Two central limit theorems and their application to the estimation of both parameters in the binomial
distribution
by Willis John Alberda
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OP PHILOSOPHY in Mathematics
Montana State University
© Copyright by Willis John Alberda (1964)
Abstract:
While investigating experimentally the dynamics of a host-pathogen system, the statistical problem of
estimating and finding confidence intervals for the probability of mutation from avirulence to virulence
of the pathogen was encountered. Assuming the probability of mutation is constant during any
experimental procedure, this problem may be considered as that of estimating the parameter p in the
Binomial Distribution function B(z; n,p). Because of the nature of the pathogen, however, to obtain this
estimator it was also necessary to estimate the total number of observations.
In the binomial setting this is equivalent to also estimating and finding confidence intervals for the
parameter n. With this experimental situation as background, two central limit theorems which give a
solution to these problems are proved. These theorems are proved under various assumptions which
appear feasible in the experimental situation. Other than the sampling without replacement scheme, the
techniques are distribution free in so far as no specific underlying distribution function is assumed.
These theorems, then, are an approach to estimating and finding confidence intervals for both
parameters in the Binomial Distribution.
A solution to the problem of finding the rate of convergence in each case is also given. This solution is
obtained using the same assumptions used in establishing the central limit theorems.
The techniques suggested by these two theorems and the rates of convergence for each case are then
applied to the experimental data obtained. two
Cekt Sal l im it theorems and t h e ir a pplic a t io n , to the
Estim a tion of both parameters in the
binomial d is t r ib u t io n
by
WILLIS JOHN ALBERDA
A t h e s i s su b m itte d t o th e G raduate I h c u lty i n p a r t i a l
f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e d egree
of
DOCTOR OF PHILOSOPHY
■ in .
M athem atics
A pproved:
(2 .
Chairm ad^ Exam ining Committee
a n . G ra d u a te .D iv is io n
MONTANA STATE COLLEGE
Bozeman, Montana
A ugust, 196E
A W
fi-op .3
(11)
HTA
The au th o rj, W illis John Alherda, was b o m i n Bozeman, M ontana,
Ite b rm ry 7 , 1936, t o Mr. an d M rs. P e te r W., A lh erd a o f M an h attan , M ontana.
He r e c e iv e d h i s seco n d ary e d u c a tio n a t th e M anhattan C h r is tia n High S ch o o l,
M an h attan , M ontana, f o r two y e a r s , and g ra d u a te d from W estern C h r is tia n
H igh .S chool, H u ll,' I c m .. I n 1959 he re c e iv e d a B ach elo r o f A rts deg ree i n
e d u c a tio n , w ith a mathematics m ajo r from C alv in C o lle g e, Grand R ap id s,
M ichigan. I n 1963 he r e c e iv e d a M aster o f S cien ce d eg ree i n M athem atics
from Montana S ta te C o lle g e , Bozeman, M ontana.
(ill)
ACKHOWLEDGMEM
The r e s e a r c h r e p o r te d i n t h i s t h e s i s was su p p o rte d hy th e U n ited
S ta te s Atomic B aergy Commission, D iv is io n o f B iology and M edicine P r o je c t
AT(45"1) - 1729.
The a u th o r i s v e ry g r a t e f u l f o r th e f i n a n c i a l a s s is ta n c e
re c e iv e d from t h i s agency d u rin g th e c o m p letio n o f t h i s work and f o r th e
funds made a v a il a b le f o r th e e x p e rim e n ta l i n v e s t ig a ti o n s from w hich th e
d a ta u s e d i n t h i s p a p e r was o b ta in e d .
The a u th o r w ish es t o th a n k D r. C. J . Mode f o r h is encouragem ent and
th e e x c e lle n t a d v ic e he o f f e r e d so w i l l i n g l y d u rin g th e tim e IrMfe work was
b e in g com pleted.
The a u th o r i s a ls o g r a t e f u l t o Mr. Gordon Chang who
perfo rm ed th e ex p erim en ts d e s c rib e d i n t h i s p a p e r and who c o n tr ib u te d th e
d e s c r ip ti o n s o f th e e x p e rim e n ta l p ro c e e d u re s and th e d a ta he o b ta in e d .
T hanks, t o o , a r e due D r. C. P . Q uesenberry f o r h i s a d v ic e i n s e t t i n g up
th e s tu d y program p re c e d in g t h i s w ork, a l s o f o r th e c o r r e c tio n s and
e x c e l l e n t s u g g e s tio n s he an d th e o th e r com m ittee members made t o improve
t h e f i n a l copy.
A word o f acknowledgment i s a ls o due M iss M ferilin McElwee f o r h e r
d ilig e n c e i n ty p in g th e f i n a l copy and th e a u th o r 8S w ife f o r h e r p a tie n c e
and work i n ty p in g th e f i r s t d r a f t'.
TABLE OF COM M ITS
Page
C hapter
I
INTRODUCTION
E x p e rim e n ta l S e ttin g
P r e lim in a r y R e s u lts from Sam pling T heory1
T h e o r e t i c a l .S e ttin g
B asic A ssum ptions and a h E x iste n c e Theorem
OOOWJ H
I.
II.
THE LIMITING DISTRIBUTION OF (&
» 3^ ) / ^
13
IIII.
THE LIMITING DISTRIBUTION OF (6
- SqTt)/ \[SqIt(I-Jt)
22
IV.
V.
V I.
The C e n tr a l L im it Theorem
P r o p e r tie s o f th e A pproxim ate C onfidence I n te r v a ls f o r rt
22
31
RATES OF CONVERGENCE
39
APPLICATIONS TO EXPERIMENTAL RESULTS
53
SUMMARY AND REMARKS
56
LITERATURE CONSUIffED
59
(v )
ABSTRACT
'
"While ;i n v e s t i g a t i n g e x p e rim e n ta lly th e dynamics o f a h o s t-p a th o g e n
systemj, th e s t a t i s t i c a l problem o f e s tim a tin g and fin d in g c o n fid en ce
i n t e r v a l s ,f o r th e p r o b a b i l i t y o f m u ta tio n from a v iru le n c e t o v iru le n c e o f
th e p ath o g en was e n c o u n te re d . Assuming th e p r o b a b i l i t y o f m u ta tio n i s
c o n s ta n t d u rin g any e x p e rim e n ta l p ro c e d u re / t h i s problem may be c o n sid e re d
a s t h a t o f e s tim a tin g th e p a ra m e te r p i n th e B inom ial D is tr ib u tio n fu n c tio n
B.(z; n > p ). Because o f th e n a tu r e o f th e p a th o g en , how ever, t o o b ta in t h i s
e s tim a to r i t was a l s o n e c e s s a ry t o e s tim a te th e t o t a l number o f o b se rv a ­
t i o n s . In th e b in o m ia l s e t t i n g t h i s i s e q u iv a le n t t o a l s o e s tim a tin g and
f in d in g co n fid e n ce i n t e r v a l s f o r th e p a ra m e te r n . W ith t h i s e x p e rim e n ta l
s i t u a t i o n a s : background, two c e n t r a l l i m i t theorem s w hich g iv e a s o lu tio n
t o th e s e problem s a r e p ro v e d . These theorem s a r e p roved u n d e r v a rio u s
a ssu m p tio n s w hich a p p e a r f e a s i b l e i n th e e x p e rim e n ta l s i t u a t i o n . O ther
th a n th e sam pling w ith o u t re p la ce m e n t scheme, th e te c h n iq u e s a r e d i s t r i ­
b u tio n f r e e i n so f a r a s no s p e c i f i c u n d e rly in g d i s t r i b u t i o n f u n c tio n i s
assum ed. These th e o re m s, th e n , a re an ap p ro ach t o e s tim a tin g and fin d in g
c o n fid e n ce i n t e r v a l s f o r b o th p a ra m e te rs i n th e B inom ial D is tr ib u tio n .
A s o lu tio n t o th e problem o f f in d in g th e r a t e o f convergence i n each
c ase i s a l s o g iv e n . T h is s o lu tio n i s o b ta in e d u s in g th e same assu m p tio n s
u s e d i n e s t a b l i s h i n g th e c e n t r a l l i m i t th e o re m s.
The te c h n iq u e s ~ su g g e s te d by th e s e two theorem s and th e r a t e s o f
convergence f o r e ac h c ase a r e th e n a p p lie d t o th e e x p e rim e n ta l d a ta
o b ta in e d .
T
CMPEER I
BraROMCEIOH
E x p e rim en ta l S e ttin g
^
Bae s t a t i s t i c a l problem s c o n sid e re d i n t h i s d i s s e r t a t i o n a ro s e in an
e x p e rim e n ta l i n v e s t i g a t i o n o f th e dynamics o f a " b io lo g ic a l h o s t-p a th o g e n
sy stem .
The s o lu tio n s t o th e s e problem s # n o t n e c e s s a r ily th e o n ly p o s s ib le
s o l u t i o n s } w ere o b ta in e d u n d e r c o n d itio n s w hich a p p ea re d m ost f e a s ib le i n
t h i s e x p e rim e n ta l s i t u a t i o n .
Ba t h i s e x p e rim e n ta l i n v e s t i g a t i o n t h e b a r le y m ildew fungus was
t r e a t e d w ith a m utagenic a g e n t, in o c u la te d on e ig h t r e s i s t a n t v a r i e t i e s o f
b a r l e y , and th e n s c re e n e d f o r m u ta n ts .
One o f th e m ain o b je c tiv e s o f t h i s
e x p e rim e n ta l in v e s t i g a t i o n was th e e s tim a tio n o f th e p r o b a b i l i t y o f m u ta tio n
from a v ir u le n c e t o v ir u le n c e a t e ig h t l o c i o f t h i s fu n g u s.
The p r i n c i p a l s t a t i s t i c a l problem a r i s i n g from t h i s work was t h a t
o f f in d in g a n e s tim a to r and c o n fid en ce i n t e r v a l s f o r th e p r o b a b i l i t y o f
m u ta tio n a t each lo c u s .
The d i f f i c u l t i e s i n f in d in g t h i s e s tim a to r and
c o n fid e n c e i n t e r v a l s , how ever, a r i s e from th e f a c t t h a t u n d e r o rd in a ry
c o n d itio n s th e fungus does n o t grow on th e r e s i s t a n t v a r i e t i e s o f b a r le y .
M oreover, when a p u s tu le a p p e a rs on a r e s i s t a n t v a r i e t y i t i s c o n sid e re d
a m u ta n t.
The e s tim a to r o f th e p r o b a b i l i t y o f m u ta tio n a t e a c h lo c u s o f
t h i s fu n g u s, q u ite n a t u r a l l y , w i l l be a r a t i o o f th e t o t a l number o f mu­
t a n t s o b serv ed on some s p e c i f i c r e s i s t a n t v a r i e t y o f b a r le y t o th e t o t a l
number o f sp o re s g e rm in a te d on t h i s v a r i e t y o f b a r le y .
T h e re fo re , s in c e
i t i s im p o ssib le t o co u n t th e t o t a l number o f sp o re s g e rm in a te d on a s e t .
o f r e s i s t a n t v a r i e t i e s , i t i s n e c e s s a ry t o e s tim a te t h i s t o t a l number o f
2
s p o re s g e rm in a te d f o r each r e s i s t a n t v a r i e t y .
The s t a t i s t i c a l problem s a r e
th e n t o f in d e s tim a to r s and c o n fid en ce i n t e r v a l s f o r th e p r o b a b i l i t y o f
m u ta tio n a t each lo c u s and f o r th e t o t a l number o f sp o re s g erm in ated on
e ac h r e s i s t a n t v a r i e t y o f b a r le y .
The e x p e rim e n ta l procedure, f o r e s tim a tin g th e t o t a l number o f sp o re s
g erm in ated pn a s e t o f r e s i s t a n t v a r i e t i e s may be d e s c rib e d a s fo llo w s .
Whenever a s e t o f p o t s ' o f r e s i s t a n t v a r i e t i e s o f b a r le y was in o c u la te d
■
w ith m ildew s p o re s , a s e t o f p o ts o f s e e d lin g s o f a s u s c e p tib le v a r ie ty
was a l s o in o c u la te d .
L et
r e p r e s e n t th e t o t a l number o f sp o re s germ i­
n a te d on l e a f i o f a r e s i s t a n t v a r i e t y o f b a r le y .
X^, Xg, .
.
X^ i s
th e n th e p o p u la tio n o f th e number o f sp o re s g e m in a te d on H le a v e s o f th e
r e s i s t a n t v a r i e t y o f b a r l e y , where H i s a known p o s i t i v e i n t e g e r .
The
s p o re s g e rm in a te d on e ac h r e s i s t a n t v a r i e t y o f b a r le y c o u ld be re p re s e n te d
i n t h i s way.
As soon a s p u s tu le s on th e s u s c e p tib le v a r ie ty , w ere v i s i b l e ,
a count o f th e number o f p u s tu le s on n le a v e s was made.
L e t x..represent
th e 'n u m b e r o f p u s tu le s co u n ted on l e a f i o f th e s u s c e p tib le v a r i e t y .
sample x ^ , x_,
x
n
The
i s th e number o f p u s tu le s which a p p e a re d and were
,
co u n ted on n le a v e s o f th e s u s c e p tib le v a r i e t y . o f b a r le y , where n i s a ls o
known and n < Ef.
The numbers x ^ , x ^ , .
X^ w i l l be, c o n sid e re d a random
sam ple o f s iz e n < U ta k e n w ith o u t re p la c e m e n t from a p o p u la tio n i d e n t i c a l
t o th e f i n i t e p o p u la tio n X., Xg,
X_ of s i z e M. I f x i s th e av erag e
number o f p u s tu le s p e r l e a f on th e s u s c e p tib le le a v e s , th e n Bx was u se d
a s a n e s tim a te o f th e t o t a l number o f s p o re s g erm in ated on th e r e s i s t a n t
v a r i e t y o f b a r le y o r th e t o t a l number o f o b s e r v a tio n s .
3
T his p ro c e d u re c o u ld be re p e a te d i n d e f i n i t e l y , say up t o q. s ta g e s , so
t h a t i f E x . r e p r e s e n ts th e e s tim a te o f th e t o t a l number o f o b s e rv a tio n s
a t th e K -th s ta g e , th e n
I
■
V
'
k=l
p ro v id e d am e s tim a te o f th e t o t a l number o f o b s e rv a tio n s o v e r q, s ta g e # ,
If z
r e p r e s e n ts th e t o t a l number o f m u tan t p u s tu le s o b serv ed over
q s ta g e s , th e n
,
^
\
sa
p ro v id e d a n e s tim a te o f th e p r o b a b i l i t y o f m u ta tio n .
The t h e o r e t i c a l problem c o n s is ts o f f in d in g co n fid e n ce i n t e r v a l s f o r
th e t o t a l number o f o b s e r v a tio n s , S^, and th e p r o b a b i l i t y o f m u ta tio n , st.
Hhving c o n sid e re d th e e x p e rim e n ta l s i t u a t i o n i n t h i s s e t t i n g , how ever,
b e fo re p ro c e e d in g t o th e d is c u s s io n o f th e t h e o r e t i c a l problem s i t i s
■
' '
a p p r o p r ia te a t t h i s p o in t t o m en tio n s e v e r a l . r e s u l t s from sam p lin g w ith o u t
re p la c e m e n t th e o ry a s found i n W ilks ( 1962) .
These r e s u l t s and r e l a t i o n s
w i l l s e rv e t o m o tiv a te c e r t a i n c o n ce p ts and symbols u sed i n th e d is c u s s io n
o f th e t h e o r e t i c a l p ro b le m s.
P re lim in a ry . R e s u lts Fpom Sam pling Theory
F o r k = I , 2 , 3 , . . . , where k d e s ig n a te s th e k - t h s ta g e i n t h i s
e x p e rim e n ta l p ro c e d u re , l e t Xjf1, Xfeip,
s iz e K. w ith mean
2
an d v a ria n c e 0^ , where
be a f i n i t e p o p u la tio n o f
k
%
I
Pfc
1=1
and.
JR.
k “ E
L e t Xjc l, Xjc2,
Xj_
Le a sam ple o f s iz e n^. ta k e n -without rep lacem en t
from t h i s p o p u la tio n w ith mean Xjc and v a ria n c e
*k m
where
'
JU l ■
and
k
k ~ rr^ l
=k"
X
i= l
W ith th e s e d e f i n i t i o n s , i t i s shown i n th e re fe re n c e o f W ilks c i t e d above
th a t
E (% ) - Mk'
-
fi .
( I - 1 .) 4
\
Y
and
E (^?)
"k7 =
“ a?k
Ifom th e s e r e s u l t s i t fo llo w s im m ed iately t h a t
k f k ' s s Ek 5
E
'k
where
S-
=
Xki = ^ k '
1=1
-
5
and
Var (H1Xk ) = * ^ ( i
-
^
.
A lso i t i s obvious t h a t E(S ) = S , where
9.
Q.
' k»l
and
9.
sa=Z
iV t -
k=l
2
an d t h a t E (s^ ) = s ^ , where
q,
^ _
I -
ZLj " k(iX
Z
k=l
^
&2
0V ;
" iK T5 k
K
and
sX
Z^<1-k*°x■
i "
^
V
F u rth erm o re , i f th e a ssu m p tio n o f independence betw een s t a g e s 'i s
im posed, th e n th e fo llo w in g theorem o f some im portance i n th e s e q u e l i s
a ls o tr u e .
Theorem I . IS
I f s*T, 's , and S a re d e fin e d a s b e fo re and th e E x . form
q.
<1
q
KK
a sequence o f in d e p en d e n t random v a r ia b le s ^ th e n
V ar(S a ) = E ($q - Sq ) 2 = s | .
P r o o f:
S in ce E (§ ) = S ,
9,
9.
V ar(Sa ) = E (^ a - Sq) 2 .
6
E(S1qi-Sq ) 2 = E
5V 2V
k=l
X
2 j) "by th e assu m p tio n o f in d ep en d en ce,
ks=l
g.
W ith th e s e p re lim in a ry r e s u l t s , th e problem s a s s t a t e d in th e f i r s t
s e c tio n can now he d e s c rib e d more p r e c is e ly i n m a th e m atica l te rm s .
T h e o r e tic a l S e ttin g
I n o rd e r t o f in d co n fid e n ce i n t e r v a l s f o r th e t o t a l number o f o b s e r­
v a tio n s , S , and f o r th e p r o b a b i l i t y o f m u ta tio n , it, i t i s n e c e s sa ry t o
d e term in e th e d i s t r i b u t i o n f u n c tio n s o f th e ' random v a r ia b le s u se d to
e s tim a te th e s e p a ra m e te rs .
I n many c a s e s , a s f o r example i n t h i s s i t u ­
a t i o n , th e e x a c t d i s t r i b u t i o n .f u n c t i o n s may be d i f f i c u l t t o d eterm ine
and n o t am enable t o t a b u l a t i o n .
When th e s e s i t u a t i o n s a r i s e , a s o lu tio n
can be o b ta in e d by d e te rm in in g a lim i t i n g d i s t r i b u t i o n which i s amenable
to ta b u la tio n .
A pproxim ate c o n fid en ce i n t e r v a l s can th e n be c o n s tru c te d
from t h i s lim i t i n g d i s t r i b u t i o n .
T his i s th e ap p ro ach u sed t o o b ta in a
s o lu tio n t o th e problem s p re s e n te d h e re .
Under f a i r l y g e n e r a l assu m p tio n s i t can be shown t h a t th e lim it in g
7
d i s t r i b u t i o n o f th e random, v a r ia b le
- S
q.
.8
g.
i s norm al w ith mean z e ro and v a ria n c e one.
S in ce in many c a s e s i t i s
p o s s ib le t o s u b s t i t u t e th e e s tim a to r o f th e v a ria n c e and s t i l l o b ta in th e
same lim i t i n g d i s t r i b u t i o n , th e problem o f f in d in g c o n fid en ce i n t e r v a l s
f o r S , th e t o t a l number o f o b s e r v a tio n s , a p p e a rs q u ite l o g i c a l l y - t o be
t h a t o f d e te rm in in g th e li m i t i n g d i s t r i b u t i o n o f th e random v a r ia b le
S
I
S in ce s
i s assum ed unknown, some e s tim a to r m ust be s u b s t i t u t e d .
.
c h o ic e ,o f 5
9.
The
2
seems n a t u r a l from th e u n b ia se d n e ss o f £ and w i l l in d eed
9
p la n a n im p o rta n t r o l e i n th e s o lu tio n o f th e problem .
F u rth erm o re , u n d e r f a i r l y g e n e r a l assu m p tio n s th e li m i t i n g d i s t r i ­
b u tio n o f th e random v a r ia b le
/ S qJt(I-Jt) .
i s norm al w ith me,an z e ro and v a ria n c e one.
S in ce S^ i s an u n b ia se d e s
tim a to r o f S , th e n a t u r a l e x te n s io n i n t h i s ease i s , i f p o s s ib le , to
<1
d eterm in e th e lim i t i n g d i s t r i b u t i o n o f th e random v a r ia b le
Zq - V
V SqJl(I-It)
8
From t h i s i t i s p o s s ib le t o c o n s tr u c t ap p ro x im ate c o n fid en ce i n t e r v a l s
f o r JC3 th e p r o b a b i l i t y o f m u ta tio n .
Once th e lim i t i n g d i s t r i b u t i o n o f th e s e two random v a r ia b le s have
b een e s ta b lis h e d and a r e am enable to t a b u l a t i o n ^ i t i s p o s s ib le f o r a
f ix e d q t o c o n s tr u c t approxim ate c o n fid en ce i n t e r v a l s f o r
th e t o t a l
number o f o b s e r v a tio n s , and Jt3 th e p r o b a b i l i t y o f m u ta tio n , r e s p e c tiv e ly .
The f i n a l problem th e n i s t o f in d a r a t e o f convergence t o th e lim it in g
d i s t r i b u t i o n i n e ac h c a s e .
B efore p ro c e e d in g t o th e s o lu tio n o f th e s e problem s i t i s n e c e s sa ry
t o make some a ssu m p tio n s u n d e r w hich i t i s p o s s ib le t o so lv e th e problem s
s ta te d .a b o v e and t o e x h i b i t th e e x is te n c e o f a p r o b a b ility space on which
th e random v a r ia b le s w ith th e s e imposed c o n d itio n s a re d e fin e d .
The e x i s ­
te n c e o f t h i s p r o b a b i l i t y space i s n e c e s s a ry i n o rd e r t o g iv e meaning t o
p h ra s e s o r symbols such a s a lm o st s u re convergence (8^ " ) o f sequences o f
random v a r i a b l e s , convergence i n p r o b a b i l i t y (5 ) o f sequences o f random
v a r i a b l e s , convergence i n d i s t r i b u t i o n
a s th e y a p p e a r i n th e s e q u e l.
(X) ), and e v e n ts
(The symbol
means
th e s e t o f Wj such t h a t th e :random v a r ia b le s a t i s f i e s th e c o n d itio n s t a t e d .
(
a s f o r exam ple, Xjfl(ur1) < x . )
B asic A ssum ptions and an E x iste n c e Theorem
The problem s a s d e s c rib e d i n th e p re v io u s s e c tio n a r e so lv e d u n d er
th e fo llo w in g a ssu m p tio n s.
'l .
F o r q ss I , 2 , J3 . . . , t h e p o p u la tio n (X . , Xq^ , . . . , Xqjj ) i s
a f i n i t e p o p u la tio n o f d i s t i n c t , p o s i t i v e in te g e r s w ith ' mean (aq and
9
v a ria n c e a , b o th unknown.
q. e I , 2# 3# .
2.
2
The d i s t i n c t n e s s im p lie s t h a t f o r
a2
> d^ > 0 and a l s o . t h e sample v a ria n c e cr^ > dg > 0.
F o r q. « I , 2 , 3> *
and i = I , 2 , .
G < m < Xg^ < M < <»,
w ith M-m > 0 .
3«
The Ng and Hg a r e known, p o s i t i v e , in te g e r s such t h a t f o r
q. = I , 2 , 3 ,
k.
2 < N < N < oo and 2 < n < N - I .
q —
— <3. — q •
NgXg, q ri I , 2 , 3 , . . . , i s a sequence o f in d ep en d en t n o t n e c e s ­
s a r i l y i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s .
N
5.
F o r q - I , 2,- 3,.
)
y ^ , th e t o t a l number, o f m u ta n ts
k=l
a t s ta g e q , and NgXg, th e e s tim a to r o f th e t o t a l number o f o b s e rv a tio n s
a t s ta g e q , a r e in d ep en d en t random v a r i a b l e s .
th e t o t a l number o f o b s e rv a tio n s a t s ta g e q .
Sw a s d e fin e d b e fo re i s
3V
The random v a r ia b le s
d e fin e d by
yk = I
a 0
w ith P 3rI f 1
i f m u ta tio n o c c u rs,
o th e rw is e ,
= rt, f o r a l l k , a re in d e p e n d e n t.
The fo llo w in g rem arks and theorem e x h i b i t th e e x is te n c e o f a p ro b ­
a b i l i t y space on w hich th e random v a r ia b le s s a t i s f y i n g th e s e assum ptions
a r e d e fin e d .
F or each q , th e p r o b a b i l i t y f u n c tio n o f NgXg and th e ra n g e , 1 ^ ,
o f t h i s random v a r ia b le a r e d eterm in ed by th e sam pling w ith o u t r e p la c e ­
ment scheme.
For q = I , 2 , 3 , . . - , s e t
10
B lg ( ^ lg ) ,
iq.J
where
I s some elem en t o f
, some f i n i t e s e t o f r e a l num bers»
As s t a t e d , p ^ ( i ^ ) f o r a l l i ^ B I 1
i s d e term in e d by th e sam pling w ith o u t
re p la ce m e n t p r o b a b i l i t y f u n c tio n , b u t f o r u se i n th e s e q u e l need n o t be
c a lc u la te d .
S im ila r ly , f o r each q , by th e d e f i n i t i o n o f th e
yk i n a ssu m p tio n 5 , th e p r o b a b ility , f u n c tio n o f z_
and th e
i s g iv e n by
N
m ^ ( 1-a )
. 'B q =
where
i s an elem ent o f
s P2q ^ 2q ^
, th e ran g e sp ace o f Z ^ ,, w hich i s some
9.
f i n i t e s e t o f in te g e r s 0 , I , 2 , . . . , 8_
.
\
.
If
Zjj )> th e n th e ran g e sp ace o f th e v e c to r , random
v a r ia b le X i s some p ro d u c t space I = I 1 X I 0 n , and by a ssu m p tio n 5 ,
9.
9,
-L9. • ^9.
th e p r o b a b i l i t y f u n c tio n o f th e v e c to r random v a r ia b le X i s
q
p
where i
q
“ T l q ^ l q ) p2q ^ 2 q) " Pq ^ V "
= ( i , , i« ) i s a n elem en t o f I .
v Iq
2qz
q
W it^ th e s e rem arks i t i s now
p o s s ib le t o prove th e fo llo w in g .
Theorem 1 .2 :
(E x iste n c e Theorem)
There e x i s t s a p r o b a b i l i t y space
(-/I, <^, p ) w ith th e sequence o f in d ep en d en t v e c to r random v a r ia b le s X^
d e fin e d on i t and such t h a t f o r each q th e two elem en ts o f X^ a re a ls o
in d e p e n d e n t.
P ro o f:
In o rd e r t o prove t h i s theorem , i t i s s u f f i c i e n t t o c h a r a c te r iz e
11
th e space -A-,. d eterm in e th e e - f i e l d
th e e - f i e l d ^
,
, and d e fin e a s e t f u n c tio n P on
Toward t h i s end th e n , l e t
o f th e form ( i ^ , i g , ± y
. . . ) where i^ E 3^,.
he th e s e t o f a l l sequences
D efine a c y lin d e r
C (i^ , i g , . • •, i j j ) . f o r some f ix e d N t o he th e s e t o f a l l sequences in -J'L
such t h a t th e f ix e d e lem en ts i ^ ,
, . . . , i^
a p p e a r i n th e f i r s t H p la c e s
V arying W o v e r a l l p o s i t i v e in te g e r s and i ^ o v e r a l l e lem en ts o f I fe.,
y ie ld s a c la s s ^
o f a l l such c y lin d e r s c ( i ^ , i g ,
/P
ta k e n t o he th e m inim al c r-fie ld o v er th e c la s s
.
f u n c tio n P on th e c la s s ^
a s fo llo w s .
i^ ).
Then ^
is
Uow d e fin e th e s e t
F o r each C (i^ , i g , . . . , i^ )
a s s ig n th e p r o b a b i l i t y
p s C^i
U
^ ig f . . o , i^ ) ,= n p^(i_j_5 .
The p r o b a b i l i t i e s d e fin e d i n t h i s way a r e c o n s is te n t.
H ence, by v a rio u s
th e o re m s, f o r example Theorem A, P g . 137 Loe5ve ( i9 6 0 ) , th e s e t fu n c tio n
on th e c l a s s ^
may be e x ten d e d t o a p r o b a b i l i t y m easure on ^
i t i s p o s s ib le t o c o n s tr u c t a p r o b a b i l i t y sp ace (-T b ,
th e random v a r ia b le s
p ).
.
Hence,
To d e fin e
on t h i s p r o b a b i l i t y space p ro c e ed a s f o llo w s .
US' i s th e sequence ( i ^ ) , th e n s e t
(o r ) = i^. f o r q = I , 2 , 3# «• • •
If
Di
t h i s m anner th e v e c to r random v a r ia b le s X a r e d e fin e d on t h i s p r o b a b ility
I
space and th e a ssig n m en t o f th e s e t
f u n c tio n P makes them in d ep en d e n t.
The a ssig n m e n t o f p
makes th e elem en ts o f th e v e c to r in d e p e n d e n t. These
9a ssig n m e n ts d e term in e a l l d i s t r i b u t i o n f u n c tio n s on t h i s sp ace and th e
theorem i s p ro v ed .
From, t h i s theorem th e n , s in c e
and
a r e sums o f in d ep en d en t
12
random v a r ia b le s d e fin e d on t h i s sp a c e , th e y to o a r e d e fin e d on t h i s
p r o b a b i l i t y sp ace-
T h e ir j o i n t d i s t r i b u t i o n f u n c tio n i s d eterm in ed by
th e sam pling w ith o u t re p la ce m e n t scheme and th e b in o m ia l d i s t r i b u t i o n o f
th e yk I n th e s e q u e l a l l e v e n ts
X < x
w i l l be elem en ts o f th e Of- f i e l d
and a . s . convergence and convergence i n p r o b a b i l i t y o f seq u en ces o f
random v a r ia b le s w i l l be convergence w ith r e s p e c t to t h i s p r o b a b ility
space.
Boundedness and d iv e rg e n c e o f sequences o f random v a r i a b l e s , though
n o t d e s ig n a te d a s such i n v a rio u s p la c e s i n th e c o n te x t, i s a . s .
boundedness and a . s . d iv e rg e n c e w ith r e s p e c t t o t h i s p r o b a b i l i t y sp ac e .
I n th e s e e a s e s , how ever, th e s e t s o f unboudedness and o f convergence a re
empty s e t s .
W ith t h i s in tr o d u c tio n and th e s e p re lim in a ry theorem s i t i s p o s s ib le
t o p ro c e ed w ith th e s o lu tio n o f th e problem s p re s e n te d i n th e p re v io u s
s e c tio n .
CHAPTER I I
THE LHHTmG DISTRIBUTION OF ( £ * S V s ,
q.
.<£ g,
U sing th e a ssu m p tio n s made in th e I n tr o d u c tio n and th e n o ta tio n
found t h e r e , th e p ro c e d u re u sed i n f in d in g th e lim i t i n g d i s t r i b u t i o n o f
th e random v a r ia b le (S ^ -S ^ )/s ^ i s th e fo llo w in g ':
f i r s t e s ta b lis h th a t
n ( o ,i) .
second show t h a t
A
S -S
P].:;'9--
A
S_ -S_
A
-* 0.
S
These two r e s u l t s a r e th e n s u f f i c i e n t t o prOve t h a t
W(OA)j
from w hich ap proxim ate c o n fid e n ce i n t e r v a l s f o r
The f i r s t s ta te m e n t i s shown by th e fo llo w in g *
Theorem 2 .1 :
th e n
If
I"
w~
<y
V ^ r 1
k=l ■
can be c o n s tru c te d .
14
and
P ro o f:
O bviously
by th e d e f i n i t i o n o f th e Y , .
To prove t h i s theorem i t i s p o s s ib le t o
u s e th e "Bounded Case" Theorem, Pg. 277 L oelve ( i 960) .
I t i s th e n
n e c e s s a ry t o v e r i f y t h a t
(I)
IYgjJ i s u n ifo rm ly bounded f o r a l l k ,
(ii)
s^-s-co a s q_ -> oo
By th e assu m p tio n o f independence o f th e
th e
d e n t random v a r i a b l e s , and s in c e E (x^) = ^
a r e a l s o indepen
th e Y ^ a re c e n te re d a t
e x p e c ta tio n s so t h a t th e o th e r c o n d itio n s n e c e s s a ry t o employ th e
"Bounded Case" Theorem a r e s a t i s f i e d .
(i)
^k^k^^c^k^
S in ce s^
> 0 and
^ ou:n^e ^ £°r a l i k .
I
i s "bounded f o r a l l k i f
But
Ifk I ^ N tk I S Hk (M-m) by a ssu m p tio n 2,
< H(M-m) by a ssu m p tio n 3 }
h e n c e , IYg k I i s u n ifo rm ly bounded f o r a l l k .
15
(ii)
<3.
V
an d s in c e by a ssu m p tio n I
2< \ <
\Z
Ifc l
p
^
^
^
K
> d^> 0 f o r a l l k and by assu m p tio n 3
\< ^
i " i ) cfS / 0
as k
co.
^
Hence,
,a
a?
Z
k=l
a s q -* co and th e r e f o r e
The
%
i
oo
’
-> co a s q -%o and th e theorem i s p ro v e d .
r e l a t i o n s ' a re h e c e s s m y t o p r o te the. n e x t theorem and
a r e a l s o r e q u ir e d itia l a t e r - theorem s', h e g te ., a te g iv en here, a s a lemma.
Lemma 2 ,1 :
o
V
i f (i)
> '
2
and
A2
a r e d e fin e d a s b e f o r e .
1
(ii)
th e n
>2
s^,
^
p
> d^ > 0 and
^p
> dg > 0 (assu a rp tio n I ) ,
(iii)
2 <
< Sfk - I (a ssu m p tio n 3 ) ,
(I)
G < I 1 < Gk < 2(M-m) 2 f o r
a ll k,
(2)
0 C d2 < Gk < 2(M~m) 2 f o r
a ll k,
i6
P ro o f:
(I)
(3 )
O<
<E
qJS2 (M
-In)2,
(4 )
O< qd2 < £2 < Eql2 (M
-Hi)2.
'2
% (ii) ^
" I 2j
E i=i
> 0 and
■ "k!2 -
§ ~ r"i ^
(H-m) 2 ky a ssu m p tio n 2 ,
it i
%
s in c e J^y a ssu m p tio n 3 > 2 < JS^«
(2)
(ii)
0Jc >
dg > 0 and "by th e same re a s o n in g a s .in ( I ) ,
1=1 :
K
s in c e hy a ssu m p tio n 3 a l s o , 2 < n. .
q
o)
s:
IT
' s? > a.I ^
k k
I 4 & Iinfe $
1k I )
kU
> qd^, s in c e "by ( i l l ) and a ssu m p tio n 3
Uy ( I ) j
.1
I
>
17
A lso
s4
“ iNk
k=l
by (2 ),
' K
< 2 qjj
s in c e by ( i l l ) and assu m p tio n 3
5 *
(4)
R ep lac in g *k by ^k and d^ by dg i n th e p ro o f o f ( 3 ) , i t
fo llo w s im m ediately t h a t
.
< 2f /( M - m ) 2 ,
and th e lemma i s p ro v ed .
These r e l a t i o n s a r e u s e f u l i n th e n e x t theorem w hich i n t u r n g iv e s
th e key t o e s t a b l i s h i n g th e second s ta te m e n t i n th e in tr o d u c tio n o f t h i s
c h a p te r.
Theorem 2.2S
I f s 2 a n d zI 2 a r e d e fin e d a s b e f o r e , th e n
9.
QL
S2
- zS 2
- 3 -------&
q.
P ro o f:
o.
Wsing Kdlmogorov.U Convergence C r i t e r i o n , Loe-*ye ( i9 6 0 ) ; P g . 238,
on th e random v a r ia b le s
- ^ 5 '
i t m ust be shown t h a t
(I)
E(Tk )
(ii)
th e Yk a r e in d e p e n d e n t,
(H i)
y
k. 1
0 w hich th e n a l s o p ro v es t h a t t h e
v a r^
k2
<».
a re .^ n te g ra b le ,
18
(i)
E(Yfe)
O s in c e E(^k ) ■= ^
.
(ii)
The Yk a r e in d e p en d e n t by a ssu m p tio n 4 .
f •■
(in )
l
Y
t= l
k
E^ >
b=l
k2
" . * , 1 _ i ,2
%
V
K=I
k
<f <
4
k = l.
'\
Y i A
k=l
(Z *
' j )
k2
E ^ A
= )2
- < £ ),
) 2
k
CO
by Lemma 2 ,1 ,
< 00
ksi
s in c e
y
i
< oo and W and M-m a r e f i n i t e .
T h is p ro v es th e theorem .
ltl *
T h is r e s u l t g iv e s th e key t o p ro v in g th e fo llo w in g
Theorem 2 .3 :
I f S . zS , s
<r
r
I
and s
A
sa ■ s I
q.
a r e d e fin e d a s b e f o r e , th e n
S - S
z<
S
q.
L,
0
19
P ro o f:
Py M arkov1S I n e q u a lity
" 8 CL1 X
1 - e] - E|^
"I
' sJ
H|<^m
pI X
I?
f o r a l l e > O and by 'Schwarz8s I n e q u a lity
i
I l|
1
V
b
*
-
I
,
- s )
2
*
s(i - I
Sq.i
But E(S^ - S g)^ s -S g by Theorem 1 .1 and
S_Sj
q.- q
s ,
■q.
- ySl
M oreover,
A
-
;)
= E
4
■
< E
*
.
;
i&zAu&iZir
q.
- S^
l \ ~ Sd
- q l n Er s q “ s ^l ^
Zemna 2 .1 .
ySr
)2
20
Shus
ECS1n
But Is2 .
1q
sa I
+ s2 < knFq(M-m) 2
by Lemma 2 ,1 so t h a t
,2 _ g.
r2
q ."
^ 2 ,-
< ^ l 2 (M-Hi) 2 <
S2
H ence, th e random v a r ia b le s — - — -
a re ln te g ra b le .
A lso by Theorem
2 .2 t h i s sequence o f random v a r ia b le s converges alm o st s u r e ly t o 0,
h e n c e , th e sequence converges i n p r o b a b i l i t y t o 0 and by th e Dominated
Convergence Theorem, Loe'v e ( i 960) j Pg= 152,
E
•* 0
a s q -» cb and th e theorem i s p ro v ed .
In o rd e r t o prove th e f i n a l theorem o f t h i s c h a p te r, i t i s n e c e ssa ry
a t t h i s p o in t t o employ a theorem o f Loe5ve ( i 960) J Pg= 1 6 8 .
The fo llo w in g theorem s t a t e d i n an e q u iv a le n t manner a s t h a t in
Loe*ve
i s g iv e n w ith o u t p r o o f f o r th e g e n e r a l c a s e .
Theorem 2 . 4 :
th e n
If
CD
<X c y
■»
4*)
x„ -
B 0,
,
21
Note':
If X
B
- Y a.s<
B
0 , th e theorem a l s o h o ld s «
She C e n tra l L im it Sheprem o f i n t e r e s t i n t h i s c h a p te r th e n fo llo w s
im m ed iately a s shown by
Theorem. 2 ,5 •
If
and yS^ a re d e fin e d a s b e f o r e , th e n
) -» H (0 ,1 ).
P ro o f:
Theorem 2 .1
A
$1( 0, 1):
and by Theorem 2 .5
0,
s_
th u s by a p p ly in g Theorem 2 .4 th e r e s u l t f o llo w s .
From t h i s theorem th e n i t fo llo w s t h a t
Iim P
A
8 - 8
I % . 9| <
_ i
/
zX
- t /2
dt
fo r z„ > 0
V&c '
so t h a t f o r q " s u f f i c i e n t l y ” la r g e a n ap p ro x im ate c o n fid en ce i n t e r v a l f o r
Sri i s g iv e n by
A
S
g.
%1 -
CHAPTER I I I
THE LIMITING DISTRIBUTION OF (z ^ - SqIt ) / ^ qTt(I-Tt)
.
The C e n tra l L im it Theorem
In e s t a b l i s h i n g th e l i m i t i n g d i s t r i b u t i o n o f (Zq^-SqTt)/ VSqTt(I-Tt)?
i t i s n e c e s s a ry t o employ m ethods d i f f e r i n g from th o s e i n C h ap ter I I .
A f te r h a v in g shown t h a t
f \
~ V
N(Oj I ) ,
V sgIt(I-It)
a tte m p ts t o show t h a t
z - S it
z - S it
J l ____ g_ _ a
q.
V S ^ it(I-It)
Q.
fa ile d .
The th eo rem , how ever, i s p ro v ed by showing t h a t f o r a l l z 6 R, R th e s e t
o f r e a l num bers.
rjjf
<
^ S qTt(I-Tt)
r z - S it
- P _a„ _...q
"I
W ^ it( I - I t)
j
i
*?,
-
I n o rd e r t o j u s t i f y th e use, o f th e p i v o t a l q u a n tity
V
A
i V
/ ^ taIf(I “It)
a s th e e x p re s s io n from w h ic h ' t o work f o r co n fid e n ce i n t e r v a l s f o r th e
p r o b a b i l i t y o f m u ta tio n it > 0 , which i s assumed t o be c o n s ta n t o v er a l l
s ta g e s , c o n s id e r th e fo llo w in g .
I f a s in, assu m p tio n 5 th e y^ a r e d e fin e d
23
- I i f m u ta tio n o c cu rs
y \ s 0 o th e rw ise
where P
ft? th e n th e p r o h a h i l i t y f u n c tio n o f y^ i s
h Cyi ) =
I f , a s b e f o r e , zq = y
y i / , ^ 1 -y I
(l-sr)
y^ i s th e to ta l'n u m b e r o f m u ta tio n s o v e r q, s ta g e s
i= l
where th e y^ a r e in d e p en d e n t in d ic a to r s a s d e fin e d ab o v e, th e n th e
fo llo w in g two theorem s a r e w e ll known and a r e g iv e n w ith o u t p ro o f.
H o te :
z q. -
and th e random v a r ia b le
a r e n o n -n e g a tiv e in te g e r s and
\ °
Theorem 3 .1 :
I f th e y^*s a r e a s d e fin e d above and
3t = P Jfi = I
S
(U )
Zi - I y t ,
■Iril
th e n th e p r o b a b i l i t y f u n c tio n o f z^ i s
Theorem 3 - 2 :
I f th e p r o b a b i l i t y f u n c tio n 6f z^ i s h(z_) d e fin e d above,
th e n th e maximum lik e lih o o d e s tim a to r o f # i s
= z ^ /s ^ and
24
(i)
E(k ^) S' if^
(U )
V a r(^ )= S iia I
Q,
Uow s in c e
\
j
it( i- it)
- V
S ^jt(I-If)
i t i s p o s s ib le t o f in d co n fid e n ce i n t e r v a l s f o r it i f th e l i m i t i n g d i s t r i A '
"bution o f th e random v a r ia b le formed by r e p la c in g
by
can be shown
t o be U ( 0 ,1 ) o
B efore p ro c e e d in g t o th e th eo rem s, th e fo llo w in g lemma w i l l be
r e q u ir e d i n th e p ro o fs o f th e s e th eo rem s. (i)
Lemma 3«1:
I f S„ and S a r e d e fin e d a s b e f o r e , th e n
q.
q
(i)
S_
(ii)
^
I
oo a s q -> co ,
-» oo a s q -4- co ,
A
(iii)
P ro o f:
(i)
S = ) S- > )
R m > q2m. -* <» a s q
q Z—i Uj,
— /—j
%
k=l K
kssl
A
(ii)
(in )
S im ila r ly
I hA
k=i
__q
3i
>
I V k
Jtei
f
~ 2_,
k=l
» by a ssu m p tio n s 2 and 3«
E m > q2m -» oo a s q
K —
—
«
ksl
V
= J
< oo by assu m p tio n 2 ,
25
a ls o ,
I
_a>
Sq "
I
Nkm
%
= M >
°"
Thus,
0
*
These r e s u l t s a r e u s e f u l i n p ro v in g th e fo llo w in g C e n tra l l im it
theorem .
Theorem 3 -3 :
I f z . S , ' Sw , and z
a r e d e fin e d a s b e fo re and
IT1,
q.'
IT1
X "-V
2CLk =
■ f SqJt(I-Jt)
th e n
z - S it
S . ■ - a ..... SL.
qk y SnI t ( I i t )
and
P ro o f:
O bviousIy
by th e d e f i n i t i o n o f Z ^ .
A lso E (z^ ) = S
Zgk a r e c e n te re d a t e x p e c ta tio n s .
it, h en ce, th e random v a r ia b le s
Again u s in g th e "Bounded Case" Theorem,
L oe0Ve ( i9 6 0 ) ; P g . 277* i t i s a p p a re n t from th e d e f i n i t i o n o f th e z,
K
t h a t th e
a r e in d ep en d e n t random v a r ia b le s and s in c e I zjj
N(M) < 00* th e Z , a r e u n ifo rm ly bounded.
qK
%r Lemma 3*I
"
jt|
S a (l-a )
9. ■
<
<»
a s q. ■* 00 so a l l c o n d itio n s o f th e "Bounded Case" Theorem a r e s a t i s f i e d
am i th e theorem i s p ro v ed .
Now i n o rd e r t o d e term in e th e li m i t i n g d i s t r i b u t i o n o f th e random
A
by S i t i s n e ce ssa ry - t o in tro d u c e a
I
4.
random v a r ia b le w hich can be u t i l i z e d b ecau se o f ( i i i ) Lemma 3 .1 .
'% / S
A
D e f in itio n 3»1: Let. "0^ =
w h e r e Sg.* M and m a r e d e fin e d
v a r ia b le form ed by r e p la c in g S
as b e fo re .
T his random v a r ia b le in tro d u c e d f o r th e sake o f convenience o f
n o ta tio n g iv e s th e key t o th e p ro o f o f th e p r i n c i p a l c e n t r a l l i m i t
theorem o f t h i s c h a p te r.
O < (m/ia) 2 <
0
< I.
Theorem 3 . 4 :
If 0
9.
The fo llo w in g i s a l s o a tr u e th eo rem .
i s a s d e fin e d ab o v e, th e n
(i)
E (0g ) = m/M,
(ii)
0.
B «S •
From Lemma 3*1 i t i s obvious t h a t
m/M.
P ro o f
mE(S )
(i)
(ii)
E(Sa ) = B
9
n /m
%
By T ch eb ich ev 1S I n e q u a lity
= m,/ m.
But E (eq - xq/ m) 2 = Ee2 - (m/M)2 = (m/M)2 - g E(S2) - (m/M)2 = (m/M)2s ^ / s 2
s in c e by Theorem 1 ,1
E(&2 ) = s 2 + S2 .
I
g.
q.
Thus by Lemmas 2 .1 and 3 .1
E(.e
- m/M)2 < (m/M)2 —
^
qkwT
th e r e f o r e K /q -» 0 a s q -* «> and th u s 8_ 5 m/M.
0 < (m/M)2 < 0
1V
< I,
e
- 6ql - ^ l - f
1
" 1I <
q
=v
iZ j . ^
ksq+ l
M oreover s in c e
q+r
i|
q
S ^
|_SL2ȣ _ -Li
^ /M S 4
1 |-
W
q+r
How l e t $
e q+r
E2 (M^m) 2
s ^- where K
q
and S s y
Hk^ k th e n
““ V "
lte:q+l '
i v ■\i
< i sA z i i
5i l <
sA
,V r
- ^ 5P
V3A
+ s r>
s in c e
< Sr ,
A
I
=| -> 0 a s q.r> <x>}
s in c e S and ^ a r e bounded a a d S and S d iv e rg e a s q -> «> "by Lemma 3 .1 .
r
r
- q
q
Thus th e 0 s a t i s f y th e Cauchy a . s . convergence c r i t e r i o n so t h a t
9. 0q
0 f o r seme 0.
However, s in c e 0^ §■ m/M, th e r e e x i s t s a subsequence
28
g,*' su ch t h a t 0^ , a °Jl° m/Mj "but s in c e th e sequence i t s e l f a l s o converges
a . s . th e y m ust converge to " th e same l i m i t m/M.
Thus 0^a *-$“m/M and th e
theorem i s p ro v e d .
H ote:
Sin ce G < (m/M)2 < 9^ <
I , l / e a "&'M/m fo r
|0 q - m/M I = (m/M) 0q | M/m ~ l / 0q | > (m/M) 5 |M/m - l / 9q | .
W ith t h i s r e s u l t th e c e n t r a l l i m i t theorem fo llo w s from th e fo llo w in g
o b s e r v a tio n .
Theorem 3 - 5 :
For a l l z £ R, R th e s e t o f r e a l numbers, and it > 0,
A
- P
|F
^ S gJt(Irst) .
iJS it( I - J f )
ti.
0n / m/M
L y.
j
' P ro o f:
< z
and
<
w here #q = z q/ s q and a = ZqZ sq .
Z
A lso , b y th e d e f i n i t i o n o f 9 .
xI ~ '
M -t'
29
!Ehus,
P 1 -S1* = p 3fq - 3t < Z8,
PL\/SqS-(W)^
-
and.
z\
2 ~ S Jl
-9 — —SL < z
■)<zs]
LV m a
™31
where z ’ = z
low l e t
A
'I
a_ - a < z*
-He
/
*q - * J < z '
Bq =
Ga .=
@a = m/M
and AC, Bc and Gc he th e complements o f e ac h o f th e s e e v e n ts .
y.
9.
9.
W 'S A^Ct , th e n W t B = ,; th u s . A ^
FB1 < BCAq U Cq) < PAq + PCq.
M oreover, i f U t£
c
B= s i a Bq c Aq
U C=
C le a rly i f
s0 t 6 a t
Thus PBq - PAq .< PCq and -PSq < PAq -PBq.
th e n 'ur s A°, so t h a t B^Ga F Aa and Aa c Ba
U Ca
Thus, PAq < P(Ba U Qa ). < PB -+ PCa Zand PAq ~ PBq < PCq w hich coupled w ith
th e p re v io u s r e s u l t g iv e s jpA^ - PBqJ < PCq «
S h is , when c o n v e rte d t o th e
o r i g i n a l n o t a tio n , provfes th e theorem .
T his hound on th e d if f e r e n c e o f th e p r o b a b i l i t i e s o f th e s e two
random v a r ia b le s f o r a l l z b e lo n g in g t o R to g e th e r w ith th e p re v io u s
theorem s o f t h i s c h ap te r, g iv e s th e fo llo w in g c e n t r a l l i m i t theorem .
30
Theorem 3 .6 :
Pi*oof:
I f z , 8^ and
st > 0 , a r e d e fin e d a s b e f o r e , th e ii
Let
- V
Hq( a) = P \
L/ S g3t(l-Jt)
and,
„
a
G(z) = —
■l/a ?
I
J~a
g
e"* Z2
d t , h e re
a = 3 .l4 l6 ' "
.
The theorem i s p ro v ed i f i t can be shown t h a t f o r a l l z £ R
|F g (a ) - G(z) I -* G.
However, |p ^ ( z ) - G(z) |- < |F ( a ) " - E ^ (z) | +
IHfl(Z) = G(z) |, and by Theorem 3 -3 ,
l^ (a )
w hich a p p ro ach es z e ro as q
|E ^ (z ) - G(z) | -> 0 .
.From Theorem 3«5
- Ea ( Z ) I < P Gq 4 m/M
oa s in c e , by Theorem 3 -4
6) ^S>° m/M, and th e
theorem i s p ro v ed .
From t h i s theorem i t fo llo w s t h a t f o r z > O'-
I im P
g.
' i z q. - y
/ ^ a 3rC1^ )
i
<
Z
if e - ^ at
o r-
31
frcm w hich ap p ro x im ate co n fid e n ce I n t e r v a l s f o r * , th e p r o b a b i l i t y o f
m u ta tio n , cari be .c o n stru c te d .
I n th e n e x t s e c tio n a d is c u s s io n i s g iv en
c o n c e rn in g th e c o n s tr u c tio n o f th e s e c o n fid e n c e i n t e r v a l s and scene o f
t h e i r p ro p e rtie s .
B ro p fe itie s o f thfe A pproxim ate C onfidence in te r v a l^ f o r . a .
I t i s o f i n t e r e s t t o in v e s t i g a t e th e .p r o p e r tie s o f th e c o n fid en ce
i n t e r v a l s f o r jt d e riv e d from th e p re v io u s th eo rem .
I n o rd e r t o do t h i s
th e fo llo w in g c o n cep t i s o f some im p o rta n c e .
D e f in itio n 3 °2s
An e s tim a to r 9
i s s a i d t o be an a lm o st su re ( a . s . )
e s tim a to r o f. 9 i f , and o n ly i f , 9 converges a lm o st s u r e ly t o 0.
n
Uote s
Almost s u re e s tim a to r s a r e c o n s is te n t e s tim a to r s and i n f a c t th e
convergence i s p o in t w ise c o n v erg en c e .
Theorem 3 - 7 ;
(i)
If
ifq - Zq f S^
u n b ia se d e s tim a to r o f jt.
P ro o f:
By th e d e f i n i t i o n o f th e
and V ar( z ) = S \ Jt(I-Jt) so t h a t
■\
• %
CO
, th e y a r e in d ep en d en t and in teg rab lfe
32
Thus, "by K oM ogorovt S a . s
convergence c r i t e r i o n P g . 238 L oe8ve ( i 960)
3f°
< I ^ 4 - .X 11I + U 4 - x | .
*1
S
Jt j
TSy th e f i r s t p a r t o f th e p ro o f
s in c e z ^ /s ^ ^ 4 * k and
A ls o , s in c e
^ 4 'm /M .
Thus
q
and
i s an a . s . e s tim a to r o f a .
i s a . s . "bounded, i t fo llo w s "by th e Dominated Convergence
Theorem P g . 1$2 Loe8Ve ( i 960) , t h a t E (a^) -> a , and th e theorem is, p rb v ed .
i""
'
• ''
'
The fo llo w in g lemmas a r e u s e f u l i n e s t a b l i s h i n g p r o p e r tie s o f th e
c o n fid e n c e i n t e r v a l s f o r a .
Lemma 3 « 2 :' In th e e q u a tio n w ith r e a l c o e f f i c i e n t s ax
if
(i)
a > 0, b < 0, c > 0,
(ii)
2a + b >. 0 , "b - 4ac > 0 ,
(iii)
a + "b > - c ,
2
1
th e n th e r o o t s ,
and Xg s
—
"b + [r h - 4ac
2a
2
+ "bx + c 3 O
33
have th e fo llo w in g c h a r a c t e r i s t i c s :
P r o o f:
© < xI < x2 <
2
S in ce "b - 4 a c > 0 # th e r o o ts a r e r e a l ; and s in c e a > 0 , b < Ojl
th e n rh /S a > 0 so t h a t
*
-Td
■v
- 4ac
x- > 0 i f -h - / b2 - 4ac > 0 s in c e a > 0 .
2
-"b +
2a
>
I-
/ 5---------c > 0 ; so -4 a c < 0 and b - 4 ac < b . Thus (/ b - 4ac < -b s in c e
2
-b > 0 and t h e r e f o r e , x^ > 0 . I f -c < a + b , th e n -4 ac < 4 a + 4ab and
b
2
” 4ac < 4a
2
2
B ut a c > 0 s in c e ,a > 0 and
2
2
+ 4alD Hh Id « ( 2a 4- "b.) > so t h a t
/
D
4 ae < 2a + b s in c e 2a + b > 0 .
Thus,
4ac < 2a ,
< i s in c e a > 0 ,
x2 F
and th e lemma i s p ro v e d .
Lemma 3*3:
I f y i s any r e a l number and x i s a n o n -n e g a tiv e r e a l number,
th e n |y | < X0Ifj, and o n ly i f , y2 < X2 .
P ro o f:
I f |y |< x , th e n |y |
|y | < |y | x < x2 o r y2 < x2 .
I f y2 < x2 , th e n y2 - x2 < 0 o r (y - x) (y + x). < 0 , w hich im p lie s
(i)
'( y ; - x) < 0 and ( y + x) > 0 , o r
( i i ) (y T k ) > 0 and ( y + x ) < 0 .
(I)
im p lie s y < x and y > -: x o r |y | < x , and
( i i ) im p lie s / y > x > 0 and y C 0 s in c e x > 0.
T h is i s im p o s s ib le , h e n c e , th e o n ly 'p o s s ib le c o n c lu sio n i s t h a t |y | < x .
The fo llo w in g theorem shows th e e q u iv a le n c e betw een th e p i v o t a l
q u a n tity
I%q " ^ q 3tI
< z
and th e f a c t t h a t th e end p o in ts o f th e co n fid e n ce i n t e r v a l f o r n,. a re r o o ts
o f a n a p p r o p r ia te q u a d r a tic e q u a tio n , and t h a t th e y a l s o s a t i s f y c e r t a i n
c o n d itio n s .
Theorem 3 .8 :
(D
F or 0 < z < oo
I v - v l
<
Z"
i SqJf(Irat)
;
■p
'
i f , and o n ly i f , a r t 4* b jt -$• c^ < 0 where
I
y.
q —
^ q * " ®q^2Zq +
(ii)
th e r o o ts o f th e e q u a tio n a j t . + b^ie + C ^ = O a r e
35
and i f S a^ s * z s a t i s f y
q q
(iii)
a .s •
O^ ’
(Iv)
ltI q " %
/
Vl
B1
«H
a •s • < .
-
= 0 , th e n Jtqq = 0 .
(V)
P ro o f:
Qt• S •
(i)
Erom Lemma 3*3
i f , and o n ly i f ,
• o r , e q u iv a l e n tly , i f , and o n ly i f ,
2
4 - 2W
/12 2
+
2A ,
< = S 14 I t ( I l t ) j
o r,
- s / 2 z ^ + z 2 ) Jt +
w hich p ro v e s ( i )
(ii)
C le a rly th e r o o ts o f th e e q u a tio n
2
a jt
q.
+ “b it + C = O
q
q
where
A ,A
2.
a 4 “ S^ S4 + 2 ) ’
\
= - V 2\
+
W O1
Sq Csq + Z2 ) . /
36
Cq " V
b
P
- j l P - 4a„,c„
Q.
are * H -
''
a^d
g - V bq 3 l(|
5tSq =
y q.
^ a •s .
(ill)
/\
S in ce i t i s assum ed t h a t
an d hy d e f i n i t i o n
2
2
z Q —"
> O and z > Ojl i t i s e v id e n t .t h a t a Q > 0 , b
> 0 and
< 0 z. and
c^Cf —> 0 .
-
He S o
a Cl
bI
\$<l ~2Zq} - Z(^\
"hq. =
*
k\ \
=
*
4y
2 +
" 2z^
Z^ )
=
“ 4\ \ ( S
“ Z2 =
q +
Z2 )
8» e S e
= 4 z ^ y 2 Cgq - z ^) + z%
2aI
/i /\
= 2Sq(S% , y
3> o S •
Thus hy Lemma 3 .2 , 0 <
—
>
/i a
+ 4a* >
3 «S o
J t,„ <
XQl —
0p
S * > 0.
Qf e S c
<
<dq. —
I.
(iv )
”aq - ltI q = —«
S a
K
af
-
'9 .
- ■Vq.' + ^ q
dA
' - a2)
*8 ♦
aX
l a ( g ; T ; V s to e e
/Ia-
-
-
^ ma
c_
Since
37
A ’
S > 1 f o r q la r g e ,
q —
T h e re fo re ,
a .s . .
-tICl S
j
p
I
L.
V
-O
,
1 + V s.
a s q -> ro s in c e
w a s q -* « and
is f in ite fo r z fin ite .
Iiiu s , I 24 - Kllj^ S - 0 .
(v )
If
q
=s 0 , th e n c
q
= 0 and
■
v
/ f
~( ~ v
=5 0 s in c e h
q
< 0,
%
and th e th e o re m .is p ro v ed .
These r e s u l t s prove t h a t '
I
■
TC
A .
V
1
<
Z
^ < «2q j
V s^st(I-Jj)
so t h a t
Iim P
stIq - 31 - 3tSq
q
where z > 0 and
and
f
i
V 2 tc
j- z
have th e c h a r a c t e r i s t i c s g iv en i n th e theorem
above.
Furtherm ore f o r a l l it
=■t / 2
'3W ' *2q
I V l V I<
(ZtgMC1 -It)-
Z
58
f o r a l l z > 0 , an d c o n v e rs e ly .
B ils i s t r u e s in c e th e f u n c tio n
Sr1 - V 2 ♦ Y
i s a q u a d r a tic f u n c tio n i n Jt and a
+ \
> 0 im p lie s th e g rap h o f th e fu n c tio n
y.
i s a p a ra b o la opening upward w ith a - i n t e r c e p t s a t
and
CHAPTER I V
RATES OF COISTVERGENCE
I n t h i s c h a p te r a n a tte m p t i s made t o d e term in e j u s t how good th e
ap p ro x im ate c o n fid e n ce i n t e r v a l s i n th e p re c e d in g two c h a p te rs a r e o r
a l t e r n a t i v e l y .to d e term in e th e r a t e i n term s o f q. a t w hich each, o f th e s e
d i s t r i b u t i o n fu n c tio n s ap p ro ach th e Wormal D is tr ib u tio n w ith mean z ero
and v a ria n c e one.
I n Theorem 2 .1 i t was shown t h a t
an d i n Theorem 3°3 i t was shown t h a t
/ SqSt(I-Jt)
B oth o f th e s e random v a r ia b le s were shown t o be sums o f in d e p en d e n t random
v a r ia b le s c e n te re d a t e x p e c ta tio n s and by th e bounded assu m p tio n th e t h i r d
a b s o lu te moment i s f i n i t e .
They th u s s a t i s f y th e c o n d itio n s n e c e s sa ry t o
employ th e theorem o f .Grame *r (1937) J P g • 77> f o r d e te rm in in g t h e i r r a t e s
o f convergence t o 'W(OjlI ) .
I n Theorem 3»5 i t was shown t h a t
\
f o r a l l z 'e R, R th e s e t o f r e a l num bers.
- V
< z
But s in c e
< P G ,4 m/M
It-O
v £ A
Qq = m/M
■ pP
S<1 ~
■ iJ
a
-
S_
where b y Theorem. 3«4
S - Srt a . s .
_9L____a _
-
A s i m i l a r r e s u l t h o ld s f o r
A
rs - s
------ S L < z
L
s
a
rS
- S
- P -a—
A
L
S
-I
a < z
a
-1
f o r a l l z £ Rj, a s i s shown "by-the fo llo w in g
Theorem 4 .1 .
I f S„» S , s and zS a r e d e fin e d a s b e f o r e » th e n
a
a .a
a
Ir r t i < ; -p
a
rif -
S c-
I<p % N
A
a
I
o
a
fo r a l l re a l z.
P ro o f:
U sing th e same p ro c e d u re a s em ployed i n Theorem 3°5> s e t
rt i <
1V
bQc s C
,
- s C^ < Z
• V1 ■
4l
■ A ■
1■]
.C _c
w ith A^j,
. and Cq th e complements o f e a c h o f t h e s e . e v e n ts .
th e n
Ifu rg A ^ C f1,
9 .1 '
th u s . A^C c
and b y com plem entation B c a U Cc so t h a t
y.
11
I
I
l
l
PBq < PAq + PCq . M oreover, i f u r £ BqCq , th e n u^£A®, th u s BqCq c Aq .,. so
ur
£
t h a t , PAq, < PBq + PCq.,
Gomhining t h e s e r e s p i t e "gives
L
^
< ,1 I < P
J
L
l l
H
J
<ra
s
i_ _CQ
Ip L
J
I .
But
-^2 - s2
- 8qH gq + V
s
However, s in c e f o r a l l q.
G<
zS + s
- S ------------- S <
CO
,
A
4 2
'Szn - S
i
-a 4 i
I-Q
A lso s q/ s q s I i f , and o n ly i f s q/ s q = I , th u s
L8 A
an d th e theorem i s p ro v e d .
M oreover, th e f o ll o w i n g .is a l s o t r u e .
Theorem 4 .2 :
If s
I
and Is
I
a r e d e fin e d a s "before th e n ,
^ 1
, o
42
Is2 - s 2
J l ____ g
2
s
q
P ro o f:
a .s .
o.
By Theorem 2 .2
As 2 - s 2
._ s ____ a
s ii*
0,
a a d from Lemma 2 .1
i
where 0 <
^ »ai
< w.
Thus
ua.
aI < I
's2 - S2
q____ qi
I - jL = - a I
a .s .
0,
and th e theorem i s p ro v ed .
Erom th e s e rem arks an d theorem s i t i s th e n a p p a re n t t h a t th e r a t e s
o f convergence i n "both c a s e s can he exam ined w ith th e same m ethod.
Toward t h i s end i t i s c o n v en ie n t t o in tro d u c e th e fo llo w in g g e n e r a l
n o ta tio n .
L et
A
Y_
V
A
S
a_fa
&
V f a
or
,
|/ s It(I-IC)
a c c o rd in g a s
p
p
a
9.
and l e t
Fq (Z) = P
A
Hq(z ) » P
*
and
-C O
Im p lie d i n th e s e d e f i n i t i o n s i s th e c o n v en tio n t h a t Yq , Yq and Xq a re
sim u lta n e o u s ly th e f i r s t random v a r ia b le o r th e seco n d , s in c e th e s e
a re
th e co m binations o f i n t e r e s t i n th e c o n te x t.
U sing t h i s n o ta tio n th e problem i n e ac h ease i s t o bound
IFq Cz) - G ( z ) |
f o r a l l z E R-
T h is i s done by- c o n s id e rin g
IFqM
- G (z )| < IFq (z ) - Hq ( z ) | + IHq (z ) - G(z) | ,
where by Theorems 3*5 and 4 .1 .
IFq(%) - Hq ( z ) | < P
\
*
°-
and
|Hq(z) -G(Z)I < e p i l o g q / /c f"
by th e theorem of: Cramer, m entioned ab o v e, where, c , i s a n a b s o lu te c o n s ta n t
w hich can be ta k e n t o be 3 a c c o rd in g t o H. Cramer- (1 9 2 8 ),
(Pr 3 - ^ l I j f r) ,
so t h a t th e second, d if f e r e n c e i s hounded.
To hound P
I -P
th e r a t e a t w hich P
2V
0
\
= 0 , i t ' . i s n e c e s sa ry o n ly t o c o n s id e r
app ro ach es one.
Py a theorem P g . l 68 Loe5ve
( i 960) } sin c e X - ^ ‘O, th e d is tr ib u tio n fu n c tio n o f X- , Fy ( x ) -» O or I
<1
• '
q
\
a c c o rd in g .a s x < 0 o r x > 0 , w hich i s e q u iv a le n t t o sa y in g P X = O
L <1
-> I .
Thus th e problem re d u c e s t o f in d in g th e r a t e o f convergence o f th e sequence
o f random v a r ia b le s X i n law t o a d i s t r i b u t i o n d e g e n e ra te a t , z e r o .
%
S i n c e ^ ( X g ) -» o \ ( 0) ; th e c h a r a c t e r i s t i c fu n c tio n f^ ( u ) - > 1 = f ( u ) ,
th e c h a r a c t e r i s t i c f u n c tio n o f th e d e g e n e ra te law .
A lso by theorem
P g. 199, Loe«ve ( i 960) ,
‘
f o r "0 < S < I and
f #
= 1 +
\
^
\ 1+8
i s a complex v a lu e d q u a n tity w ith [ SgJ < I .
Taking 8 =J I g iv e s
= 1+ 5
From t h i s e x p an sio n o f f ^ ( u
.
45
2 ( f a (u ) - I )
i s a f u n c tio n o f u .
From, t h i s i t fo llo w s t h a t
-R Sq (U) <
uX
s in c e R f^(u) - I < 0 and | f^ ( u ) - l | < 2 .
(H ere R d e n o tes th e r e a l p a r t
o f th e complex f u n c t i o n .)
U sing th e s e o b s e rv a tio n s i t i s th e n p o s s ib le t o prove
Theorem - 4 . 3 .
F o r a l l q an d r e a l u ,
-R Stl(U) < - 2
U■
where -R ,S^(u) -is d e fin e d a s ab o v e .
P ro o f:
u,
The r e s u l t i s t r i v i a l f o r - I < u < I s in c e 16^ (u ) | < I f o r a l l
Thus th e r e s u l t m ust be p rov ed f o r th e c ase |u | > 1 .
From th e
In c re m e n ts I n e q u a lity P g. 195; Loe5ve ( i 960) j w hich s t a t e s t h a t f o r any
r e a l ti and h
I fq(U ) - f q (u + h) I2 < 2 f q (0 ) ( f q (0 )
i t fo llo w s s in c e f ( 0 ) = I t h a t
2 (1 - R fq ( h ) ) > | f q (u ) - f q (u+h) | 2 ,
and s in c e | l - f^ ( u ) | > I - Rfq Cu),
2 1f q (h ) - l | > | f q (u ) - f q (u+h) | 2 .
Rfq ( h ) ) ,
46
But
f q (u ) - 1
+ 1
e j u ) U2EX^ - ,
and
f^Cu+h) - 1
+ | e q (u+h} (u+ h)2EX^ >
Thus
2 |f ^ ( h ) - 1| > J(EX2)2I Sq(U)U2 - 0q (u + h)(u+ h) 2 | 2 ,
J(EX2 ) 2 IR 0 ( u ) u2 - R 0 (u+h)(u+h)
g.' '
2,2
q.
Efow assume t h a t f o r |u | > 1 and f o r a l l q
-R e (v.) > ± 2 .
■
U
Coupled w ith th e f a c t t h a t
" B
V")
>
i t fo llo w s t h a t
jj(E x |)2 |H eq Ca)u2 - B Sg (W h )(M b )2 I2 > j ^ k ^ 2 | i 2 a2
t
( Wh ) 2 I2 = I ( b ^ ) 2 I1 - r ^ l 2 ,
" (W h )2E ^
or fo r a l l h
2 | f (h ) - i | > ^ I ex2 - if| 2 .
^
However f o r h = 0 , t h i s im p lie s
0 > ||E X ^ - 4 | 2 > o
T
Hence, -R e (u ) < - p f o r a l l q and a l l u .
f o r a l l q , a c o n tr a d ic t io n .
T his r e s u l t th e n g iv e s th e key t o th e fo llo w in g .
Theorem 4 .4 :
X f c i f (X^)
(O ), th e n
P
P ro o f:
< 7 ® t| •
Ry th e second T ru n c a tio n I n e q u a lity P g . 196, Loe8ve ( i 960); f o r
a l l u > 0,
u
d5L
(x ) <
-U J
0
|x | > l / u
f
Vq
I
5
/■
Rft ( v ) )
dv
■R 6_ (v) EX?v2dv
q.
0
u
<
s / e^'dv
s in c e -R 6^(n) <
■ TEX2 .
S in ce th e bound i s in d ep en d en t o f u , and th e c h a r a c t e r i s t i c fu n c tio n i s
d e fin e d f o r - <», < u < <»«, ta k in g th e l i m i t on u g iv e s
LX 1 4 °
and th e theorem i s p ro v e d .
K
x
> 0
48
Prom t h i s theorem and th e p re v io u s rem arks i t fo llo w s t h a t f o r a l l
re a l z
A■
-S -S
-2 -3
L r
- 2 - s 2\ 2
< z
/ ^ / 2 a t | < T ^ s )
S a) +
^
s
✓ g-
where
q
11
I
<°!>3 /2
and
'V V
- 8_\2
- t ^ /2 d t I < T O - S - S uj + 3 lo g q
i /(
< a
\/SqJt ( I - J t ) '
where
i
Z , e |? _
q Sfcwi
%
- a . "I
x
e3q =
i
(S4lt( I X ) )3 /2
W ith th e s e r e s u l t s i t i s p o s s ib le t o p ro v e th e fo llo w in g .
Theorem 4 .5 :
(I)
I f th e c o n s ta n ts H, M, and m a r e d e fin e d a s b e f o r e t h e n
rs - s
I? ^
a) I
<z
q
5 q (k -m A n -2 )2
T
t 3 lo g q. W(M-m)
Y5!
A
(Ii)
=z f?S it
q qIP
/ ^ ^ ( H jj)
< Z
G(z) I < ~ | + 3-— ^ --S W3 m +
I
T
v— ■*&-*>
f o r a l l r e a l z , where. G(z) i s th e . d i s t r i b u t i o n fu n c tio n o f th e norm al
D i s t r i b u t i o n w ith mean z e ro and v a ria n c e on e.
P r o o f:
(i)
th e independence a ssu m p tio n .
E^ t
“
*q} 2
=Z
k=l
' &
\ )E ^
" ffk )2
&
By th e sam pling w ith o u t rep la ce m e n t p r o b a b i l i t y f u n c tio n .
<
-
^ )2
Thus by Lemma 2 .1 and a ssu m p tio n 3,
„x2
- | k ) E ( ^ - ^ ) 2 < 5(M™m)^(u-2)^ ,
V
- 7
-
^ 5dCM-m)4 (H -2)2
'<
U sin g th e boundedness o f th e
I 51kA
k=l
-and th e Xjs^ , i t fo llo w s t h a t
- a M I3 < I
e ^k A
ksl
so t h a t
3a
< M e ™1
s
w hich p ro v e s ( i ) .
(ii)
S in ce E (S^ - S j i 2 = s 2 ,
9
X
-ml a s
1
q . .
50
>
and i n t h i s c a s e , u s in g S c h m rz 6s I n e q u a lity ,
- S11 sr)2E(
k
<
9
s ,T .jf(i-jt) y » t ( i " j f ) + 1 ,
•i'll.
so t h a t
3EM
"*"■jf(l-jt)
and ( i± ) i s p ro v ed .
Erom th e s e r e s u l t s , to g e th e r w ith th e r e s u l t s p roved e a r l i e r t h a t
ft* > q.2 q ?, S^ >
q.
I
q.
and s^ <
<3.
i t fo llo w s t h a t b o th o f th e
d i f f e r e n c e s , ( i ) and ( i i ) , i n Theorem 4 .5 w i l l he l e s s th a n e > 0 , i f
lo g q
>
max
" (M -m )V
aA 2
-/d ^ e
m ejt(l“3t ) J
These r e s u l t s a r e o b ta in e d w ith th e added assu m p tio n s t h a t (M=m) > I ,
cL > I and lo g q, > I .
However, th e s e a r e v e ry c o n s e rv a tiv e b o u n d s.
I f th e sum o f th e two bounds i n ( i ) o r ( i i ) o f Theorem 4 .5 can be
made l e s s th a n some p r e s c r ib e d number, th e n th e s o - c a lle d I-= 5C p e r c e n t
c o n fid e n ce i n t e r v a l s have a d eg ree o f c o n fid e n c e w hich can be d eterm in ed
51
F o r exam ple, i f th e sum o f th e two hounds i n ( i ) o r ( i i ) o f Theorem 4 .5 i s
l e s s th a n o r e q u a l t o .0 5 , th e n th e c o n s tr u c te d 95 p e r c e n t co n fid en ce
i n t e r v a l s have a t . l e a s t an 85 p e r c e n t c o n fid e n c e .
Under th e assu m p tio n t h a t th e JlfcXfc a r e a l s o i d e n t i c a l l y d i s t r i b u t e d ,
th e r a t e o f convergence i n each case i s f a s t e r a s can he see n from th e
fo llo w in g .
Theorem 4 .6 :
I f , i n a d d itio n t o th e p re v io u s a ssu m p tio n s, th e BfcXfc a re
i d e n t i c a l l y d i s t r i b u t e d , th e n
c (M-m)
< Z
4
W
(ii)
A
V
|P
,
< 2
3%
G (a )| < — 3 -
T sr
^ i aJt(I-SC)
J 5 m + St(I-St)
f o r a l l r e a l z where G(z) i s th e Mormal D is tr ib u tio n w ith mean z e ro and
v a ria n c e o n e, and
P ro o f:
» M H, M,M and m a re a s d e fin e d b e fo re and c < 3«
I n th e i d e n t i c a l l y d i s t r i b u t e d case a l l th e Mfc, Hfc a n d Xfci a re
th e same. f o r a l l k .
I n t h i s c ase lo g q / /c fb e c a m e s l / /"q" a c c o rd in g t o
Crame’r ( 1957)•
(i)
In th e i d e n t i c a l l y d i s t r i b u t e d case 8
8N'
, H(M-m)
(M-m)
52
a2 ) 2 < 5<l(M-m)^(K-2)2
so t h a t
^ 5(M~m)4 (H -2) 2
(ii)
1 IzHfc " 8H3t I5
3®M +
(S j^ C i-ic ))3^
A
I
st(l-jc)
a2
T h is p ro v e s th e theorem .
I n th e i d e n t i c a l l y d i s t r i b u t e d case th e r a t e o f convergence i s
f a s t e r ,, b u t t h i s a ssu m p tio n a l s o r e q u i r e s , a s m entioned, t h a t a l l th e
Ufc, and Xfc,. a re th e same.
T h is c o n d itio n would be I m p r a c tic a l in th e
e x p e rim e n ta l s i t u a t i o n d is c u s s e d h e r e .
However, i f th e e x p e rim e n ta l
s i t u a t i o n c o u ld be s ta n d a rd iz e d so t h a t t h i s assu m p tio n i s m e t, th e r a t e
o f convergence would be f a s t e r .
These theorem s do n o t n e c e s s a r ily s o lv e th e r a t e s o f convergence
p ro b lem , s in c e much s h a r p e r bounds may e x i s t .
The. bounds d e riv e d in
Theorems 4 . 5 and 4 . 6 , how ever, do g iv e some in fo rm a tio n a s t o how f a s t
e a c h o f th e s e d i s t r i b u t i o n fu n c tio n s a p p ro ach es th e Hormal D is tr ib u tio n
w ith mean O and v a ria n c e I .
CHAECBR V
APPHCATXOMS TO EXPERBMTAL RESULTS
An e x p erim en t d e sig n e d t o e s tim a te th e p r o b a b i l i t y o f spontaneous
m u ta tio n from a v iru le n c e to . v ir u le n c e a t th e A lg e ria n lo c u s o f th e fungus
was preform ed w ith th e fo llo w in g r e s u l t s .
: -
The. e s tim a te o f S , th e t o t a l
g.
number o f sp o re s g e rm in a te d on th e v a r i e t y A lg e ria n , was S
b u t no m u ta tio n was o b serv ed .
= 4 , 587, 088,
U sing th e te c h n iq u e s su g g e s te d i n th e p r e ­
v io u s c h a p te r s , th e approxim ate 95 p e r c e n t co n fid e n ce i n t e r v a l f o r S
i s ( 4 ,4 4 o ,5 4 lj < 4 ,7 3 3 ,6 3 5 ).
q.
The p o in t e s tim a te o f it i s z e ro and an ap p ro x ­
im ate 95 p e r c e n t c o n fid e n ce i n t e r v a l f o r at, th e p r o b a b i l i t y o f m u ta tio n ,
i s ( 0 ; 8.7201 x 10
).
r e p e t i t i o n s , was 59*
The number q i n t h i s e x p erim en t, th e number o f
The numbers H, M and m can n o t be d e term in e d e x a c tly
from th e e x p e rim e n ta l d a ta , b u t from th e d a ta o b ta in e d i t is-know n t h a t
IR > l l 8l , M > 366, m < 6 and zS^ = 5 ,5 9 4 ,0 2 4 ,2 1 9 .
S u b s titu ti n g th e s e
e s tim a te s i n th e bounds d e riv e d i n Theorem 4 . 5 , g iv e s th e fo llo w in g
bounds;
f o r ( i ) a p p ro x im a te ly 4 and f o r ( i i ) ap p ro x im a te ly .3 -
E xperim ents were a l s o made t o in d u ce m u ta tio n s from a v ir u le n c e t o
v ir u le n c e a t th e A lg e ria n lo c u s i n th e fungus u s in g gamma r a d i a t i o n .
In
th e e x p erim en ts u s in g gamma i r r a d i a t i o n th e fo llo w in g e x p e rim e n ta l
te c h n iq u e was u s e d .
P la n ts o f th e s u s c e p tib le v a r i e t y Compana were in o c ­
u l a t e d w ith m ildew sp o re s when th e y were a b o u t seven days o ld .
Two days -
" a f te r in o c u la tio n th e Conrpana p la n ts w ere t r e a t e d w ith gamma i r r a d i a t i o n
c o n tin u o u s ly f o r 48 h o u rs .
dosage o f 4 8 o b r.
D uring t h i s p e rio d th e y re c e iv e d a n e s tim a te d
I n a b o u t sev en days a f t e r th e i r r a d i a t i o n tre a tm e n t,
th e m ildew on th e Compana p la n ts began t o s p o r u la te .
These sp o re s were
54
th e n u sed t o in o c u la te th e r e s i s t a n t v a r i e t y A lg e ria n i n o rd e r t o s c re e n
f o r m u ta tio n s from a v ir u le n c e t o v ir u le n c e .
E h is ex p erim en t produced no
im ita tio n s and th e f i n a l e s tim a te o f S , th e t o t a l number o f o b s e rv a tio n s
on th e v a r i e t y A lg e ria n , w a s 's ^ = 431,931*
U sin g th e same, te c h n iq u e s a s
b e fo re g iv e s th e approxim ate 95 p e r c e n t c o n fid en ce i n t e r v a l f o r S a s
■■ q.
(4 0 5 ,7 7 5 ; 458,077) • The p o in t e s tim a te o f jc i s z e ro w ith approxim ate
95 p e r c e n t c o n fid e n ce i n t e r v a l f o r x b e in g (0,* 8 .8 3 9 % i o " ^ ) »
e ase q. = 2 2, I > 309, M > 227, m s I and ^
s= 1 7 8 ,0 8 6 ,3 4 1 .
In t h i s
S u b s titu tin g
th e s e e s tim a te s i n t o th e bounds o b ta in e d i n Theorem 4 .5 g iv e s th e ap p ro x ­
im ate bounds f o r ( i ) and ( i i ) , r e s p e c tiv e ly a s 3 a n d ,. 5 « M oreover,' i n
t h i s e x p erim en t i t was p o s s ib le t o c o n s id e r each p o t o f A lg e ria n p la n ts a
s ta g e i n th e q -s ta g e e x p e rim e n ta l p ro c e d u re .
Uhen th e d a ta o b ta in e d 'is
a n a ly z e d i n t h i s way, th e e s tim a te o f th e t o t a l number o f o b s e rv a tio n s i s
A
a g a in S s 4 3 1 ,9 3 1 , b u t th e approxim ate 95 p e r c e n t co n fid e n ce i n t e r v a l
y.
f o r S^ i s (4 2 0 ,5 4 7 , 443,315)«
The p o in t e s tim a te and th e approxim ate
95 p e r c e n t c o n fid e n ce i n t e r v a l f o r x a r e a g a in 0 and (Oj 8 .8 3 5 % 10 ^ ) ,
re s p e c tiv e ly .
In tM s c a s e , how ever, q = ICO, H > 6 7 , M > 227, m s I and
s 3 3 ,7 3 0 ,9 7 2 .
Wsing th e s e e s tim a te s i n th e bounds d e riv e d i n Theorem
4 ;5 g iv e s approxim ate bounds f o r ( i ) ; 5 .8 and f o r ( i i ) j .3*
I t is in te r­
e s t i n g t o n o te t h a t w hat is .g a in e d , i n s h o r t e r co n fid en ce i n t e r v a l s f o r
S , i s l o s t i n , t h e approxim ate bound o b ta in e d i n th e r a t e o f convergence
. q_
1
"
problem .
, F u r th e r e x p erim en ts w ere made t o in d u ce m u ta tio n s a t a l l e ig h t l o c i
o f th e b a r le y m ildew fungus u s in g d i e t h y l s u l f a t e ;
I n th e e x p erim en ts
55
u s in g d i e t h y l s u lf e te ^ Campana "barley p la n ts were sp ray ed f iv e tim e s a t
30 m inute i n t e r v a l s w ith f r e s h 0 . 0076m s o lu tio n s one t o two days a f t e r
in o c u la tio n .
When p u s tu le s a p p ea re d on th e Campana p l a n t s , th e s e were
th e n u sed t o in o c u la te e i g h t r e s i s t a n t v a r i e t i e s o f'" b a rle y .
Each r e s i s t a n t
v a r i e t y o f "barley c a r r i e s a d i s t i n c t gene f o r m ildew r e s i s t a n c e .
A f te r
a h e s tim a te d 20 ,2 7 0 sp o re s had been t e s t e d on th e r e s i s t a n t v a r i e t y Bana
m /n AB 62-=4, a p u s tu le a p p ea re d on t h i s v a r i e t y .
E his p u s t u l e ," a f t e r
f u r t h e r s c r e e n in g , was c o n sid e re d a m u ta n t. A pplying th e same te ch n iq u e
A
i n t h i s case g iv e s
s 2 0 ,2 7 0 and ap p ro x im ate 95 p e r c e n t c o n fid en ce
in te rv a ls fo r S
a s (1 8 ,6 3 0 ; 2 1 ,9 1 0 ).
The p o in t e s tim a te o f n is .
.
4.904: x 10 ^ w ith ap p ro x im ate 95 p e r c e n t co n fid e n ce i n t e r v a l (8 .4 8 5 x lo ~ ^
2 .8 7 5 x 10
)i
I n t h i s case q = 1 5 .
Eroin th e e x p e rim e n ta l d a ta i t i s
known t h a t I > 5 4 , M > 79, m < 10 and t h a t 's ^ = 672, 3 8 9 .
U sing th e s e
numbers t o d eterm in e th e bounds i n Theorem 4 .5 g iv e s f o r ( i ) ap p ro x im a te ly
12 and f o r ( i i ) a p p ro x im a te ly 1.5»
Eram a p u re ly i n t u i t i v e p o in t o f v iew , th e c o n fid en ce i n t e r v a l s
o b ta in e d f o r th e p a ra m e te rs
re a s o n a b le .
and # i n th e s e ex p erim en ts a p p e a r q u ite
The bounds o b ta in e d , how ever, a r e n o t v e ry s h a rp .
T his i s
n o t s u r p r i s i n g , how ever, s in c e th e bounds o b ta in e d i n Theorem '4 .5 a re v e ry
c o n s e r v a tiv e .
I t i s p o s s ib le t h a t s h a r p e r bounds e x i s t and t h a t i f th e s e
c o u ld be found th e y w ould p ro v id e b e t t e r bounds even i f e s tim a to r s were
s u b s t i t u t e d f o r p a ra m e te rs o r c o n s ta n ts ^w hich a p p e a r.
CHAPTER VI
SUMMARY AND REMARKS
The two c e n t r a l l i m i t th e o re m s, 2 .5 and 5 .6 , a s s t a t e d and proved
i n C h ap ters I I and I I I when c o n sid e re d i n a g e n e r a l s e t t i n g , a r e an
approach, t o e s tim a tin g and f in d in g c o n fid e n c e i n t e r v a l s f o r b o th p a ram eters
i n th e B inom ial D i s t r i b u t i o n .
The assu m p tio n s made t o prove th e s e theorem s
a r e g e n e r a l enough t o make th e theorem s a p p lic a b le t o many more e x p e r i­
m e n ta l s i t u a t i o n s th a n th e one g iv e n i n C h ap ter I .
The m ethods o f p ro o f ,
t o o , may be a p p lic a b le t o o th e r p ro o fs o f c e n t r a l l i m i t th e o re m s.
The f i r s t th eo rem .
a t f i r s t g la n ce may n o t a p p e a r t o be a new th eo rem .
However, t h i s theorem
d i f f e r s from o th e rs o f t h i s ty p e i n t h a t th e c o n d itio n ^ / s ^
i is .
l e s s r e s t r i c t i v e th a n t h a t o f r e q u ir in g a c o n s is te n t e s tim a to r o r th e
a ssu m p tio n o f th e summands b e in g i d e n t i c a l l y d i s t r i b u t e d .
I n f a c t , th e
c o n d itio n o f th e r a t i o o f th e e s tim a to r t o th e p a ra m e te r c o n v erg in g
a lm o st s u r e ly t o one i s im p lie d by th e c o n d itio n o f a c o n s is te n t e s tim a to r .
The second th e o re m .
i s t y p i c a l l y b in o m ia l e x c e p t f o r t h e .f a c t t h a t
v a ria b le .
i s a l s o a random
I n t h i s s e q s e , u n d e r th e assu m p tio n s s t a t e d , th e theorem
t o th e a u th o r ’ s know ledge, new.
is ,
In t h i s th eo rem , to o , th e r a t i o o f th e
57
e s tim a to r t o th e p a ra m e te r, zS / s , c o n v erg in g a lm o st s u r e ly t o one p la y ed
cl q.a n im p o rta n t r o l e .
T h is id e a , h av in g worked more th a n once, m ight be
c o n sid e re d a d e v ice i n p ro v in g o th e r c e n t r a l l i m i t theorem s i n o th e r
s itu a tio n s .
T his i s p a r t i c u l a r l y tr u e when b o th e s tim a to r and' p aram eter
d iv e r g e .
The method o f p ro o f o f th e s e two theorem s co u ld have b een th e same,
nam ely, showing th e d if f e r e n c e o f two d i s t r i b u t i o n fu n c tio n s converge t o
zero .
However, i t sh o u ld b e .n o te d t h a t a more ”r e s t r i c t i v e " c o n d itio n
c o u ld be shown i n th e f i r s t case and n o t i n th e second.
The r a t e s o f convergence theorem f o r th e d e g e n e ra te law i s , t o th e
a u th o r ’s know ledge, new. . T h is , th e n , combined w ith th e c l a s s i c a l r a t e s
o f convergence theorem o f C ram etr can be c o n sid e re d a new r a t e s o f co n v er­
gence theorem f o r random v a r ia b le o f th e ty p e d is c u s s e d i n C h ap ter IV.
A c lu e t o an ap p ro ach t o th e r a t e s p f convergence f o r t h i s s e t t i n g became
a p p a re n t i n th e p r o o f o f th e second c e n t r a l l i m i t th eo rem , nam ely, t h a t
i t was p o s s ib le t o bound th e d if f e r e n c e o f two d i s t r i b u t i o n fu n c tio n s
u n ifo rm ly by th e p r o b a b i l i t y t h a t a n a lm o st s u r e ly co n v erg en t random
v a r ia b le i s n o t e q u a l t o i t s l i m i t .
T h is le d t o th e c o n s id e r a tio n o f
th e r a t e o f convergence o f a random v a r ia b le t o th e d e g e n e ra te law ,
(o ).
V arious theorem s from Loe’ve ( i 960) , on d i s t r i b u t i o n and ch arac
t e r i s t i c f u n c tio n s were o f g r e a t im portance i n fin d in g a s o lu tio n to t h i s
problem .
Even though th e r a t e s a s th e y a p p ea re d in C h ap ter IV and in th e
e x p e rim e n ta l a p p lic a tio n s seem f a i r l y re a s o n a b le , and i n some in s ta n c e s
2
v e ry good, i t sh o u ld be k e p t i n mind t h a t i n t h i s s i t u a t i o n EX^ and
-
58
c o u ld n o t "be e v a lu a te d e x a c tly and c o u ld o n ly "be bounded=
The r a t e s a re
th u s v e ry c o n s e rv a tiv e .
The c o n c e p t"o f a lm o st s u re e s tim a to r , f o r example
, p la y e d an im9.
p o r ta n t r o l e i n e s t a b l i s h i n g th e p le a s a n t p r o p e r tie s o f th e c o n fid en ce
i n t e r v a l s f o r a , th e ^ p r o b a b ility o f m u ta tio n .
Almost s u re e s tim a to r s .
X
a s n o te d , a r e alw ays c o n s is te n t and i f bounded, a s y m p to tic a lly u n b ia se d
e s ti m a to r s .
T h is co n cep t i s , i n a s e n s e , a n e x te n s io n o f K olm ogorov's
S tro n g Law o f Large !Numbers.
F u r th e r work c o u ld be done i n i n v e s t i g a t i n g th e p o s s ib le t r u t h o f
th e s e theorem s when th e assu m p tio n o f boundedness i s dropped by w orking
w ith tr u n c a te d random v a r i a b l e s .
A ls o , i t may be p o s s ib le t o d e riv e
s h a r p e r bounds by e v a lu a tin g th e second and f o u r th moments e x a c tly , and
r a t h e r th a n bounding th e s e e x p r e s s io n s , le a v e them i n th e expanded form .
I n th e a p p lic a tio n s th en , i t would be p o s s ib le t o use Tukey1S m u ltip le -k
s t a t i s t i c s t o p o s s ib ly ob ta in sharper bounds.
R ec o g n itio n sh o u ld be made o f two a r t i c l e s w hich a p p e a re d i n Pub.
M ath. I n s t . H ungarian A cad. S cien ce w r i t t e n by ErdHs
(1 9 5 9 ), and H ajek , J . ( i9 6 0 ) .
P . and Re’n y i, A.
These a r t i c l e s w ere n o t a v a i l a b l e f o r
p e r u s a l a t th e tim e o f t h i s w r i t i n g / b u t from th e r e fe re n c e made t o th e
a r t i c l e s i n C h ap ter H o f Sam pling T echniques by Cochran/ 2nd E d itio n , i t
was f e l t t h a t th e c e n t r a l l i m i t theorem s o b ta in e d i n t h i s p a p e r were p ro v ed
i n a n e n t i r e l y d i f f e r e n t s e t t i n g and a r e new r e s u l t s .
59
LEEERATURE CONSULTED
Crasadr, H a ra ld , "On th e C om position o f E lem entary E r r o r s ,"
A k t u a r i e t i d s k r ; 13-7k# l k l - l 8 o , 1928. .
Skando
Crame8T , H a ra ld , Random V h ria h le s and P r o b a b il ity D i s t r i b u t i o n s ,
Cambridge T r a c ts i n M athem atics and M ath em atical p h y s ic s No. 3 6 ,
Cambridge U n iv e rs ity P r e s s , 1937»
L oe8Ve, M ich e l, P r o b a b il ity T heory, Second E d itio n ^ D. Van N o stran d
Company, IncTJ i 960.
W ilk s, Samuel S . ,
1962 .
M ath em atical S t a t i s t i c s , J . W iley and S ons, I n c . ,
MONTANA STATE UNIVERSITY LIBRARIES