-RANDOM HYPERPLANE INTERSECTING A FIXED
LINE IN Rn AND THE CAUCHY DISTRIBUTION
by
Ibrahim A. Salama
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1466
September 1984
RANDOM HYPERPLANE INTERSECTING A FIXED
LINE IN Rn AND THE CAUCIIY DISTRIBUTIO~
Ibrahim A. Salama
Department of Biostatistics
University of North Carolina
Chapel Hill, NC 27514
Key Words and Phrases:
Cauchy distribution; product and quotient
of independent Cauchy random variables
ABSTRACT
n
Let L be a fixed line in R given by the equation L(t)
n
n
B + tV, B,V E R and t E R. Let P(c) be a hyperplane in R given
n-l
by the equation c.P = 0, where c E S a n d c.V # o. Let L(t )
c
be the point of intersection of Land P(c), and define X (P(c))
n_ln
=
Sgn (t ) II L(t ) - L (0) II . Consider X as a map from S
{c
S
n-l
v
c
E Sn-
•
1
c
Ic.v #
v
n
o} to R, and assume a uniform distribution over
We show that for every n > 2, X
-
n
has the same distribution
as aX + b, where X is standard Cauchy and a,b are constants.
Some
consequences of this result are discussed.
1.
INTRODUCTION
n
Let L be a fixed line in R given by the equation L(t) =
n
B + tV, B,V E R and t E R. For simplicity of exposition we
assume for the moment that II BII = Ilvll = 1 and B.V
n
P(c) be a hyperplane in R given by the equation c.P
o.
Let
0, where
2
n
c E Sn-1 (the unit sphere in R ) and c.V
i
o.
Let L(t ) be the
c
point of intersection of Land P(c), and define X (P(c»
n
Sgn(t )IIL(t ) - L(O)II.
c
c
n-l
Let P = {p(c)lc E S
,c.V i O}, then X may be viewed as a
v
n n-1
{c ~ Sn-1
map from P to R, or equivalently as a map from S
~
1
v
v
c.V i O} to R. Assuming that P is given a uniform probability
v
structure, which induces a uniform distribution over Sn-l, the
v
question is: what is the probability distribution of the random
n-1
variable X (c), c E S
.
n
v
2.
THE PROBABILITY DISTRIBUTION OF X
n
AND RELATED RESULTS
First, note that the probability distribution of X
n
is invari-
ant under the group of transformations representing rotations in
n
R . Hence, in answering our question, there is no loss in
generality by assuming that the line L is given by L (t) = e +
1
n
te , e , ... , e are the standard basis for R . Second: X (c) is
1
n
n
n
given by X (c) = - ctn/ctv. Finally: for n = 2, we have the well
n
known result that X (c) is Cauchy.
In fact, this will be the case
2
for every n as given by the following.
Theorem 1:
For n > 2, we have
P{X
1
1
2
7T
< t} = - + - Tan
n-
-1
(t)
'
-
00
< t <
00
,
that is
1
,
-
00
< t <
00
•
Pmo f:
the equator).
=
+ ten' Let s:-l = the upper half of (Sn-l 1
For given t E R, consider the corresponding point
Assume L(t)
e
e'
3
n-1
Then we have X (c) = t for all c E S
n P(L(t».
n
n-l
u
n-l
bounded by Pee ) and
Let S
(t) be that cross section of S
u
n
u
n
n
P(L(t» and intersects R (the positive orthant of R ). Then we
n
L( t) E R •
+
have
p{X
< t}
p{c
n-
E
Sn-l(t)}
u
where A(o) is the area function.
and P(L(t», 0 < e
nn/2 - 1/f(n/2) e.
< n.
Let
e
be the angle between Pee )
n
Then it is easy to see that A(Sn-l(t»
Let ¢ =
e-
u
n/2, then since Tan(¢)
n/2
n/2-l
n
n
2 f(n/2) + nn~
=
t, we get
n/2
n/2-l
-1
n
+ n
2 f(n/2)
r(n/2) Tan (t)
Hence
p{X
1/2 + lin Tan
< t}
n-
-1
(t) , t E R
n-1
nn/2
(since A(Su ) = f(n/2) )
In the general case, L is given by L(t) = B + tV, such that
Ilvll
=
1 and the angle
e between
B and V satisfies 0 <
Define Y (c) = sgn(t ) II L(t ) - L(O) II.
c
e
n
1/ VB II = 1, V. VB = 0, and B =·11 B II Sin
VB +
II B II
Since
Y (c)
n
II B 1/
we have
Sin
e
X (c) n
e<
n
Let VB be such that
e
II B II
Cos
e ,
COS
=
e v.
4
TJzeoY'em 2:
For n > Z, we have
II B II COS 8)
~
II BII Sin 8 J
t +
p{y
<
n-
t
p
{
X <
n -
= -1Z + -1T1
Tan
II B II COS
II B II Sin 8
t +
-1
[
8] , t
E
R
that is
f
(t)
Yn
=
II B II
1T[IIBI1 2
Sin 2
8
Sin 8
+ (t +
IIBII
,
COS
which is cauchy with location parameter - II
parameter
II B II
BII
t E
R
8)2]
COS 8, and scale
Sin 8 .
COY'O UaY'y 3:
n-l
If Al and A are any two non-colinear vectors in S
, then
Z
the random variable Y = A~C/A~C , where C is chosen uniformly in
'
h
X'1S stan d ar d
ut10n
as a X + b , were
Sn-l , h as t h e same d'1str1' b
Cauchy and a,b are constants.
COY'O ZZaY'lJ 4:
n-l
Let AI' A be any two orthogonal vectors in S
. If
Z
t
/
t
t
/
t
t
t
/
t
t
Y = ClA I ClA Z • CZA I CZA Z
CI (AlAI) Cz Cl(AZA Z) Cz ' where CI and
C are chosen independently and uniformly in Sn-l, then fy(y) is
z
the density function of the product of two independent standard
Cauchy,
That is
log (y2) , _
00
< Y <
00
•
5
PY'Q,][:
The probability density function of the product of two independent standard Cauchy is given by Springer and Thompson (1966).
Under the saffie notations and assumptions of Corollary 4, the
t
t
t
t
random variable Y = C (A A ) C /C (A A ) C has the probability
2
2 l
l
l 2
2 l
density function fy(Y) given above.
n
Let Xl'···' X be n independent standard Cauchy. Set Yn
n
X..
Springer and Thompson (1966) gave the probability density
IT
i=l ~
function of Y , n < 10, explicitly. This result may be applied in
n
t
t
t
t
our situation by choosing X. = c. AI/C. A or X.
C. Al.IC. A , ,
2~
~
~
~
2
~
~
~
~
where Ali' A
are orthogonal unit vectors. Let
2i
-e
Z
n,k
k
n
IT X.
IT
x~l
i=l ~ j=k+l ~
and noting that Z k has the same distribution as Y , thus we may
n,
t
t
n
choose some of the Xi's as Xi
C A /C Al .
2 i
i
Let A ,A be any two orthogonal unit vectors. If Y
l
2
t
t
t
n-l
C (AlAI) C/C
(A A ) C
where C is chosen uniformly in S
,
2
2
then fy(y) = l/n(l + y)
0 < Y < 00. That is, Y is S2(1/2,
t
IY ,
1/2).
3.
CAUCHY AS PRODUCT OF INDEPENDENT RANDOM VARIABLES
n-2
= (-n, TI), II = (-n/2, n/2) and r2 = 1 x II . For
0
..
T : .6+
n
n
consi d er t h e parametr~zat~on
Sn-l , wh ere f or G
- E: .6
we
-l Let 1
S
n
0
have
T(G)
U
n
(Ul, ... ,U )E:S
n
nn
n-l
,and
-
6
Sin 8
n-2
Cos 8
U
n, n-Z
=
Cos 8
Cos 8
U
nn
Cos 8
n-2
£
n as
n-
3
Cos 8
n-2
U
n, n-l
Regard 8
Sin 8
n-2
Cos 8
n-2
Cos 8
2
1
1
Sin 8
Sin 8
Cos 8
1
0
0
a random vector, and let g be the probability
density function (p.d.f.) over n which is carried by T to the unin-l
form distribution over S
. Noting that the area element
n-2
.
l
associated with T is given by \(0) = IT Cos 8. d8, thus g(8)
n-l
.
i=l
1
IT Cos l 8./A(Sn-l). Hence, the marginal p.d.f.'s go'···' gn-2
i=l
1
of 8 , ••• , 8 _ are given by
0
n 2
1
21T
=
g. (8)
1
where a. =
1
-1T < 8 < 1T
-1
i
a.
Cos
1
zi Sri; 1
8
,
1T/2 < 8 < 1T/2
, i +z
Assume that L(t) = e
i
1, ... , n - 2 ,
I)
, and 8 , ... , 8 _ are independent.
n 2
0
n-l
+ ten' and for Un £ S
let X =
l
e Jut e
By
theorem 1, X is standard Cauchy,
unn /Un 1.
n n n l
general for i :; j u . /U . is standard Cauchy. Thus we have
nl nJ
u
t
In
Theorem 7:
For every m ~ 1, there exists m independent (nondegenerate)
m
random variables Xl' ..• ' X such that IT X. is standard Cauchy.
m
i=l 1
Coro Uary 8:
Let X be standard cauchy, then:
(i) for every m ~ 1 and every integer k :; 0, there exists m indep~ndent
IT
i=l
(nondegenerate) random variables Xl' ... ' X such that
m
X. has the same distribution as ~.
1
7
(ii) for every m ~ 1 and every real a
I 0, there exists m independ-
ent (nondegenerate) random variables Xl' ••. ' X such that
m
m
a
IT X. has the same distribution as Ixl .
i=l 1
4.
SOMB RELATED RESULTS
Consider the random variables X.
= Sin
= Cos
8, Y.
1
8, and
Z. - Cot
e,
2,
Let f.(h., q.) be the p.d.f. of X.(Y., Z.) respectively.
1
where
e-
1
g. (- means has the p.d.f.) and i = 0, 1,
1
1 1 1 1 1
1
Thus we have
i-1
-1
f. (x)
a. (1
hO(X)
fO(x)
1
1
-1
h. (x) = 2 a.
1
-e
2
2
x )
x2
xi/II
1
-1 < x < 1
i = 0,1,2, ... ,
o<
i
x < 1
1,2, ... ,
and
qi (x)
.
i+ 1
a~l x 1 /(1 + x 2)2
_00
< x <
00
1
i
0,1,2, ....
For n > 2, write (with the conversion X_I - 1)
n-2
un1"
•
= X
n-i-1
Defining Y
nk
U
n,n-k
IT
Y.
1
j=n-i
n
IT Y.
i=k
i
1, ... , n .
k
0,1, ... , n, then
k
0, 1, ... , n-l .
1
n-2,k • ~-l
Y
Lemma 9:
For every n > 2, the U . 's are identically distributed.
-
•
their common p.d.f. is f
n1
n-
2'
Hence
8
Proof:
The fact that the U .'s are identically distributed follows
n1
n-1
at once from the symmetry of S
. Hence the common p.d.f. is that
of U .
nl
Notes:
(1)
~i
1
(2)
is the beta p.d.f. over (-1, 1) with parameters
+2 1 '
+
211
)'
-(i
+2 1
1w1ll
.
) and
be denoted here by B
Y~
- Bl(i ; 1,
t).
Since the product of
i
(j ; 1,
independ~nt
Bl(a, b) and Bl(a + b, c) is Bl(a, b + c), then IT y2
k=i i
2 _ B (k + 1 n - k + 1)
Ynk
1
2
2 ' Hence
O<y<l.
(3) Let D = {WI"'"
W } be a set of random variables.
n
*
the random variable D
Define
n
IT W., D may be partitioned into
i=l 1
arbitrary disjoint sets D , ... , D . For every partition
m
1
of D we have a corresponding representation of D (in D)
*
given by D*
m
* We say D - f * if D* f * If D
IT D..
l
i=l 1
and D are two sets of random variables, then we say D
2
l
*
)~
D if D and D are identically distributed.
2
1
2
This notion provides us with an economical way to state our
results.
Statements concerning product of random variables may be
considered through this notion.
For example, lemma 9 is a state-
).~
ment about the equality (=) of n sets.
Let P be the number of
partitions of a set D, then the statement D
f * is in fact P
statements concerning product of random variables.
corollary is formulated in this notion.
The following
9
Corollary 10:
Consider the following sets of independent random variables:
then
i
= 1, 2, . . . .
In particular, if X and Yare independent such that X - S(i ; 1 ,
i
+2
lJ
and Y - qi+l' then for every integer i > 0, XY is standard
Cauchy.
If D
= {WI"'"
2
W }, we define D
n
Corollary 11:
With the same notations and assumptions of Corollary 10, we
-e
have
i
•
= 1,
2, . . . .
and Yare independent such that X
;
2
i),
then for every integer
Theorem 12:
The content of Theorem 7 and Corollary 8 applies
i ;
1)
and 13
2(i ' i)
+
1
Z
(with minor modifications).
The following statements are stated here because they are
easily obtained from our results.
Corollary 13:
.
Let C and C be two points chosen independently and uniz
l n-l
formly in S
Let dn be the distance between C and C . Then
2
l
Hence
d~ 13
1 , n ;
.i
1
1) .
(n ;
2)-2
4"""
n-3
d
-1
n
- a n_ 2 x
n-2 (
1 -
x
,0<x<2.
10
Proof:
Without loss of generality fix C = e and set C
2
1
1
n-1
be chosen uniformly in S
. Thus
d~
= II e l - Un 11
2
U to
n
= 2 - 2 Sin e, e - gn-2 .
Now, we generalize lemma 9 as follows:
Coro Uary 14:
Let P
n
n
be a fixed subspace of R of dimension k, and
k
Let C(U ) be a
n
k
n-l
point (vector) chosen uniformly in S
, and d k be the length of
Tf : R
k
-+
Pk be the orthogonal projection onto P .
T~en d~k - Sl(~
the orthoronal projection of C(U ) onto P k .
k - 1
n
n -
2
,
•
Proof:
We may assume that P
k
is spanned by {en _k + l ,···, en}
n
2
U .
i=n-k+l n1
L
Thus
n-2
2
Y.
i=k-l 1
L
..
From Corollary 14 we have,
Theorem 15:
Let A be an n x n symmetric idempotant matrix such that
Rank (A) = k < n - 1.
CtAC _
Sl(~
,-n -
~
If C is chosen uniformly in Sn-1 then
- 1) .
Coro ZZary 16:
For every integer n,m > 0 we have
(i)
fTI/2 Cosne de = 2n s(n ; 1 , n ; 1)
-Tf/2
Q(!2 '
IJ
n
+
2
1)
.
11
Tf /
(ii)
2
f
o
Pr>oof:
2
Tf /
(i) For n
then
=
0, 1, 2, ... , let a
Xis S[n; 1 , n; 1].
X2 _ 8l (t '
n ;
1) ,
cosne de.
f
-Tf/2
n
If
x-
Hence the first equality.
f ,
n
But
hence the second equality.
= (U n+ 2 , 1'···' Un+ 2 , n+2) .
(ii) Consider the random vector Un+ 2
now
Tf 2
/
-1
a
n
I
I
·
Sln
e1
m
Cos n e de .
-Tf/2
I
Tf
2a -1
n
o
and
=
I
n
II a~
i=l
1
1
Tf
I
/
2
Cosm+.1 e de]
-Tf/2
•
n+rn
II a.
i=m
i=O
1
n
II a.
i=O
n+rn
II a.
1
II a.
i=O
1
m-l
n
1
II a.
i=O
1
12
By lemma 9 we have
I n+ 2, 11 m)
E[ U
thus
n+rn
II a.
i=O 1
m-l
n-l
2 II a. II a.
i=O 1 i=O 1
TI/2
fo
Sinme Cosne de
k+2
-2-
k
noting that
Note:
II a.
i=O 1
TI
r(k ; 2)
, the results follows.
Other integral formulas and relations may be reached using
similar arguments.
Corollary 16 is known but the proof is new.
We conclude with the following remarks.
1.
The problem of finding the distribution of product of random
variables may be solved using the Mellin transforms.
A complete
coverage of the subject may be found in Springer (1979).
In
fact our first attempt to answer some of the questions raised
above was tried using the Mellin transforms.
The computational
involvement led us to the geometric approach used in this
paper.
2.
Laha (1959, theorem 2.1) showed: If X and Yare (i) independent,
(ii) identically distributed with common distribution function
F, and (iii) W = X/Y is Cauchy, symmetric about 0, then F has
the following four properties.
13
(1) Symmetric about 0 •
(2) F is absolutely continuous and has continous p.d.f. f(x) > 0 .
(3) X has unbounded range •
(4) f satisfies
2
Joof(X) f(ux)xdx = CO/(l + u )
o
for all u. where Co is a constant.
Consider the following problem:
"Let X and y be two random
variables with joint distribution function F
and Fy • such that X/Y is standard Cauchy.
Y and marginals F
X•
X
What kind of general
conditions must hold on F y' F ' and Fy ".
X•
X
Some partial answers:
Theorem 17:
Let X and Y be identically distributed random variables with
common distribution function F such that X/Y is standard Cauchy.
Then X and Yare independent iff F has the four properties mentioned
above.
Proof:
The if part follows from Laha's theorem.
X
= U2l and Y = U22 .
For only if:
Let
That shows if X and Yare not independent.
then F does not satisfy these properties.
On the other hand, if we assume that X and Yare independent
(but not identically distributed) such that X/Y is standard Cauchy.
then F and Fy do not have to satisfy these four properties
X
simultaneously. This may be seen by considering U /U .
3l 33
14
5.
MOTIYATION: ESTIMATION IN ONE LESS THAN
FULL RANK LINEAR MODEL
The problem posed in section 1 was motivated by the following.
Consider the linear model E(Y)
=
XS, where Y is an m x 1 random
vector, X is m x n design matrix, and S n x 1 parameter vector.
Sof
The least square estimate
t
Rank(X X)
=n
B satisfies
(XtX)S
=
Xty.
If
- 1, then in order to obtain an "estimate" for B we
must add one more constraint.
If there is no theory or previous
knowledge to suggest the form of this new additional constraint, it
is chosen more or less at random.
random constraint such that
II CII
tAt
= O.
fying (X X)Sc
= X Y and
t
C Sc
Let CtS
0 be the additional
=
A
=
1.
Let Sc be the solution satis-
Let Y be the eigenvector corre-
sponding to the zero eigenvalue of XtX. Then B may be written as
c
Sc = S __ (CtS/cty)y. Thus, the stochastic properties of B are
c
essentially those of the quantity CtS/Cty, C € sn-l .
15
ACKNOWLEDGEMENT
This research was partially sponsored by grant 5-T32-ES07018
from the National Institute of Environmental Health Sciences.
BIBLIOGRAPHY
Laha, R.G. (1959). "On the laws of Cauchy and Gauss".
Math. Stat., 30, 1165-1174.
Springer, M.D. (1979).
Wiley, New York.
The Algebra of Random Variables.
Ann. of
John
Springer, M.D. and Thompson, W.E. (1966). "The distribution of
product of independent random variables." J. SIAM Appl. Math.,
Vol. 14, No.3, 511-526 .
•