1
. Department of Mathematics, Temple University
This research was sponsored in part by the U.S. Army Research Office,
Durham, under contract No. DAHC04-7l-C-0042.
A VECTOR VALUED t-1ULTIVARIATE HAZARD RATE - I I
N.L. Johnson &S. Kotz l
University of North Carolina, Chapel Hill
and
Temple University, Philadelphia
Institute of Statistics Mimeo Series No. 883
August, 1973
A VECTOR VALUED MULTIVARIATE HAZARD RATE - I I
By
N.L. Johnson, University of North Carolina at
Chapel Hill
and
S. Kotz, Temple University, Philadelphia, Pa.
1.
INTRODUCTION
This paper is a continuation of [1], wherein can be found the definition
of a vector valued multivariate hazard rate, some general properties and applications to specific distributions.
In the present paper we first discuss some
further general properties, and then consider some further specific distributions.
Throughout it is assumed that the variables are continuous and possess
joint density functions.
FX(~)
~
2.
= Pre
We use the notation
m
m
n (X.sx.]
j=l
J
J
GK(~)
._
= Pre j=l
n (X.>x.)]
J J
RELATION BETWEEN HAZARD RATES AND ORnlANT DEPENDENCE
In section 3(i) of [1] it was shown that if Xl""'Xm are mutually
independent then
where
hx(~)
is the (vector) multivariate hazard rate of Xl"",Xm and
hX (x.) is the (univariate) hazard rate of X.. Another way of expressing
j J
J
this result is
,...,
2
Lehmann [2] has defined positive (negative) quadrant dependenae between
(1)
(~r
l
He has shown that these definitions are equivalent to tllose obtained by
Dykstra et. aZ. [3]
(negative)
o~thant
have extended these definitions to define positive
dependenae among
m(~2)
variables
X ,X 2 "",Xm to corresl
pond to
(3)
for all
x.
'"
We, however, will work with the equally natural extended definition
(4)
for all
~,
which we will call
G-positive (negative)
o~thant
dependenae.
For m = 2, the two definitions are equivalent; for m > 2 this is not
so.
(A counterexample is easily constructed (for m = 3) by noticing
that (3) is satisfied if the conditional joint distribution of
Xl
and
X
2
3
given (X 3 S x ) has positive (negative) quadrant dependence, while (4)
3
is satisfied if the joint distribution given (X s > x ) has positive
3
(negative) quadrant dependence.)
We will now develop some relations among
rates.
RG(~)(~)
and various hazard
We first note that
(5)
lim RG(X , ... , X )(x1 ,···,xm)
1
m
= RG(X
xl~-oo
2"'"
X )(X 2 ,···,x )
m
m
In particular, for m = 2
(6)
Since
-e
a10g RG
(X)
(7)
(~)/ax.
J
= alog
GX(~)/dXJ'
~
- alog GX (x.)/ax.
j J
J
""
= hX (x.)
j
J
- hX(~)
~
.
J
we see that for bivariate distributions (m=2), if h
xj (x.)
J
for all
~
and j
= 1,2
>
«)
hx(~),
~
J
then (remembering (6)), the distribution has posi-
tive (negative) quadrant dependence.
The converse is not necessarily true.
We cannot immediately extend this result to cases m > 2.
In view of
(5), we do have results like:
"If X "",X _1 have G-positive (negative) orthant dependence and
m
1
h (xm) > «) hX(~)m for all ~, then XI"",Xm have G-positive (negative)
Xm
orthant dependence.,11
~
4
For the Morgenstern-Gumbel-Farlie bivariate distributions defined by
or equivalently
(see equation (6) in [1]),· we have
= l+a.{l
(9)
(10)
RG(X ,X ) (x l ,x 2) ~ (~) 1 according as a. ~ (~) O.
l 2
is true of RF(X X )(x l ,x 2)). Also, in this case, from equation
l' 2
[1] , hX (x.) - hX(~). has the same sign as a., so in this case
j J
,..,
J
verse result referred to above (that positive (negative) quadrant
In this case
_A
•
ence implies
hX. (x j )
>
(The same
(7) of
the condepend-
«) hX(x)j)' does hold.
J
3.
TRANSFOR~~TION
OF VARIABLES
We first note that if
! have positive (negative) orthant dependence
(in the sense of Dykstra et. aZ.) then
-!
have
G-positive (negative
orthant dependence, and conversely.
If
21
= -Xl
then (remembering the variables are continuous)
X (zl'x 2"",x )
Gz X
l' 2'"'' m
m
= Gx2"
X (x ,···,x )
.. , m 2
m
- GX
X (-zl'x2 ,···,x )
1" .• , m
m
5
and so
(11.1)
while for j
(11.2)
~
2
X (zl'x 2 ,···,x ). = hx
X (x ,···,x ).- hX(-Zl,x2""~x )~
m
mJ
2'" " m 2
mJ
'"
mJ
hZ X
"1' 2"'"
Similar (though more complicated) results can be obtained when several of the
Xj'S are changed in sign.
X.
J
If Y.
is a continuous increasing monotonic function of X., so is
of
and we can write
J
Y.
J
J
X'
J
Since the events (Y.
J
>
y.)
J
= F;. (Y . )
and
J
(j=l, .•. ,m) •
J
(X.
J
> ~.(y.))
J
J
are equivalent we have
and
(12)
If !
is IHR (OHR) and
function of Yj
for j
particular, if !
£>
is
a~j/aYj
= 1,2, ... ,m
is a non-decreasing (non-increasing)
then
X
IHR (OHR) then so is
also is
IHR (OHR).
CblXl+a1, .•. ,bmXm+am)
In
if
Q• .: fSoo a1s.0 section 3 (iv) of [IJ).
It is )nown ([4] p. 37) that mixtures of univariate OHR distributions
are also OHR.
(A similar property is not valid for IHR distributions.)
Since
is the hazard rate of the conditional (univariate) distri-
h (;1S)l
X
'"
bution of Xl
given
m (V
j~2 ~j >
Xj)' (see Section 3(iv) of [1]), it follows
6
that mixtures of multivariate DHR distributions are also DHR.
4.
SOME FURTHER SPECIAL DISTRIBUTIONS.
Details of many of these distributions
will be found in [5].
(i)
Bivariate Logistic
We consider the joint comulative distribution function:
FX X (x l ,x2) = (1
1 2
+
e
-xl
+
e
-x2-l
)
for which
=1
Gx X (x l ,x2)
l' 2
(13)
- (l+e
-Xl -1
-x2 -1
)
- (l+e
)
+
(l+e
-xl
+ e
-x2-1
)
We find
= (1 +e
-x 2 -1
-xl -x -1
-xl -x 2 -1
-xl -1
) (2+e +e 2) { (1 +e
+e
)
+ (1 +e
) }
which is an increasing function of xl'
For general
The joint distribution is IHR.
m with
FX
X (xl,···,x )
1 ,· .. , m
m
= (1
+
m -x. 1
I e J)-
. 1
J=
it has not yet been possible to establish that the distribution is IHR,
though it seems very likely.
(ii)
Gumbel's Bivariate Exponential Distributions.
If the joint cumulative distribution function is
then
7
(15)
whence
hX(~)
(16)
= (1+ex2~
'"
l+ex l ) .
These components are constant with respect to variati.on in the
corresponding variable (ie.
hX(~)1
does not depend on
xl' nor
'"
hX(~)2
on x2) but not with respect to variation in the other variable.
'"
In section 3 (v) of [1] it was shown that if h~(~) = £, with £
constant (with respect to aZZ variables) then XI ,X 2 , ..• ,Xm must have
independent exponential distributions. For convenience we might term
=£
h~(!)
a str>iatZy constant hazard rate; the distribution (10), on
the other hand, may be regarded as having a ZoaaZZy constant hazard
rate.
If we assume
hX(~)
'"
= (fl (x 2),
f
2 (x l »
then
AI(o), A2 (o) are arbitrary functions subject to the standard
conditions on Gx(~).
wl~re
'"
=0
gives Al(O)
= -x2f 2 (0)
A2 (x2), so A2 (x 2) must
be a linear function of x2 • Similarly Al (xl) must be a linear function of xl' Now putting xl = a F 0, we obtain
Putting xl
+
8
so that
f l (x 2) must be a linear function of x2 . Similarly f 2 (x l )
must be a linear function of xl. It follows that the joint distribution of appropriate linear transforms of X1,X Z is of form (10).
To summarize, Gumbel's family of bivariate exponential distributions includes all distributions with locally constant bivariate
•
hazard rates •
..
(iii)
Bivariate Exponential Distributions of Marshall and Olkin.
These have joint survival functions of form
(17)
For these distributions
h~(~) =
(18)
(A 1 ,A 2+A lZ )
{(A +A ,A2)
l I2
They are not strictly IHR, but for
ent
x2 (xl ) fixed the first (second) compon-
is a non-decreasing function of xl (x ).
2
The analysis for multivariate (m > 2) tmrshall-Olkin distributions
(see [5]) follows similar lines.
(iv)
Freund's Bivariate Exponential Distribution.
For this distribution, the joint density is
= {~B'eXP{-B'X2
PX X (xl ,x2)
I' 2
a'Sexp{-aix
l
- (a+B-S1)x l }
- (a+B-a')x }
2
9
with a,S,al,S' >
o.
We will assume
a+8
~
ai,S', and
a l > a, S' > S ,
We find
(a+i3-S') -1 [exexp{ -13' x2 - (ex+S-S' )x l }+ (13-13 I )exp{ - (a+s)x Z}
(19)
(0~xI<x2)
(a+S-8') -1 [Sexp{ -ex I x1 - (a+S-ex' )x2 }+ (a-a') exp{ - (a+s)x I }
and
a(a+S-@')
(20)
13ex'-(a'-a) (a+(3)exp{-(a+S-a l ) (Xl-X2)}
13-(a'-a)exp{-(a+i3-a ')(x l -x Z)}
Taking the case
a+S>a'>a; a+S>S'>S we find that for
I
xl < x2 ;
h~(~)l is of form c l {c Z-c 3 exp(c 4 xl )}-1\,!here c j > 0 (j=1,2,3,4), and
so is an increasing function of xl'
For xl > xZ;
hX(~)l
,...
is of form
CXP(C 6 x 1 ) - c 7
gexp(c 6 xl ) - c 9
where c j > 0, and is an increasing function of xl
c
S
c
if
c 7c 8 - cSc g > O.
Now c 7c8 - cSc 9 = [S(exl-a)exp{(a+s-a')xZ}][a+S-a l ] > 0" So
hX(;S) I is also an increasing [unction of Xl for Xl > xZ" Similarly,
,..
h~(~)z is an increasing function of Xz for all xZ" The joint
10
distribution is IHR.
The same result is obtained for other inequalities between a+e,
a' and S' (but always keeping a' > a, 8' > 8).
(v)
A ~~ltivariate F-Distribution.
We first consider the j oint distribution of
G
j
=
V/V 0 (j =1, ... ,m)
x2 ,s with
where VO'Vl""'Vm are independently distributed as
\),2,2, ••• ,2 degrees of freedom respectively. The joint density is
and the joint survivor function is
m
(21)
(1 +
1
}'
-2 v
... gJ')
j=l
whence
(22)
The joint distribution is DHR.
In this case all elements of the vector
multivariate hazard rate are equal.
These results also apply to the multivariate F-distribution which
= ~VGj
is the joint distribution of F
j
For the joint distribution of
r.
J
(j2 1 , .•• ,m).
= +/G.
J
we find (directly, or
from (9))
(23)
hr (tJ i = vt. (l +
_
1
m 2-1
j=l J
It. )
(i=l, •.. ,m)
(t. > 0)
J
Recalling that
d~ (--';"J = (A+t 2) -2
A+t
(A_t 2), we see that
hrCtJ
_
is an
11
increasing function of t.
1
t~
> 1 +
1
(vi)
I t~ - t~ .
j=l J
Bivariate
2
for
t. < 1 +
1
m 2 2.
I t.-t. , decreasin~'0 for
j=l J 1
The joint distribution is neither IHR nor DHR.
1
Fr~chet
Distributions
For given marginal cumulative digtribution functions
FX (Xl)'
1
family of bivariate distributions has joint
FX (x2) the Fr~chet
2
cumulative distribution function
FXl,X2(Xl,X2)
= w max{O,F x (Xl)
1
+ FX (x 2)-1} + (l-w)min{F x (xl),F X (x 2)}
2
1
2
(0 < w < 1).
Hence
(24)
and
GX(fS)hX(?£)
"'"
=
(wfX (xl),fx (x 2)) for FX (xl)<F X (x 2);
1
2
1
2
FX (xl)+F X (x 2) < 1
"'"
'(25)
(,\lith f X (x.) = dF X (x.)jdx.).
j J
j J
J
1
2
12
Note that in general there are discontinuities in the components
of
hx(~)
even when
""
hX (xl)
and
hX (x 2) are continuous.
1
The
x.
component of
J
2
GX(~)hX(~)
""
0, wf
For x2 fixed, as
(l-w)fx (x.),
Xj (x.),
J
j J
xl
increases
If FX (x 2)
2
>
o~
f
Gx(~)hx(~)
""
over successive intervals of xl
can be
""
X.
J
takes the values
,...,
if Px (x2 )
1.2 ' the sequence is
(x.) •
J
<
21 .
2
is a continuous function of x, it is clear these cannot
'"
be increasing or decreasing functions over the whole range of xl'
for any x2 ' if f (xl) t: o. So, genera:i.ly, Frechet distributions cannot
X
1
be IHR or DHR, using the vector multivariate hazard rate we have defined.
(vii)
Multinormal Distributions
Numerical evidence in Section 4.1 indicates that bivariate normal
distributions are IHR, at least when the correlation coefficient is not
negative.
so ..
Dr. J. Galambos (private communication) haz
sho~m
that this is
Here we note a few
points in connection with general m-variate multinormal distributions.
If x ,X , ... ,x
have a joint multinorm&l distribution, the condil 2
m
tional distribution of Xl' given X2"",Xm is normal with a fixed
13
standard deviation (which we can arrange to be 1) and expected value
m
I b. x. + bI' say).
2
m (Z =j=2
J J1
The first element of the vector multivariate hazard rate is thereX , •.. ,X
equal to a linear function of
fore
E[¢(xl-Z)]
=
(26)
where
~(t)
= (1:rIT)-le
E[l-'P(xl-Z)f
.1t 2
2
'P(y)
=f
t
~(t)dt
and expectations are taken
_00
with respect to
Z, distributed conditionally on
m
n (X. > x.).
j=2 J
J
Although
is an increasing function of xl' for any
is an increasing function of Xl
-e
s.
Z it does not follow that
for all
x•
....
CONCLUDING REMARKS
We feel that
enou~h
results have been presented, in [1] and the present
paper, to provide evidence of the usefulness of our proposed vector definition
of multivariate hazard rate.
Using the definitions of IHR and DHR based on this concept we are able to
classify a number of important multivariate distributions, comparable with
those so classified by other definitions of IHR and DHR.
We feel however that the criteria of overall IHR and DHR are too sweeping
to be of general usefulness.
Even for univariate distributions, the DHR
property implies tha.t (if continuous) the density must be a decreasing function
of X over its support (because
14
dhX(x)
dx
GX(x)fX(x)
+
=
{fX(X)}2
{G v (x)}2
J\.
where
fX(x)
= -dGX(x)/dx
is the density function of X).
Rather, the hazard rate function itself
~.$
of value as a description of
the distribution - in particular of the intervals in which hazard rate is
increasing or decreasing.
From this point of view, the desirability of some form of vector hazard rate
for multivariate distributions is evident.
Whether the particular one we have
chosen is, in any sense, optimal, is an open question.
We believe that we have
shown it to be useful.
REFERENCES
-e
[1]
Johnson, N.L. and Kotz, S. (1973). A vector valued multivariate hazard
rate - I. Institute of St&tistics, University of North Carolina,
Mimeo Sel'ies No. 873.
[2]
Lehmann, E.L. (1966).
37~ 1137-53.
[3]
Dykstra, R.L., Hewett, J.E. and Thompson, W.A. (1973).
are almost independent, Ann. Statist.~ 1, 674-81.
[4]
Barlow, R.E. and Proschan, F. (1965).
New York: Wiley.
[5]
Johnson, N.L. and Kotz, S. (1972). Distl'ibutions of Statistics: Continuous MultivaPiate Distl'ibutions~ New York: Wiley.
Some concepts of dependence.
Ann. Math.
Stati8t.~
Events which
Mathematical Theol'y of
Reliability~
MIMEO SERIES NO. 873 - CORRIGENDA
The following corrections and changes should be made to Mimeo Series
No. 873 (A Vector Valued Multivariate Hazard Rate - I by N.L. Johnson and
S. Kotz).
..
Page
1
Line
2
6
2
11
·e
Insert "of" between "rate" and "Xl'"
g(x) should be g(X)
4
11
x l ,x 2 , ••. ,xm for aLL xi' ... 'x~ (including xj<xj )
Replace "monotonic" by "increasing"
4
14
g
2
..
2
6f
Should read:
-1
(Xl) should be g
-1
4
4f
X should be
8
7f
Insert "bivariate" between "joint" and ildistributions"
10, lower table
Interchange
-1
13
6
e. X.
13
8
Should read:
J
J
X
"'"
xl
should be
and
16
17
II
x2
-1
e. x.
J
-a log Gx(~)/ax
"'"
r
•
(Yl)
J
= ae
r
(
m
I e~
j=l J
1
x.-m
J
+
1) -1
3-2f
Insert lISince, for all g > 2, 2 < K. (g) < log g, the
1-3
upper bound is, in fact [exp{K (gO)}
j
Delete
J
1]-1