)
n
= 3,
Proof:
k
= 1:
In (9.1.1.9), let a
= -~,
c
= i, and
rep1~ce x by -~ to
obtain
The result follows from (9.1.3.1) above and (9.1.1.4).
(iii)
n = 1, k = 2:
~
F(-~;l;-~) = e- 2 [(~+1)Io(~)+~I1(~)]'
Proof:
The result follows from application of (9.1.1.8) to
(9.1.2.1), followed by (9.1.1.3).
(iV)
n
= 3,
k
= 2:
A.
F(Proof:
~;l;-A.) = ~e- 2 [(2A.2+6A.+3)Io(~)+A.(A.+2)I1(~)]'
In (9.1.+.9), let a =~, c
= 1,
(9.1.3.4)
and replace x by A. to
obtain
F(2·1·A.)
2' ,
= g[(A.+2)F(~·l·A.)-~F(~·l·A.)]
3
2"
2
2"
•
The result follows from (9.1.2.1) and. (9.1.1.8), and, subsequently, from
(v)
n = 1, k =
Pr~of:
3:
In (9.1.1.9), let a =~, c =~, and replace x by -A. to
obtain
The result follows from (9.1.1.4) and (9.1.1.5).
(vi)
n
= 3,
Proof:
obtain
k
= 3:
In (9.1.1.9), let a =~, c =~, and replace x by -A. to
69
F(- ~.2._~) = -21(~+ 2)F(--21.2._~) - *F(~·2._~)
2'2'
2
'2'
~
2'2'
.
The result follows from (9.1.3.5) above and (9.1.1.5).
9.2 A Convenient Approximation
Theorem 9.2.1:
For r
~
0, and x large, x > 0,
r(r-1)
2
r(f))'
r
r x =x e
Proof:
i
x
Employing Stirling's asymptotic series for log r(x) [see
National Bureau of Standards (1964)], we have that
log
r~(;»)
-
[(x+r-~)log(x+r-1)
-
[(x-~ )log(x-1) - (x-1) + 12(;-1) +
- (x+r-1) +
= (x+r-~)log(x+r-1) -
12(X~-l)
0 (
+
O(~)]
~p
(x-~)log(x-1)-r
-=-=-r---:--:-.r~::-\" + 0 ( 1 )
- 12(x+r-1)(x-1)
~
= (x+r-i)logx(l+ !:l)
x - (~-~)logx(l- 1)
x
= ( x+r-'21)[ logx
2
+ -r-1 - '21(r-1
- ) + 0 ( -1 ) ]
x
x
.;5
-
(x-~ )[logx - 1 - ..J:...2
= r 1ogx
+ r(r-1)
2x + 0 ( 2 '
x
and the result follows.
"!"J
- r + 0{ 2
x
2x
J:..)
x
+ 0(
1...)
]-r
.)
+ 0 ( .1..
2)
x
70
Table 9.2.1 gives some indication of the quality of the approximation, as measured by the smallest integer value of x such that the
relative error of approximation is less than
ular, surprisingly good agreement for r
any value of r in the closed interval
= ~;
We observe, in partic-
€.
this is also the case for
(0,1). The quality of the
approximation drops off rapidly as r increases beyond unity.
However,
there are two convenient deVices that will enable one to inmrove the
approximation when r is "too" large or x "too" small.
When r is "too" large:
We may separate r into its integer part,
[r], and its nonnegative fractional part, f
r(C»
r
x
=r
r
- [r].
Then
r[(x+[r])+f ]
= (x)[r]
r(x+[r] {
This device has the desirable feature of yielding a larger "effective"
II
value of x while giving a smaller
•
If
effect~ve
value for r.
Example:
r(x+
~)
rex)
and the
.
t'
approx~ma ~on
= (x)3
i s a-npl{ed
to
~
...
When x is "too" small:
r(C»
r x
=
r[ (X+3)~]
r(x+3 )
r[(x+3)~]
r (x+3 )
Then
(x)n
(x+r)
n
r[ (x+n)+r]
r(x+n)
,
71
Table 9.2.1.
Values of x, A, and B such that
r(r-1)
A = r«~)
and B = x r e 2x
r x
x
0·5
1
1.5
4
2·5
11
3.5
21
A
8.862x10-1
8.724xlOO
4. 714x102
5.179x104
=
where
= .001
€
x
B
€
€,
-
= .01
€
r
JA-BL
A $
8 .825xlO-1
2
8.786x100
4. 759xl02
11
35
4
66
5·227x10
A
B
O
1.3293xlO
1
3·7710x10
1.3285xlOO
3.7748x101
7.6384x1a3
2.4932xl06
7.646ox1a3
2.4957x106
.0001
€
=
.OOOO~
r
x
0·5
4
1.5
36
2·5
112
3·5
208
A
B
O
o
1.93862xlO
1.93847xlO
2.1824lx102 2.18262x102
1.34981xla5
1.32530xl08
1.34994xla5
1.32543x108
x
A
B
2.958643x100
2.95862lx100
354
1.189260xla3
2.370304xl06
1.189272xla3
2.370328xl06
661
7.474381x109
7. 474456x109
9
112
72
Example:
x
= i;
r =~; n
= 2,
say (n chosen to suit one's needs).
1
(4)2 r(t4) = 2-
=(3)
42
ret)
21
,
r(t4)
9 .
and the approximation is applied to
r("4)
And, of course, these two devices may both be used to 6ive an
approximation of desired quality.
f(~~)
-rTxJ-
In such a case,
(x+[r])
r[(x+[r]+n)+f]
= (x)[r] (x+r) n
r(x+[r]+n)r
n
_ (x) n+[r] r[(x+[r]+n)+fr ]
r(x+[r]+n)
- (x+r)
n
9.3 Proof of Equation (5.6.19)
Note:
Throughout this appendix, unless otherwise indicated,
the limits on all integrals are zero to infinity and
may not be shown explicitly.
9.3.1 Introduction
Let
(9.3.1.1)
where r > 0, r
f 1,2, ... ,
a ~ 0, ~ ~ 0, N is some positive ~ integer
larger than r, and recall that
f
1
(t)
= SiC-(,)
p t
and
f (t)
2
= 1(1 _ cost(,»
t
p t
with A(t) and pet) being given by (5.3.2) and (5.3.3).
'
73
We shall prove that
ff
F(t,q,o,o,r)dtdq =
[cf. (5.5.3) and (5.6.19)].
J
H(t,o,r)dt
In particular, we shall prove the fo1lowlim) .
ing sequence of steps (where L means
'1_0
'1
(true by hypothesis)
ff
J~J ~
~
ff
~ ff
~
LL F( t, q,a,t3,r )dtdq
at3
2
LL
F(t,q,a,t3,r)dtdq
F( t, q,a, t3, r )dtdq
at3
LL
F(t,q,a,t3,r)dqdt
at3
1L
J
~
J
a
~
(obviously true)
L
a
h(t,a,r)dt
LJ
a
[H( t,a, r )+K( t,a, r)] dt
H(t,a,r)dt +
LJ
a
K(t,a,r)dt
(obviously true)
10
--
11
--
J
J
LH(t,a,r)dt
a
H(t,o,r)dt.
N
The ~ convergence factor, e-~t , has been introduced in order to
establish the absolute convergence of the iterated integral (dtdq),
required for application of Fubini's theorem [change of order of integration (step 4)].
In the proof that follows, we shall call upon many properties of
various functions.
not.
Some of these properties are obvious, others are
If, for example, a bracketed expression such as [9~.4.3] should
follow some claimed property of some function, it means that that
property is established in Subsection 3 of Section 9.3.4of this appendix.
If some function is said to be ~(o,oo), say, we mean that that function is Legesgue integrable over (in this case) the positive real line.
Also, for convenience, we shall simply write L.D.C.T. when appealing to
the Lebesgue dominated convergence theorem [see ~.~., Kolmogorov and
Fomin (1960), p. 56, Theorem 1].
Finally, before proceeding with the proof itself, the author
wishes to point out that the proof was established under the direct1qn
of Dr. W. R. Haseltine, many of the essential ideas being his.
9.3.2
Proof, Part I
Let
J
g(q,a,~,r) = F(t,q,a,~,r)dt.
It will be shown in Section 10.3.3 that
L g(q,a,~,r) ;;;: g(q,a,o,r),
~
and that there exists an A > 0 and a B > 0 such that
Ig(q,a,~,r)1 <
(0)0, a~o, O<~<~l <00).
Al +o
B+q
(9·3.2.2)
Further, (9.3.2.2) holds for ~ ;;;: 0 by (9.3.2.1).
(continuity)
Clearly
A
~(o,co)
:is
in q.
We have that
; ;: L[L g(q,o:,(3,r)dq
(by L.D.C.T.)
~
0:'
J
;: J
g(q,o:,o,r)dq
;;;: L
0:
L g(q,o:,o,r)dq
(by L.D.C.T.)
0:
; :; J J
L
F( t, q,o:, o,r )dtdq
0:
=
ff F( t, q,
0,0,
r )dtdq.
Thus, conditional upon (9,).2,1) and (9,),2.2), the propriety of passing
the limits outside the integral signs (steps 1 through
4) is established.
Now the bracketed expression in F (9,),1.1) is clearly bounded
since fl(t) and f 2 (t) are bounded, and
76
l/r
e-aq
is
)
L:L(
0,00
in q,
Thus, with a > 0 and t3 > 0, it is clear that the iterated integral
ff
exists.
LL
at3
ff
\F(t,q,a:t3,r)1 dtdq
Hence, by Fubini's theorem [see, ~.~., Munroe (1953), p. 207],
F(t,q,a,t3,r)dtdq
= LL
at3
= LL
at3
= LL
at3
ff
J [J
(step 4)
F(t,q,a,t3,r)dqdt
t3tN
e-
J
e
-t3tN
F(t,q,a,o,r)dq]dt
(step 5)
h(t,a,r)dt,
where h(t,a,r) is, as previously designated in Section 5, the integrand
of (5.6.2).
Clearly,
and, for a > 0, h(t,a,r) is L1(0,00) in t.
LL
at3
J
e-
t3tN
h(t,a,r)dt
J
a
=L
=L
a
L et3
t3tN
[9.3.4.3] Hence by L.D.C.T.,
h(t,a,r)dt
J
h(t,a,r)dt.
(step 6)
(step
7)
We have seen (Section 5.6) that we may write
h(t,a,r)
= H(t,a,r) + K(t,a,r),
where H and K are given by (5.6.15) and (5.6.16).
(step 8)
It turns out that
77
H(t,a,r) is L1 (0,00) for all a ~ o. [9.3.4.3]. Because h is ~ for
a > 0 and H is L1 for a ~ 0, it follows that K is L for a > O. Thus
1
step 9 is valid. But we have also seen (Section 5.6) that
J
Hence L
a
NOli
K( t,a,r )dt
J
(for a > 0; r > 0, r
0
E
K(t,a,r)dt
=
J
1,2, ... ).
O.
clearly IH(t,a,r)1 ~ !H(t,o,r)l,
L
a
f
=J
H(t,a,r)dt
Hence, by L.D.C.T.,
(step 10)
L H(t,a,r)dt
a
J
= H( t,o,r )dt.
(step 11)
Thus, conditional upon (9.3.2.1) and (9.3.2.2), the proof is complete.
9.3.3
Proof, Part II
We now turn to establish the truth of (9.3.2.1) and (9.3.2.2).
Let us define
and
I 2 (v,l3)
Jg~2
=
sin(vt)dt,
N
, f 1 = f 1 (t), f 2 = f 2 (t), and v
Note that, apart from the (v,q) change of variable,
where ~
= gN(t,l3) = e- I3t
subject to the separate existence of II and 12 ,
=~
q1/r.
78
Before proceeding further, it is perhaps well to remark that,
whereas fl(t) is Ll (and thus well behaved), f 2 (t) is not ~, and,
because of this fact, the proofs to follow are not as neat as they
otherwise could have been.
9·3.3.1 Proof of (9.3.2.1).
all v ~ 0, ~ ~ 0.
Il(v,~)
II1(v,~)1 $~ Iflldt,
Clearly,
But fl(t) is Ll(o,oo).
[9.3.4.2]
for
Hence by L.n.C.T.,
exists and
For v > 0, 12 (v,0) exists as an improper Riemann integral.
[9.3.4.4] For ~ > and fixed v > 0, we see that
°
=
~
c/v
r
f 2 sin( vt )(gN-l)dt +
J
-~
sin(vt) dt +
c/v
$
J
c/v
t
r
J
gN
sin(vt) dt
t
c/v
o
00
(X)
00
c/v
cosA( t )sin( vt)
tp( t)
(X)
[l-gN(~' 13)] If 2 sin( vt) Idt
+
c/v
o
+
J
r
J
BN
sin(vt) dt
t
(X)
00
Sintvt) dt
c/v
+
Jc/v
dt
tp(t)
We may choose C so large that each of the last three integrals is less
*'
Now If 2 sin(vt) I $ 2v.
of these four integrals is less than
than
€
>
0.
[9.3.4.2]
Therefore the first
79
(9·3.3.1.2)
It is clear that there exists a
€
less than 4 for all
~l
so small that (9.3.3.1.2) is also
such that 0 <
~
~
<
~l'
Hence, for v > 0,
It is readily seen that I2(0,~) ; 0 for all ~ ? O.
holds for all v ?
nlUS (9.3.3.1.3)
o.
It follows from (9.3.3.1.1) and (9.3.3.1.3) that
L g(q,o,~,r)
; g(q,o,o,r),
~
and since
ex llr
g(q,ex,~,r) ; e- q
g(q,o,~,r),
the result (9.3.2.1) also follows.
9.3.3.2
types for
Proof of (9.3.2.2).
g(q,o,~,r),
We shall establish bounds of two
from which the result (9.3.2.2) will be seen to
follow.
Clearly IIl(v,~)1 is bounded for all v? 0, ~ ? 0 (cf. first
paragraph of previous subsection).
For v > 0,
II2(v,~)1
;
J
clv
00
Jclv
_J
grf2 sin (vt)dt +
a
gN
00
<
J
a
clv
clv
If 2sin( vt) Idt
+
IJ
~
CD
clv
sin(vt) dt
t
t
cosA( t Sin(vt) dt
tp t)
~ Sin(;t)
I Jclv t~{t)
CD
dt +
80
The first integral
~
J
c/v
Given C > 0, the third integral
2vdt ::: 2C.
o
is bounded uniformly in v for 0 < v < vI <
but finite), because
tP(t)
::: 0
(tl~T)
(vI arbitrarily large,
00
), for t large.
[9.3.4.2]
In
the second integral, let C ::: 2nrr, where n is some positive integer, and
make the transformation x ::: vt to obtain
J
00
e
.
(x,
"
N
_~(c:)
t3 ::v)
-' 2 e
<..'
:::
N
.
," 1
"
;;.:; ' -nrc- .
2n;rc
2n:rr
Hence, recalli.ng that I (o,f3) ::: 0 for all f3 ~ 0, it follows that
2
(for all v, 0 ~ v < v1 <
From (9.3.3.2.1), and the fact that
and f3
~
II1 (V,f3)1
00;
f3 ~ 0).
(9.3.3.2.1)
is bounded for all v ~ 0
0, it follows that
Ig(q,o,f3,r)1 < Bl
(0 ~ q < ql <
00,
f3 ~ 0).
Thus a simple bound on g(q,o,f3,r) has been obtained.
(9.3·3.2.2)
Now we seek a
bound of a different character.
In what follows, we shall be concerned with derivatives of the
products g~j' j ::: 1,2.
By Leibniz' rule,
It may be shown (by mathematical induction) that
;e
( )
g P ::: gN
N
P
~ A(k,p,N) f3
k=l
N
N
(f3t)
Nk-p
N
.
(9·3.3.2.4)
81
g~p)
Clearly, for f3 > 0, each term in
is
~(-oo,oo) in
t if we require
that Nk-p ~ 0, thus if P ~ N (recall that N is an even positive integer).
~
Also, it may be shown that, for p
f3 P ),
1, the
j
= 1,2,
are s.(o,oo).
[9.3.4·5]
Because the integrands in I (V,f3) and I (v,f3) are seen to be even
1
2
functions of t, we may write
I 1 ( v, t)
= ~. ~
00
J
-00
and
I 2 (v,t)
=t
,
J
00
-00
Let v and f3 be strictly positive.
(u
= g~j)'
Then, by integration by parts
we obtain
L'" eivtBrfjdt
=
[ei:Brf{Xl
+
~
J
00
eivt(g~j)' dt
-00
Now the term in brackets vanishes.
Similarly, integrating by parts N
times, we have that
00
= (~)
J
N
J
00
-00
-00
Because N is an even positive integer, it follows that
.NJ
00
I l (v,f3)
= ~ (~)
e
ivt
(g~l)
(N)
dt,
o
and
I (v,f3)
2
=
1 (~)N
J
o
00
eivt(g~2)(N)dt.
82
Now, were we to substitute for
(~j)(N),
from (9.3.3.2.3), in
(9.3.3.2.5), and then consider termwise integration, it is clear that
all integrals, with one possible exception, are bounded independently
of
p
~,
~
for 0 <
~ < ~l <
1, are L ].
l
00
[this follows because f
l
is L and the
l
f~P),
The sole possible exception is
o
We now show that even this integral is bounded independently of
o < ~ < ~l <
00.
Let
x
D
k
=
max [
N
x k-l e-
2J,
x=13t
and let
N
=
~ IA(k,N,N)ID ·
k=l
k
Then
(N)I
I lSN
< ~ D(N) e
We have seen that f (t) is bounded, \f (t)1 < b 2 , say.
2
2
Then
~,
for
{'O
o
(y=
13 N
2"
t
)
b2
=
1
-N
N-l
-N
N 2 13
D(N)
J
(X)
1
--1
N-y
Y
e
dy
o
and this function of 13 is certainly bounded for
° < 13 < 131 <
(X) •
It follows then that we have established that
(for all v> 0,
° < 13 < 131 < (0),
from which it further follows that
(q > 0,
(recall that N > r and v
= -i
° < 13 < 131 <
ql/r), where 0 > 0,
00 )
Thus we have a second
bound for g( q, o,l3,r).
We have left only to see that (9.).).2.2), together with (9.).).2.6),
yields (9.).2.2).
Let q be the solution of
o
1
B
q
1~0 = Bl , .!.~.,
(9·)·)·2.2)];
qo
=
(B2 ) 1+0
B
(implicit, here, is that ~ ~ ql [cf.
1
recall that ql may be arbitrarily large).
84
For 0
~
q
~ ~,
Ig(q,o,~,r)1
Ig ( q, 0, ~, r ) I
< Bl
<
B
Thus, letting A
2
= 2B2 and B = --,
B we have
l
Ig(q,o,~,r)1 <
A
1+0
B+q
(0 < ~ < ~l <
00),
and the desired result (9.3.2.2) follows since
Ig(q,o:,~,r)1 ~ Ig(q,o,~,r)1 .
9.3.4 Properties of Functions
We shall now establish the several properties of the various functions of t, 0
~
t <
00,
cited in Sections 9.3.2 and 9.3.3.
Clearly,
all of the functions with which we deal are continuous in the open
interval (0,00).
It follows that all such functions are Ll(O,oo)
provided that they are 0 ( t l :
) , 01 > 0, for t small, and a ( 1;5 ),
Ol
t
2
02 > a, for t large.
9.3.4.1 The Functions A(t) and pet).
A(a)
= a.
A(t) = a(t), t small.
pet) is a monotonic increasing function of t.
85
p(O) :::: 1p(t) ::::
0(1), t small.
pCt) :::: 0 ( :TJ ), TJ > 0, t large.
follows:
This property may be seen as
The exponential part of p(t) behaves like a constant (in fact,
!A.
like e ~) for t large.
Vi
The product part,
2
n (l+o-~t2)4 , is larger than
(~lt)
~
V
l /2
,from which the result follows.
9.3.4.2
The Functions fl(t) and f 2 (t).
fl(t) is a bounded function of t.
fl(t) ::::
0(1), t small [see (5.6.5)].
If1 ( t) I
~
tp Ct) :::: 0 (
t
i
+1) ) ,
1)
> 0, t large.
fl(t) is Ll(o,oo).
f (t) is a bounded function of t.
2
f (t) :::: O(t), t small [see (5.6.5)].
2
o<
f
2
( t)
2
< t'
0
<t <
00.
9.3-4.3 The Functions h(t,a,r) and H(t,a,r).
Because fl(t) and
f (t) are bounded, it is clear that there exists a constant C > 0 such
l
2
that
C
:::: 0 ( 1:.. ) t small.
Ih(t,a,r)1 < _ _-.;l~_
r
r'
(if+ ~ )2
"
Also, because Ifl(t)\
~
tP(t)
~
i and If2 (t)1 < t ' it is clear
that there exists a constant C2 > 0 such that
Ih(t,a,r)\ <
Thus it follows that h(t,a,r) is Ll(O,oo) for all a > 0 (recall
that r > 0).
Now, with respect to H(t,a,r), it follows from (5.6.13) that
IH(t,a,r)1
= 0 ( tl~e ), e >
0,
t small.
Also, from (5.6.12), it is clear that
01
>
02
> 0, t large
0, t large,
and
(recall that r
f 1,2,3, ... ).
Therefore,
03
>
0, t large.
Hence it follows that H(t,a,r) is Ll(o,co) for all a ~ O.
9.3.4.4 The Function 12 (v,o). We have that
12 (v,o )
=
J
o
co
[Sin(vt) - sin(vtlcosA(t)] dt
t
tpt)
.
86
Clearly,
<
sin(vtlcoSA(t)
tp t)
tP(t)
:;; 0 (
tl~Tj
), Tj > 0,
t large.
Therefore
sin(vt)cosA(t)
tp(t)
l'S
L (0,00) in t, for all v > 0,
l
00
For v > 0,
J
sin~vt)
dt exists (=!E.) as an improper Riemann
t
2
o
integral, but sin(vt)/t is not L ,
l
Hence, for v > 0, it follows that I (V,o) exists as an improper
2
Riemann integral.
9.3.4.5 The Functions f~P)(t), p ~ 1,
J
f' (t):;; 1
1
f'(t):;;
2
PTtJ
1
ptt}
[A' (t)cosA(t)
t
j :;;
1,2.
Consider the first
sinA( t) _ sinA( t) f)~lJ;J ]
t2
t
(PTtJ),
[COSA(t) + A'(t)sinA(t) + cosA(t)
t2
t
t
It is readily seen that A'(t) and
functions of t, being 0 ( 1 ) and 0
2
p~~~j
[:;; d10~(t)]
are rational
(t),
respectively, for t large.
t
~'
Because they are rational functions, it follows that A"(t) and (~)
are 0 ( t~ ) and 0 ( t ~ ), respectively, for t large, and so on.
88
Thus it is clear that the terms of fi(t) and f;(t) are, apart from
the factor p(t)' rational functions or products of sines and cosines by
rational functions, all rational functions being 0 ( t; ), t large.
Clearly, the factor p(t) and its derivatives can only aid convergence
because
1
_ 0 ( 1 )
PrtJ -
etc.,
Hence,
for p
t1)'
1)
> 0, t large.
f~(t) and f;(t) are 0 ( :3)
~
,
t large.
Similarly, we conclude,
1, that
f~P)(t) = 0(_1_),
p l
t +
J
t large, j
= 1,2.
Because fl(t) and f (t) are analyti.c at t = 0 [in particular,
2
their power series expansions (5.6.11) are valid in some neighborhood
of the origin], it follows that
Hence, for p
~
1,
f~P)(t)
J
is 0(1), t small, j
= 1,2.