On non-negative unbiased variance estimation for midzuno strategy

(1)

P a k . J . S t a t i s t .

1 9 9 4 V o l. 1 0 ( 3 ) p p 5 7 5 - 5 8 3

O N N O N - N E G A T I V E U N B I A S E D V A R I A N C E E S T I M A T I O N F O R M I D Z U N O S T R A T E G E Y

D. S. T ra cy

D e p a rtm e n t o f M a th e m a tic s , U n iv e rsity o f W in d so r, C a n a d a

A nd P. M u k h o p a d h y a y1

In d ia n S ta ti s tic a l I n s titu te , C a l c u t t a , I n d ia (R eceived: S e p te m b e r , 1993 A c c e p te d : A p ril, 199-1)

A B S T R A C T

Vijayan (1975) and R a o (1970) o b ta in e d t h e n e c e s s a r y fo r m s o f n o n - n e g a t i v e q u a d r a ti c unbiased e s t i m a t o r s o f m e a n s q u a re error o f a linear e s t i m a t o r o f p o p u  la tion total. Here we co n s id er d ifferen t u n b iased v a ria n c e e s tim a to r s w h i c h s a t i s f y th o s e n ecessity c o n d itio n s for M i d z u n o strategy. T h e i r p r o p e r tie s , viz., p r o b a b i l i  ties o f their ta k in g n e g a t iv e values a n d s ta b i l i t y h a v e b een s t u d i e d e m p ir ic a lly. T h e m o d i f i e d n o n -n eg a tive e s t i m a t o r s , as in R a o a n d V ija y a n (1977), h a v e a lso b ee n consid ered. T h e pre sen t s t u d y co v ers six te e n difTcrent e s t i m a t o r s .

K E Y W O R D S

N o n -n e g ativ e U nbiased V a ria n c e E s tim a tio n ; m e a n s q u a re e r r o r M id z u n o s tr a t e g y ; 1. I N T R O D U C T I O N A N D P R E L I M I N A R I E S

C o n sid er a fin ite p o p u la tio n V o i N id e n tifia b le u n i t s la b e lle d N . A sso c ia te d w ith i are tw o re a l q u a n titie s (Y;, x ;) , v a lu e s o f a m a in v a ria b le ‘y ’ a n d a closely re la te d au x ilia ry v a ria b le ‘x ’ resp e c tiv e ly ( i = 1 , . . . , N ) . In a s a m p le su rv e y

N

for e s tim a tin g th e p o p u la tio n t o t a l T = Yj (m e a n Y = T / N ) , a s a m p le s ( a p a r t l

o f V , w ith u n its re p e a te d o r w ith o u t r e p e titio n ) is se le c te d a c c o rd in g t o a s a m p lin g p la n p w ith p (s) as th e p r o b a b ility o f se le c tin g s ( p ( s ) > ° > X ] p ( s ) = 1 , 5 = { s } )

s e s

'R e se a rc h carried o ut at th e U n iv ersity of W indsor, o n leave fro m th e In d ia n S ta tis tic a l In s ti

tu te , C a lc u tta .

J U L

(2)

a n d an e s tim a to r e (s , y ) , a fu n c tio n o n S x Rⁿ su c h t h a t its value d ep e n d s on y = i/ v ) only th r o u g h th o s e m for w hich i € s , is e m p lo y e d . A com bination (/>, 0 is c a lle d a sa m p lin g s tr a te g y .

For e s tim a tin g T , M id z u n o (1950, 1952) - L a h iri (1 9 5 1 ) - S en (1952) p roposed th e follow ing sa m p lin g s tr a te g y . T h e firs t u n it in th e s a m p le ( o f size n ) is chosen

N

w ith p r o b a b ility ;>,(= x , / X , X = ] P x . ) a n d th e s u b s e q u e n t ( n - 1 ) u n its b y sim ple l

ra n d o m s a m p lin g w ith o u t r e p la c e m e n t (srsw o r) fro m V - {£}. T h u s

p ( s ) = (1.1)

w here x , = M i = ( .* ) , * = 1,2. T h e r a ti o e s tim a to r eR = X y , / x , , i€a

w here y , — is u n b ia se d fo r Y w ith v a ria n ce

*€*

1 O*

an d as a n u n b ia s e d v a ria n ce e s tim a to r

N v N v

E y?E ^r +EEY>

^{. J'.t}

Y> E

. . . x 5

f

i / j = l

- T l (1.2)

v (e fi) = e \ X

E>1 + ^ EEra

«€* es

(1.3)

We sh a ll c o n s id e r th e ab o v e M id z u n o stra te g y . T h e e s tim a to r v ( e n ) can o fte n ta k e n e g a tiv e v alu es, an u n d e s ira b le p r o p e r ty for a v a ria n c e e s tim a to r . R ao (1972, 1977) an d C h a u d h u r i (1976) c o n s id e re d n o n -n e g a tiv e u n b ia s e d e s tim a to r o f V ( e n ) .

V ija y a n (1 9 7 5 ) a n d R ao (1 9 7 9 ) s tu d ie d th e n e c e s s a ry f o rm o f a n o n -n eg a tiv e q u a d r a tic u n b ia s e d e s tim a to r ( n n q u e ) o f m e a n s q u a re e r r o r (M S E ) o f a lin e a r u n biased e s tim a to r o f T . T h e ir r e s u lt m a y b e s ta te d as follow s:

T h e o r e m 1 . L e t Y = bSiY i , b , t = 0 for i £ s , b e a lin e a r e s tim a to r o f T . If M S E (Y ) = 0 w h e n Yi = cto,-, i = 1, . . . , N , u?,’s b e in g so m e k n o w n c o n s ta n ts a n d c an a r b itr a r y c o n s ta n t, th e n

N

M S E ( Y ) = - £ W V j ( Z i - Z j f d i j i <j = 1

(1.4)

w here Z{ — Y i / w i, <fi;- = E ( b Si — l ) (6Sj — 1). F u r th e r , a n n q u e o f M S E ( Y ) is n e c essarily o f th e fo rm

m ( ^ ) = ~ Y ^ w i wi ( Zi ~ zi )2e*J'(s ) (1.5)

(3)

w here

E ( eiJ( s ) ) = dij (1 .6 )

and d e n o te s ^ ^ . I t m a y b e n o te d t h a t th e e q u a tio n (1 .5 ) only p ro v id e s a

> i < j < e ■»

n ecessary c o n d itio n for n n q u e o f M S E (Y ). H ow ever, all e s tim a to r s o f th e fo rm (1 .5 ) are n o t n ecessarily n o n -n e g a tiv e , i.e. th e co n d itio n (1 .5 ) is n o t su fficien t to e n s u re n o n -n e g a tiv ity .

M u k h o p a d h y a y a n d V ija y a n (1990) in v e s tig a te d e x p lic itly th e d iffe re n t fo rm s o f n n q u e o f V ( y ) . W h e n Y = bs Y{ is u n b ia se d ,

= E ( b s% bs .) - 1 = hij - 1 (sa y ) . Now, fro m (1.4),

= ( i- 7 )

i<j =1

w h ere gij = WiWj(Z{ - Z j ) 2. O n e m a y th u s g e t d iffe re n t fo rm s o f n n q u e o f V ( Y ) as

« « = £ S i }l (t) - E S i jO ' - < = 0,1,2,3 , (1.8) w h ere

a n d

**'■Si* =**

¹ ^M

ih-s'

²^{p (s)}

<110>

A j? = — - _7Tjj (1-1 1)

(3) = h j j P j s | » , j )

•J p(s) ’

^[ ^’

a n d sim ila rly fo r 1^ , 1^2\ 1^3\ w h en 7r,;- = p ( s ) a n d P ( s | i , j ) d e n o te s th e c o n d itio n a l p ro b a b ility o f s e le c tin g s g iv e n t h a t i a n d j w ere s e le c te d a t th e first tw o d ra w s . In p r a c tic e , m a n y o f th e s e 16 e s tim a to r s v k i w o u ld co in cid e.

I t m a y b e n o te d , how ever, t h a t th e e s tim a to r s , v ^ i ( k , £ = 1 , . . . , 4 ) d o n o t fo rm a n e x h a u s tiv e s e t o f e s tim a to r s o f th e fo rm (1 .5 ). P a d m a w a r (1982) h a s g iv e n o th e r (m o re co m p lex ) e s tim a to r s o f th is fo rm . H ow ever, th e e s tim a to r s (1 .8 ) a re in te r e s tin g fo r th e ir sim p licity .

(4)

2 . N N U - V A R I A N C E E S T I M A T I O N F O R M I D Z U N O S T R A T E G Y

Since r ( f / < ) = 0, for t/.aar,-,

i =

1 , . . . ,

N,

it follows from (1.4) that for Midzuno s t r a t e g y

N

w here

* '< « > = E . E ^ j 1 - ^ E ^ 7

■<.7 = 1 I

co- = I T ~ 7 - l X‘XJ'

(².¹)

D ifferen t e s tim a to r s v u { k (. = 0 , 1 , 2 , 3 ) s a tis fy in g t h e n ec e ssa rily n o n -n e g a tiv ity c o n d itio n s (1 .5 ), (1.6) are:

d , = i>u = v31

{N

- 1 ) X

(n - l)a r,

^2 — ^20

^ 3 = v 23 = *>21

V A - V n

I

I n — 1 A l2 x ,

v - v A' 2 D5 = V 0 0 = ^ C 0 —

\

M j

x V -

- 1

^6 — V01 = ^03 _ 2^ C*J---

x

M i ___1 x

X

E f

V7 = V02

V S = VlO

(5)

= 1 3

l>9 — 1^22

M i3 '.J )

Of th e 16 p o ssib le e s tim a to rs , n in e a r e d is tin c t, d e n o te d as Vi . . . , vg.

R ao a n d V ija y a n (1977) c o n s id e re d th e e s tim a to rs v, a n d vg a n d s tu d ie d th e ir stabilities a n d p ro b a b ility o f g e ttin g a n e g a tiv e value e m p iric a lly .

In th is n o te we consider th e p e rfo rm a n c e o f all th e n in e e s tim a to rs v i , . . . , v g em pirically. 2 2 p o p u la tio n s , o f w hich 10 a re show n in ta b le 1, w ere c o n s id e re d , including th e 14 p o p u la tio n s c o n s id e re d by R a o a n d V ija y a n (1 9 7 7 ). T h e c a se s n = 3, 4 a n d 5 w ere in v e stig a te d . F o r th e cases n = 4,5 to save c o m p u te r tim e , sa m p le s were d ra w n fro m m odified p o p u la tio n s , w here th e p o p u la tio n s re m a in u n c h a n g e d if N < 10, b u t w ere re s tric te d to firs t 10 u n its if N ex c ee d ed 10.

S ince v$ w as fo u n d to hav e s m a lle r v a ria n c e ^ (1)4) = V4 in m a n y of th e cases, efficiency, e,- = V4/V ; o f th e e s tim a to r v; was c a lc u lte d w ith re s p e c t to w h e re Vi

= V ( v i ) , i ( # 4 ) = 1 , . . . , 9 .

As in R a o a n d V ija y an (1 977), V{ w as m o dified to a b ia s e d n o n -n e g a tiv e e s tim a to r v* as follow s:

v ‘, = Vi, w hen vis > 0,

= v , X2 if Vi, < 0.

H ere v , is th e le a st sq u a re s e s tim a te s (£se) o f V{j3, ) = £((3S — f} ) 2 u n d e r th e m o d e l

Y{ = (3xi + e j

£ { e , I * > ) = 0 , £ ( e ? | n ) = <r2 x?

£ ( e i 6 j | n x j ) = 0, i ^ j

w here Y- is a ra n d o m v a ria b le w h o se o n e p a r tic u la r v alu e is Y { , £ , V d e n o te re s p e c tively, th e e x p e c ta tio n a n d v a ria n c e o p e r a to r w ith re s p e c t to th e m o d e l a n d /3S is th e lse o f (3. T h u s

v, = n l n —, — .1t) £ ' ~ 2x f f a ~ P>x i ) 2-

v ' igj 1

T h e m o d e l is a p p r o p r ia te for s itu a tio n s w h en th e ra tio e s tim a to r is a p p r o p r ia te . T h e r e la tiv e efficiency o f v* w ith re s p e c t to v^, d e n o te d by e* jV% — V^*, (w h e re V? — M S E 4) = 1 , . . . ,9 ) , a n d th e re la tiv e b ia s b* w h ere b* = | E ( v * ) — 7 ( e fi) | / J u S E ( V j) , 3 = 1 , . . . , 9 w ere also c a lc u la te d for th e s e 2 2 p o p u la tio n s .

T a b le 2 p re s e n ts th e p ro b a b ilitie s p,- o f ta k in g n e g a tiv e v alu es (given b y th e rela tiv e fre q u e n c y o f n u m b e r o f sa m p le s y ie ld in g n e g a tiv e v a ria n c e e s tim a te s ) a n d th e r e la tiv e efficiency e* o f th e e s tim a to r s v ;(i = 1 , 4 , 5 a n d 9) for sa m p le s o f sizes

(6)

ii = 1 a n d n — 5 d ra w n fro m th e 10 n a t u r a l p o p u la tio n s lis te d in t a b le 1. T he full d e ta ils for a ll th e 22 p o p u la tio n s for n = 3, 4 a n d 5 a r e a v a ila b le w ith th e authors.

3 . D I S C U S S I O N

T h e follow ing co n clu sio n s m a y b e d ra w n fro m th e d e ta ile d ta b le s :

For n = 3, v4 can be c o n s id e re d to b e a lm o st n n u e o f V ( e n ) . I t h a s g o t uniformly lower p r o b a b ility o f b ein g n e g a tiv e th a n all th e o th e r e s tim a to r s c o n s id e re d . Then com e t’6, V7, , t>9, vs, V3, r j in th e o rd e r o f d e c re a sin g d e s ira b ility in te r m s of taking n e g a tiv e v alu es m o re fre q u e n tly (a s m e a su re d b y th e n u m b e r o f p o p u la tio n s for w hich th e y a re n o n -n e g a tiv e alw a y s a n d th e low er a n d u p p e r lim its o f values of p ro b a b ilitie s in case th e se a re n o n -z e ro ). T h e e s tim a to r V4 is a g a in , in general, the m o st efficien t o f all th e e s tim a to r s co n sid ered . T h is s u g g e s ts t h a t V4 is th e most p re fe ra b le o n e, b o th fro m th e p o in t o f view o f n o n - n e g a tiv ity a n d efficiency.

For th e m o d ifie d e s tim a to rs , re la tiv e b ia s o f v % is a lm o s t a lw a y s z e ro , t>g takes the n ex t p o s itio n . A g ain is, in g e n e ra l, th e m o st efficient o f all th e b ia s e d estimators.

T h is s u g g e s ts t h a t is th e m o s t p re fe ra b le of all th e m o d ifie d e s tim a to r s . For n = 4, V4 is ag a in se en to be ta k in g n o n -n e g a tiv e v alu es m o re frequently th a n th e o th e r e s tim a to r s , v2 is se en to b e ta k in g n e g a tiv e v a lu e s m o s t frequently.

E x c e p t for v \ , v$ a n d vg, it is fo u n d to b e a lm o s t alw a y s m o re efficien t th a n the ot h e r e s tim a to r s . T h e sa m e tr e n d is o b se rv e d in r e s p e c t o f th e m o d ifie d estimators also.

For n = 5, 1)5 is seen to b e ta k in g n o n -n e g a tiv e v a lu e s m o re fre q u e n tly than th e o th e rs . In cases it ta k e s n e g a tiv e values, th e p r o b a b ility o f ta k in g negative values is se en to b e u n ifo rm ly low er ( b a r rin g o n e ca se ) t h a n th e o th e r s . T h e next d e s ira b le e s tim a to r s a re t>j a n d vg. A g a in v-i ( a n d also vg) is se e n t o b e th e least p re fe ra b le o n e in te rm s o f n o n -n e g a tiv ity . A lso, V5 is m o s t efficien t o f all th e other e s tim a to r s . T h u s vs is th e m o s t d e s ira b le o n e b o th fro m t h e p o in t o f non-negativity a n d s ta b ility . T h e sa m e t r e n d is o b se rv e d fro m th e v a lu e s o f th e b ia s ratio s. For coefficient o f v a r ia tio n o f x less t h a t 15% , all th e e s tim a to r s a r e a l m o s t alw ays non

n e g a tiv e for all valu es o f n . T h e r e is se en to b e a c o n s id e ra b le r e d u c tio n in th e value o f V f o v er Vi th ro u g h o u t.

T h e a b o v e a n a ly a is su g g e sts t h a t :

(i) F or n = 3, 4, v4( v ^ ) is th e m o s t p re fe ra b le a m o n g {v,-( f *), i = 1 , . . . , 9}.

(ii) F o r n = 5, v s (v $ ) is th e m o s t p re fe ra b le a m o n g {j/,-(t>,*), i — 1 , . . . , 9}.

(iii) T h e e s tim a to r v ^ v ^ ) is th e m o s t u n d e s ira b le on e.

I t is su g g e ste d th a t for la rg e values o f n ( > 5),i>5(i>j|) s h o u ld b e u s e d , while for sm a ll v a lu e s o f n ( < 4), ^4(^4) s h o u ld b e used, sp e c ia lly if th e c v ( x) is low , say, less th a n .20. H ow ever, if cu(a;) is m o d e ra te to larg e, th e n 115 m a y b e u s e d fo r anysize of th e sa m p le .

(7)

A C K N O W L E D G E M E N T S

C o m p u tatio n al h e lp receiv ed fro m T h o m a s H an so n is a c k n o w le d g e d . P a r tia l support from N S E R C G r a n t A 3111 is g r a te fu lly a c k n o w le d g e d . W e a re th a n k fu l totherefrees for u se fu l c o m m e n ts to im p ro v e th e q u a lity o f t h e p a p e r .

R E F E R E N C E S

(1) C h a u d h u ri, A . (1 976). A n o n - n e g a tiv ity c r ite rio n fo r c e rta in v a ria n c e e s tim a to r s . M e lrik a , 2 3 , 201-205.

(2) C o c h ra n , W .G . (1977). S a m p l i n g Techniques. 3 rd e d n . W 'iley, N ew Y ork.

(3) K o n ija n , U .S. (1973). S ta ti s t i c a l T h e o r y o f S a m p l i n g S u r v e y D esign a n d A n a ly s is . N o rth H o lla n d , A m s te rd a m .

(4) L a h iri, D .B . (1951). A m e th o d o f s a m p le se le c tio n p ro v id in g u n b ia s e d r a tio e s tim a tio n . B u ll. I n te r . S ta tist. In s t. , 3 3 , 133-140.

(5) M id z u n o , II. (1950). A n o u tlin e o f th e th e o ry o f s a m p lin g s y s te m s . A n n . In st. S ta tist. M ath., 1, 149-156.

(6) M id z u n o II. (1952). O n th e s a m p lin g s y s te m w ith p r o b a b ility p r o p o r tio n a l to su m of sizes. A n n a l s o f the I n s t i t u t e o f S ta ti s tic a l M a t h e  m a tic s , 3, 99-107.

(7) M u k h o p a d h y a y , P. a n d V ija y a n , K . (1990). O n n o n -n e g a tiv e u n b ia s e d e s tim a to r o f q u a d ra tic fo rm s in fin ite p o p u la tio n s a m p lin g . T e c h  nical R eport No. 41, U n iv e rsity o f W e ste rn A u s tr a lia , P e rth . (8) M u rth y , M .N . (1967). S a m p l i n g T h e o r y an d M e th o d s . S ta tis tic a l P u b

lish in g S ociety, C a lc u tta .

(9) P a d m a w a r, V .R . (1982). O p tim a l s tr a te g ie s u n d e r s u p e r p o p u la tio n m o d els. U n p u b lish e d P h .D . th e s is s u b m itte d to th e In d . S ta t. I n s t t . , C a lc u tta .

(10) R a j, D . (1972). T h e D esig n o f S a m p l e S u r v e y s . M c G ra w Hill B o o k C o m p a n y , N ew Y ork.

(11) R a o , J .N .K . (1 9 6 3 ). O n th r e e p ro c e d u re s o f u n e q u a l p r o b a b ility s a m p lin g w ith o u t re p la c e m e n t. J. A m e r . S ta tist. A s s o c ., 5 8 , 202-215.

(12) R a o , J .N .K . (1 9 7 9 ). O n d e s ig n in g m e a n s q u a re e r r o r s a n d th e ir n o n n e g a tiv e u n b ia se d e s tim a to r s . J. I n d i a n Soc. A g r i. S ta ti s t. , 1 7 , 125-136.

(13) R a o , J .N .K . a n d V ija y an , K . (1 9 7 7 ). O n e s tim a tin g th e v a ria n c e in s a m p lin g w ith p ro b a b ility p r o p o rtio n a l to a g g re g a te size. J. A m e r .

S ta tist. A s s o c ., 7 2 , 579-584.

(14) R a o , T .J . (1 9 7 2 ). O n th e v a ria n c e o f th e r a tio e s tim a to r . M e l r i k a, 1 8 , 209-215.

(15) R a o , T . J . (1 977). E s tim a tin g th e v a ria n c e o f th e r a ti o e s tim a to r fo r th e M id z u n o -S e n s a m p lin g sc h em e . M etrika , 2 4 , 203-208.

(16) S en, A .R . (1 9 5 2 ). P re s e n t s t a t u s o f p ro b a b ility s a m p lin g a n d its u se in

(8)

th e e s tim a tio n o f a c h a ra c te ris tic . ( A b s tr a c t ) . Econom etrica, 20, 103.

( IT) S u k h a tm e , P.V. a n d S u k h a tm e , B .V . (1 9 7 0 ). S a m p lin g T heory o f Surveys with A p p lic a tio n s. 2 n d . ed n . A sia n P u b lis h in g H ouse, B om bay.

(1 8 ) V ija y an , K. (1 9 7 5 ). O n e s tim a tin g th e v a ria n c e in u n e q u a l p ro b ab ility sa m p lin g . J. A m e r . S ta tist. A s s o c., 7 0 , 713-716.

( 19) Y a m a n i, T . (1 9 6 7 ). E l e m e n t a r y S a m p l i n g T heory. P re n tic e lla ll, New Jersey .

T a b le 1 L I S T O F P O P U L A T I O N S I'u p ijl.v

t ion S o u r c e y ^X N c v ( x ) f

1 M n r th v { I!1*)?), p . 2 2 8 o u t p u t n u m b e r o f w o r k e r s 8 0 .0 5 6 0 3 0 S 0.822

K o m j n ( 1 9 7 3 ) , p . 49 f o o d e x p e n d it u r e

t o t a l e x p e n d it u r e

16 0 .0 7 8 0 .1 1 1 0.954

3 M tir th y (1 9 67), p. 178 ( v illn ^ o 1 -1 0 )

a r e a u n d e r p a d d y g e o g r a p h ic a l a r e a

10 0 .0 6 5 0 .3 4 4 0.254

•\ K o m jn (1 £*73),

p. 3 8 9

m e a s u r e m e n t o b t a in e d in r e - in t e r v ie w

m e a s u r e m e n t o b t a in e d in fir st in t e r v ie w

10 0 .1 6 0 0 .1 5 1 0.998

5 S u k h a t m e S u k h a t m e ( 1 9 7 0 ) , p. K,6

n u m b e r o f b a n a n a b u n c h e s

n u m b e r o f b a n a n a p i t s

2 0 0 .1 7 5 0 .2 4 0 0.774

<; Y a m a n e ( 1 9 6 7 ) p. 33*1

n u m b e r o f v a c a n c ie s

n u m b e r o f a p a r t m e n t s

10 0 .3 5 3 0 .3 4 4 0.983

M i n t l iy (1 & 6 7 ), p. 132 ( b lo c k n o . 7)

t im b e r v o lu m e s t r ip le n g t h 13 0 .3 6 S 0 .3 5 1 0.945

ft S u k h a U n e U S u k h a t e m e ( 1!>70), p . 51

a r e a u n d e r r ic e

t o t a l c u l t i  v a t e d a r e a

10 0 .3 9 1 0 .3 9 7 0.874

•4 K a j ( l ' J 7 2 ) , p . 7 0

n u m b e r o f c a t t l e

n u m b e r o f fa r m s

15 0 .4 0 2 0 .4 2 3 0.894

10 R a o ( 1 9 7 3 )

p. 2 0 7

c o r n a c r e a g e in 1 9 6 0

c o r n a c r e a g e in 1 9 5 8

14 0 .4 7 2 0 .3 7 9 0:926

(9)

-negative Variance E s tim a ti o n

T a b le 2. P ro b a b ility o f ta k in g n egative values a n d relative efficiency o f th e e s tim a to rs v l , v4, v5, v9 for sam ples ___________ o f sizes n = 4 a n d 5 for 10 n a tu ra l p o p u la tions.

n = 4 P op.

si.

no.

Pi Pa P5 P9

rela tiv e efficiency o f t>;

e i es eg

1 .0 0 0 .0 0 0 .0 0 0 .0 0 0 1.006 1 .0 1 0 0.999

2 .0 0 0 .0 0 0 .0 0 0 .0 0 0 0.987 0.993 0.978

3 .0 0 0 .0 0 0 .0 0 0 .0 0 0 1.009 1.014 1.001

4 .0 0 0 .0 0 0 .0 0 0 .0 0 0 1.062 1.064 1.034

5 .0 0 0 .000 .0 0 0 .0 0 0 0.934 0.975 0.928

6 .0 0 0 .000 .0 0 0 .0 0 0 1.369 1.266 1.473