APPgNDIX
I
DSEIYATIQN OF A GLA33 QF COMPOUND FREQUBMCY FUNCTIONS
FROM THE GLA33 OF DISTRIBUTIONS
ADMITTING SUFFIGISNT STATISTICS
1, I n t r o d u c t i o n :
I t ^^as remarked in
§ Z. 5 t h a t i f -TL
s t a n d s for t h e g e n e r a l c l a s s of d i s t r i b u t i o n s a d m i t t i n g
s t a t i s t i c s , then f o r t h e raembers of a s u b c l a s s o)
p r o p e r t y of a d m i t t i n g s u f f i c i e n t
sense explained t h e r e .
of a s u b c l a s s ^
of -TL
of _n.
the
s t a t i s t i c s i s t r a n s i t i v e in the
This Mas a l s o d e s c r i b e d as ' f o r members
the p r o p e r t y of a d m i t t i n g
s t a t i s t i c s i s i n v a r i a n t under compounding'.
of 00
sufficient
vjas given, but i t was remarked t h a t i t
i m p o s s i b l e t o determine t h e s u b c l a s s
co
of
sufficient
The d e f i n i n g
property
seems t o be a c t u a l l y
-CL due t o some com-
p l e x i t i e s i n h e r e n t in t h e problem.
In t h e p r e s e n t appendix we s h a l l f o n n u l a t e t h e g e n e r a l
problem of o b t a i n i n g a c l a s s of compound frequency f u n c t i o n s
o r d i n a r y frequency f u n c t i o n s ,
and s t u d y t h e problem of
compound d i s t r i b u t i o n s from the c l a s s of d i s t r i b u t i o n s
sufficient
statistics.
a r e appended.
from
deriving
admitting
S e v e r a l examples i l l u s t r a t i n g t h e t h e o r y
113
2• ^2B^ preliiraiiary o b s e r v a t i o n s :
be t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n
On /S unknown p a r a m e t e r s
Let
i-(x | 0, ,Q^ , • • • , O^A, j
(P.D.F. ) of ^<^
0| , Q^ ^ • . • , Q^
.
Then
j-fj',
t h e p r o b a b i l i t y t h a t a random o b s e r v a t i o n made on JC
the interval
( x - '^^ ^ -^c ^ dx ',
f u n c t i o n of a.
.
.
depending
is
falls
in
This p r o b a b i l i t y i s a
I t i s a l s o a function of t h e unknown p a r a m e t e r s
0,, «^ , . . , , •'^j
.
Thus i f one or more of t h e parameters
change, t h i s p r o b a b i l i t y a l s o c h a n g e s .
It i s clear , therefore,
t h a t t h e s e parameters can themselves be regarded as o r d i n a r y
v a r i a b l e s , assuming c e r t a i n values ^vith c e r t a i n p r o b a b i l i t i e s .
Let t h e i r j o i n t p r o b a b i l i t y be ""r^f^:,'^,, • >'^ij AO,->.^\
i^'^i .
Then t h e theorem on t h e m u l t i p l i c a t i o n of p r o b a b i l i t i e s
"^f' - i^Vo,,^^_
•,^,
r^^.,^_,
which i s t h e j o i n t p r o b a b i l i t y of
x
gives
• ^%, 'i-- i « / ^ \
and t h e
G^
.
^^
J
1
Then t h e
P . D . ? . of -• i s given by
J-'^
^
( Jl'^/a.A ,, ., , 9 j , ^h(n,,0 ,• • , 0 , ; a<5, iO^.
i9i ,
vjhere t h e r e g i o n of t h e m u l t i p l e i n t e g r a l on t h e r i ^ht of
i s t h e one i n which
'"j-^
i s defined.
n h e compound frequency f u n c t i o n '
called,
calls,
_.c
,
' t h e o v e r a l l frequency f u n c t i o n '
of
'contagious d i s t r i b u t i o n ' o f
of
Here
,;. .
';| (.y"
is
(1)
called
Sometimes i t i s
^
.
(1)
also
Neyman (1939)
In t h i s c o n n e c t i o n ,
see
Gurland (195^^) for an e x c e l l e n t b i b l i o g r a n h y t o t h e lAjork done on
114
contagious distributions.
Instead of dealing with arbitrary frequency distributions
f
and 'f^
, let us confine ourselves to -.he class . .0_
of distributions admitting sufficient statistics.
Let i-('^/G,,Q,, ••,0^
be the P.D.F. of JC depending on i unknown but independent parameters 0^^^ •}>•--. },z,. .. , i
, and admitting a set of ,6
sufficient statistics for the parameters.
depending on %
jointly
Let the P.D.F. of Q^ be
unknown but indeoendent oarameters f^,^ ,k - i,v, • •/i .
Then it follows that the overall distribution of x
tion that the %
vary according to the la\'i 'f:,
parameters (-^^^
. Suppose now that i^^
on the assump-
deoends upon '1 '-i''^-/.,
itself admits a set of
jointly sufficient statistics for the parameters f^^,^ , v">,2.,--- ,.5,
That is, we assume that the functions f and %^
members of the class
CI
.
v - K z ,• • • , ^i
are
It is very interesting to note
it
that in some instances, the overall distribution ^ of oc admits
a set of j ointly sufficient s t a t i s t i c s for the parameters (\
lIK
themselves, or atleast for some of them when the rest are known,
so that ^
also belongs to .0- .
that the class -^
In such a case we might say
is closed under 'compounding' or in other
words, that the property of admitting sufficient statistics is
invariant under 'compounding'.
Unfortunately, this property is
not enjoyed by all the members of the class JX. , but only by the
\
115
members of a subclass oo
of -0.- due to the intractability of
some integrals occurring in the problem.
In the sequel x
variable.
may be considered as a vector random
To make the arguments free from unnecessary complica-
tions, we shall consider only distributions depending upon a
single unknown parameter.
The additional complication in the
proof for the rnultiparametric instance is in writin?: down the
various results only.
Once this is done, the rest of the argu-
ment is the same. In what follows, we suppose that the distributions under consideration are continuous.
When they are discrete,
the modification needed is obvious.
3* The class of distributions admitting sufficient
statistics:
Let -fl- denote the class of uniparametric distribu-
tions admitting sufficient statistics.
Then as shown by Koopraan
the most general form of a distribution depending on an unknown
parameter
0
and admitting a sufficient statistic for
when the range of JC
0
is
is independent of 0 , and inust be of the
form
when the range of y. depends uoon G , where the functions
and
^7!'^'
satisfy the monotonic restrictions of
§ 6.2.
-.fo,
For
116
convenience (2) and (3) w i l l be c a l l e d " P a r e n t
populations".
I f now t h e p a r a m e t e r S i s d i s t r i b u t e d a c c o r d i n g t o
t h e law
"^"(o 3
depending on t h e unknown p a r a m e t e r
admitting a sufficient
s t a t i s t i c for
(i
, then
p
and
V" must be of
t h e form:
when t h e r a n g e of 0
i s independent of
f- , and of t h e form
h ^ i (^' •
when t h e range
'-i'?.
(5)
of
0
depends on
are monotone f u n c t i o n s of
(b
.
.
Here a g a i n
a,/(;,,
For c o n v e n i e n c e ,
and
(4) and
w i l l be c a l l e d "auxiliary p o p u l - . t i o n s " .
^ ' Two i n t e r e s t i n g p r o p e r t i e s of t h e c l a s s
- C^
'.ie
s h a l l in t h i s s e c t i o n n o t i c e t'v;^ i n t e r e s t i n g p r o p e r t i e s of t h e
class
vL
.
In t h e f i r s t p l a c e , so f a r as t h e form i s concerned
(2) i s symmetric i n -J: and
a v a r i a b l e and
b u t i o n of ^
Here
0
0
, so t h a t when G i s r e g a r d e d a s
as a p a r a m e t e r , t h e n (2) sho/is t h a t t h e
admits a s u f f i c i e n t
s t a t i s t i c f o r t h e parameter
v a r i e s in some s u i t a b l e ran'Te independent
where
)'<^^ -^h^o. ,
i-(0/,c,
and
){O:/.MI
t-i(-^'
and h(y<.^ are monotone f u t i c t i o n s of :x
of
''
a l s o a dmits a s u f f i c i e n t
, ,
< .
of '--
The symmetry h o l d s even in t h e case of ( 3 ) .
may be w r i t t e n in t h e form
distri-
For,
(3)
^'^ '^ © '' bf:-/ ,
-• ^;!'-.'V
and
, so t h a t t h e
distribution
s t a t i s t i c for .,; .
A(>.
I t i s curious
117
to note t h a t the univariate-multiparametric d i s t r i b u t i o n of DC
i s symmetrical vjith the m u l t i v a r i a t e - u n i p a r a m e t r i c
of the parameters.
distribution
Thus for t h e raembers o f JX , t h e property of
admittin;'^ sufficient s t a t i s t i c s i s symmetric in v- and ^ .
Secondly we observe t h a t for t h e members of only a
subclass (A3 of .(\ , the property of adrnittin/'; s u f f i c i e n t
tics i s transitive.
statis-
Unfortunately t h i s property i s possessed by
the members of a subclass «
of O- .
No general method seems
to be possible for the actual determination of o^ ,
Thus (^ i s
characterized by the property t h a t
if
'''"'^^
) c <\) ' . X L ,
then
'/(-/(i-
- ^-•
;/e s h a l l consider \^he followin.^ cases and report the
encountered.
GA3S (1):
difficulties
Some examples w i l l be included i n the next s e c t i o n .
Treating (2) as the parent and (i^) as t h e a u x i l i a r y
d i s t r i b u t i o n , the o v e r a l l
P.D.F. of -- i s given by
1*(-y(i: •-- M^-'-/p' t'®/>-.,. HA -. cxp^ hO,. rb^m \- ^^^'~cji,\
(6)
T»?here
./fv U(.9 •;9f.r, r C f 8: -f''h(^,
'^V^ H- A,(^^ \ dO
( 7)
is the basic integral, in this i^ay a new class of compound frequency
functions
'^ (
•
'
'
-
of -
can be derived starting from the class
of distributions admitting sufficient statistics.
the integral (7) is in many cases intractable.
Unfortunately,
But vjhen the
if-
integral can b e evaluated, it f r e q u e n t l y happens that
'^ ("'-t/fi
lis
d e f i n e d i n (6) can be w r i t t e n i n t h e form
which i s i n Koopman's form of a d i s t r i b u t i o n a d m i t t i n g a s u f f i x
c i e n t s t a t i s t i c for
e
, so t h a t
j (a/(3'
.. O-
parametric case i s considered, i t t u r n s out t h a t
of j o i n t l y s u f f i c i e n t
a r e known.
.
I f the m u l t i -
^
admits a s e t
s t a t i s t i c s f o r some of t h e
This may b e t r u e ; but i t
'^ ^
when t h e
rest
seems t o be i m p o s s i b l e t o
o b t a i n a g e n e r a l proof i-Jhich i v i l l c o v e r every t y p e of c a s e , o r
even t o l a y down c o n d i t i o n s on
that
GA3J^
j ^ .(2);
and
':''''^''^
:
-
of --
as:
• '-:•; h:''^'
and
' cxp I Ui'^: •9:.>: •}- an, + ^^-j hi^^ ' ^^
i s again t h e basic i n t e g r a l .
Koopman' s form f o r
((3) may be w r i t t e n i n
Taking (3) as the p a r e n t and (5) as t h e
3•
-
' f ( - ' ; • ) ; ^-^i^/Ti"' -i^'^
:>^^
r
i i [<:
,
••- 0, :-, n- "i
an <
auxiliary
i s given by
where
)V.^;
admits
'^
p o p u l a t i o n , the o v e r a l l P.D.F. of
-1^
(9 )
As b e f o r e i f t h i s i n t e g r a l
of e v a l u a t i o n , i t f r e q u e n t l y happens t h a t
% '^
in order
Considering (2) as t h e p a r e n t and (5) as t h e a u x i l i a r y ,
h, '-^
CASB ( 3 ) ;
'^"^/.'^}
- ^-
we o b t a i n f o r t h e P.D.h\
where
iY-v'^
. a,.,^/ •
•*'ii'^'''
^
,
l-^ .
119
In t h i s case,
'^(x/p)
i s a c t u a l l y in Pitman's form of a
d i s t r i b u t i o n admitting a s u f f i c i e n t s t a t i s t i c for
^ ' ' ' > - / ^ '
-
^ ' -
and hence
•
The l i m i t s of
>
-which depend on P) ma/ conveniently be
expressed ?.s those depending on
and since
P
a (o
i s incr-^^'asine; and
e
.
For, since
'\,,{(i; i. 0 :^ 'r,((^y
<7(0; i s decreasing v,'ith 0
, "we
have in one case
'
•'
If
^ (Q'
'
J
i s decreasing and ir((^) i s increasing, we have since
ai ft) s ^ -^ s'P'
i.e. ,
xi b/Pj •. ;
, ^^,_.i ^f{.-, ; .
I t thus appears tliat the property of qdiaitting;
sufficls--nt s t a t i s t i c s i s i n v a r i a n t under"compounding'*, or in other
•vords,
'•- i s closed under compounding^.
As remarked e a r l i e r t h i s
property i s possessed 'oj only a subclass
properti/ of the
CIPSS
-'»^
'J
of
Ti- ,
This
i s used i n Jhaoter 7 I I I in proposing
a unique solutioxi to Herbert Robbins' problem with some asymptotic
optimum p r o p e r t i e s .
In the next section -we s h a l l rive some examples to sho^^
t h a t i t i s possible t o choose
r-/;5~
and
f ^ ^/f^ '
of
such a way thet t h e o v e r a l l frequency function of >. also
to
C\.
0.
in
belongs
120
5. Example (1); Consider the normal distribution:
'
ir
In the theory of errors
•precision constant'.
n.
that
•
^ is usually referred to as the
Since
n .- ^
, it is reasonable to assume
is distributed in Pearson's Type III population:
\' A .
First note that both
r
and f
overall P.D.F. of
isO C
are members of
i- . Then the
r-,
, ,
f / •
'
'
I
which admits a sufficient s t a t i s t i c for
- vjhen
-< i s known.
Consider next the normal distribution with
mean
f^
and precision constant h :
K^7 isn j
Let
I'
-
J ^-. -^
, '• '1 -• ^ ,
^-^ < . "^ '''•'
be distributed in the normal distribution \vith mean A^-
and precision constant
>
'c :
so that the overall frequency of .' is
f(^/h,^,h.,
f
• |i(-/AA,n, y-(^^//^, ,n.. i/'
I -,1- where
"
" '^'"'^ •'ni-^o.
- M ( >
c,
fir
. This shovjs t h a t
9^ i s a l s o normal
121
vjith mean
/^^
and precision constant
H
, so that
admits a pair of iointly sufficient statistics for
(when one of
h
and
h,
But when both
/^
is known).
and
K
Ilote that
if
<^-
/^o
also
and
H
H,< '^
vary simultaneously,
the
double integral seems to be rather intractable.
Bxample (2):
A simple type of model for accident proneness
(see Johnson and aar^Jood (1957) and Arbous and Kerrich (1951)
)
is obtained by supposinc;-- that the number of accidents sustained
by a given individual in a period of fixed 'unit' time may be
represented by a Poisson variable
i-(>/e,
and that
--
0 ^o
-
i
oc vjith
, ^. . o, / , ^ ,
varies from individual to individual '^ith
Pearson's Type III function
T ((^J x ,/,)
de first observe
that both X and f" admit sufficient statistics for G and
and
A
respectively,
-^
./e shall first note some interesting
properties of the model.
It is well-known that the cumulants of -x are given by
^-J^^ =
^^
for all
If the parameter 0
/
.
i^
=
has the P.D.?. 7^
't.f-'^Ao
1, 2, 3,
then
'
(10)
defines a class of compound frequency functions of -c for various
functions ^^ .
;,'e observe first that the moments of *. can, in
general, be expressed as functions of the moments of
0
.
Let Uiv]
122
denote the f a c t o r i a l moment of the order
/^Lvj^'- '^'
•
D
, then
-
Ur y^ { x]
-
•
i (•?:
:
v:^ ^•. / ' ^ O t . •
Thus the f a c t o r i a l moment of order
;)
of
x
i s equal t o the
raw moment of the same order of ©
•,'iJe note another i n t e r e s t i n g property.
From (10)
vje
have
•^:.:-..,-
so t h a t
::
.^,..
vi
'- f .
' .M
A,
' - • ' - '
•^
c'^
t
,
n'
, 0 .
.
.
!
.
t^
J '-^ ^";. . n
J
•where
-! ' • x . Q
- '~^'• ••"
•'^•,
the conditional frequency of
lA
,
]1-f
.
No-w s i n c e
3 (- ,^' > <f(j('
is
•"' for any given :r , the i n t e g r a l
on the right represents t h e regression
Tn,;/'.-
of
O on ^'
so t h a t the above difference equation can be vjritten as
i'",
J ( ^
,
(-:
.
, - . . ,
! .^ - i - :
*
'
-
.
•
1 / _;
Substituting
.,r. / A / ^•
' " ' '^
/ -< .
•^f ^/ y.,-<'
the overall frequency of
x
•
for '- we find that
i.e., of the observed nuraber
•- of
accidents per individual in a given period of unit length is given
by
,
•'J ( D<
• ,
:
-
•
or
-
G
' • '^
' i *^
123
•which is called the Negative Binomial distribution admitting a
sufficient statistic for
regression of 9 on x
loc. cit)
-^ when
A
is linear
is knoi-vn. In this case the
(see Johnson and Garwood,
and the regression line is
0 = (xt-A, / ^/^ n
Example (3); Consider next the exponential population:
v*hich admits a sufficient statistic for 6 .
random variable with the P.D. x^. T
.
Let
6
be a
Then a class of compound
frequency functions of ;•:. are defined by
tio.)
:--
) i\ 0- c"^^ a 9
.
(11)
In this case the raw moment of order •.' of x
is
•>
so that
:• v[ f 4v e"' 'i^
.
F u r t h e r , t h e r e g r e s s i o n l i n e of j . on
0
is the rectangular
hyperbola:
and t h e r e g r e s s i o n of
/
. P-c
0 on
x
i s given by
,'
^j^e s h a l l now o b t a i n a siraole formula f o r
„
.;i r,
C) ' - > ,
'H; '^',, x)
(12)
in terms
124
of
^ fa)
only.
Differentiate • (11) with regard to x
under
the integral sign and obtain
1 (x
^
•i-x
•' '•
-
- ^^
9": *^
dO
'
f(^,
. rn,(V>:
from (12), which gives immediately
Substituting
'vi'O'' ^',.«,
overall P.D.?. of
x
for
^'^ , Vv'e observe that the
is given by
which admits a sufficient statistic for
A when '^ is known.
It will be seen that if vje choose
•»n<
0
,,''^ ,, QJ-
f(;^/o,To; - - >
ir).
and
'v^'^,- '*, A,
*
<|/-^/•..,.. V
as b e f o r e , •'ve find
•
I I •'V
'
.
^ ,;'^'---^-.
^.---^^A
in
i^.-1 A. w .-i': -
. '''; '• ^-^
M 1- ^ '
f.(-'--
which adiTiits a pair of j o i n t l y sufficient s t a t i s t i c s for m and A
for a known