The Power of Tests Based on the Length of Rune

(1)

A C T A U N I V E R S I T A T I S L O D Z I E N S I S F O L IA OECONOMICA 48, 1905_____________________

C ze slaw Dom nriski*, A n d rz e j Tom aszewlcz

THE POWER OF T EST S BASED ON THE LENGTH OF RUNS

1. In t r o d u c t io n

The p a p e r le a c o n t in u a t io n o f the r e s e a r c h c o n c e rn in g the f o llo w in g t e s t e based on the le n g th o f runs (s e e [ 2 ] ) :

- a t e s t based on the maximum le n g th o f run s on one of the median ( S A ) ,

a t e s t based on a s m a lle r among the maximum le n g th s of runs above and below th e median ( S ^ ) ,

a t e s t based on a b ig g e r among the maximum le n g t h s of runs above and below the median ( S ^ ) .

These t e s t s c o u ld be a p p lie d I n v e r i f i c a t i o n o f h yp o th e se s on in d ep en d en ce o f the sequence o f o b s e r v a t io n s , In d e te rm in a

t io n o f th e tre n d In th e tim e s e r i e s , in v e r i f i c a t i o n o f the h y p o th e s is on the l i n e a r i t y o f the e c o n o m e tric model w it h one o r more In d e p e n d e n t v a r i a b l e s .

The aim o f t h i s p a p e r i s to fo rm u la te some c o n c lu s io n s con c e r n in g th e power o f t e a t s based on th e le n g th o f runs w hich a re a p p lie d in v e r i f i c a t i o n o f the h y p o th e s is on in dependence o f sub seq u en t e le m e n ts in a sam p le.

I Ye s h a l l c o n f in e o u r c o n s id e r a t io n to th e ca se o f s t a t i o n a r y Markov c h a in w it h two s t a t e s w h ich a re t r a d i t i o n a l l y denoted

' * as A and B and th e t r a n s i t i o n m a tr ix PAA PAB _■ 1 cr O H L « o ' P BA PBB «1 1 - q * * 0 r . , L o c t u r e r a t th e I n s t i t u t e o f E c o n o m e tric s and St8 ' t l s t i c s . U n i v e r s i t y o f Ł ó d ź .

(2)

t.pt Pn ^ be a d i s t r i b u t i o n of t h i s c h o in f o r each ^ e ft « * { ‘ V V ’ 0 < % < 1 * 0 < q l < 1 i and l e t An " { A ' B } n be a s e t o f a ll- n - o le m o n t seq u ences formed from e le m e n ts A and 13. Thus, wo s h a l l c o n s id e r th e p r o b a b i l i t y sp a ces

( 1 ) M„ v - ( a .2 _{n ,J- "} _* Pm _{n , ^}a) f o r * e e .

The fo rm u la te d c o n c lu s io n s a re based on the n u m e r ic a lly de term in ed power o f t e s t e b a s in g on the d i s t r i b u t i o n o f the leng th- of ru n s f o r n * 1 ,2 , . . . .

100

and on e e v e r o l dozen p a i r s chosen from th e s e t ft. The c o m b in a t o r ia l fo rm u la e o f p r o b a b i l i t y co n n ecte d w it h run s d i s t r i b u t i o n p re s e n te d in the l i t e r a t u r e ( c f . [ 3 ] , [ 4 ] ) a re in c o n v e n ie n t f o r n u m e r ic a l c a l c u l a t i o n s . I t i s more e f f i c i e n t to use r e c u r s iv e fo rm u la e e s p e c i a l l y in the caso when th e c a l c u l a t i o n s a re made f o r sub seq uent v a lu e s o f n,

2. R e c u r s iv e Fo rm u lae f o r U n i v a r i a t e Runs D i s t r i b u t i o n s

Wo s h a l l a s s ig n to each sequence

u * ( xA , Xg# . . . .

the f o llo w in g num bers:

Na(u ) - number o f e le m e n ts A in sequence u , La (l>) number o f rune c o n s is t in g o f e le m e n ts A , L ( u ) - t o t a l num b er.of ru n s .

Assume th a t seq u ences u e ^ a re th e r e a l i z a t i o n s o f the s t a t i o n a r y Markov c h a in w it h th e t r a n s i t i o n m a tr ix

p p

AA BA

P P

AB BB

w here 0 < PA B , P ^ < 1. H ence, the s t a t i o n a r y p r o b a b i l i t i e s

a re g iv e n by the fo rm u la e „

P P

( 2 ) pA « R (X « A ) - p---v V ~ * P B “ P ^X 1 " B ) “ P--- + ~W ~

A J AB BA a J AB + BA

(3)

Under th e se assu m p tio n s th e p r o b a b i l i t y d i s t r i b u t i o n on the a e t can be p re s e n te d by the fo rm u la

1 (Na - L ) L ( L - L ) (n - N - L ♦ La ) ,

AB BA

where n d e n o te a sample s i z e , ” th e number o f A - ty p e o-le m en ts in the sam p o-le, l (w) - t o t a l number of ru n a , l a^ ' number o f runs c o n s is t in g of A - typ e e le m e n ts .

E q u a tio n ( 3 ) r e s u l t s im m e d ia te ly from th e i d e n t i t y

( 4 ) P ( u ) - P C X j- X j) p C X jj- X jJX j. X j) . . . p ( xn ^ n l xn- l " xn - i )

a f t e r ta k in g in t o a c c o u n t e q u a tio n ( 2 ) ,

D e term in e f o r g iv e n n th e f o llo w in g rundom v a r i a b l e s ;

f>A - th e maximum le n g th o f run » c o n s is t in g o f A - typ o e l e m ents,

S _ - the maximum le n g th of runs c o n s is t in g o f e - ty p e e le - D

m ents,

s o * min { s a* s a í* S G - max { s A . S Q} .

H avin g t h i s n o t a t io n we can fo rm u la te th e f o llo w in g theorem s whose p r o o f s , as o f l i t t l e i n t e r e s t , a re o m itte d . T h e o r e m 1. The d i s t r i b u t i o n o f v e r i a b l e SA d e te rm in ed on M » i s e x p reaso d by the r e c u r s iv e fo rm u la n | v ( 5 ) P ( S A - s ) - Q ^ in . s ) ♦ Q ^ tn .s ), where n-e Q ^ C n .s) - Y ] Q j ( n - v , s ) q0 ( l - q 1) v " 1, v-1 8-1 8 Q i ( n , 8 ) • /..J ( n - s , v ) qA ( l - q 0 ) a_1 ♦ 1 v »0 w «l ' ' under i n i t i a l c o n d it io n s q£(

0

_.

0

_{> . Qj(O.O) . - - Í - .}

(4)

T h e o r e m 2. The d i s t r i b u t i o n o f v a r i a b l e S „ d e te rm ln -6 ed on ^ l a ex p ro ssed by th e r e c u r s i v e fo rm u la ( 6 ) P ( S G- s ) - Q ® (n ,s ) ♦ Q j ( n . s ) , where s Q ® (n ,s ) - X j Q? h ( n - s , v ) qh ( l - q 4 h ) 8" 1 v

«0

" i “ n * X Ql - h (n “ w' 8^ w «l f o r h ■ 0 , 1 , a t i n i t i a l c o n d it io n s ■fi/n \ _ nG/ Q“ ( 0 , 0 ) - Q ^(O .O )

*

0**1

The d i s t r i b u t i o n o f v a r i a b l e S0 can be d e te rm in e d on th e ba s i s o f th e f o llo w in g r e l a t i o n

( 7 ) P ( s < s ) - P ( S A < s ) ♦ P (S _ < s ) - P ( s _ c s ) .

3. Power E v a lu a t io n

On th e b a s is o f th e r e c u r s iv e fo rm u la e p r e s e n te d in § 2 th e pow er o f random ized t e s t s was d e te rm in e d n u m e r ic a lly f o r n « 1 , 2 , . . . » 100 end f o r some p a i r s ( p . j o ) , where

( 8 ) p - pa ^ - ^ q ^ » e m 1 -

0 V

The p ro c e d u re was as f o l l o w s t A A - i f S < ea - 1 , th e n HQ »• g » 0 i s a c c e p te d , - i f S > Ba , th en Hq I s r e j e c t e d in f a v o u r o f H j i j > 0 , - i f S * ■ » £ - ! , th e n H0 i s a c c e p te d w it h p r o b a b i l i t y r „ . L e t f o r th e d e te rm in e d n , p , s i g n i f i c a n c e l e v e l a and g * o C'9) Fa ( b ) " P ^S A < 8 ^» 8 “ ° » 1 *

(5)

The c r i t i c a l v a lu e o f the t e s t based on SA s t a t i s t i c w i l l be t h e r e f o r e

(1 0 ) s^ • min | s j F a ( » ) > 1 - a } •

To t h i s v a lu e th e ran d o m iz in g p r o b a b i l i t y c o rre s p o n d s

w hich was p re s e n te d in [ 2 ] .

I n the same way random ized t e s t s based on s t a t i s t i c s S g , Sp and S G w ere d e te rm in e d .

The r e s u l t s o f power c a l c u l a t i o n f o r the t e s t s S ^ , S Q ( v e r i f y i n g the h y p o th e s is HQ : £ « 0 ) f o r a ■ 0 .0 5 , p * 0 . 5 , 0 . 7 , 0 . 9 . and £ - 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 .9 end n - 5 , 10, . . . , 100, as w e l l as f o r a s im p le a l t e r n a t i v e h y p o th e s is H j i g ■ ^ p re s e n te d in T a b le s 1-4, a llo w to fo rm u la te th e f o llo w in g con c lu s i o n s .

1 . The power o f t e s t s b e in g c o n s id e re d l e th e h ig h e s t f o r p ■ 0 . 5 . (T h is io co n flrra o d by a lr e a d y quoted r e s u l t o b ta in e d by B a t e m a n [ l ] ) .

2. The t e a t based on SA s t a t i s t i c p ro ve d to be s tr o n g e r than the t e s t based on S 0 s t a t i s t i c f o r p > 0 , 5 , e x c lu d in g th e c a s e s o f v e r y s tro n g c o r r e l a t i o n ( g > 0 . 7 ) .

3. Among the t e s t s o f th e ru n s le n g th the most f r e q u e n t ly used t e s t S G p ro ve d to be s tr o n g e r th an t e s t s S A and S B , e x c lu d in g th e c a s e s of b ig asym m etry ( g > 0 . 6 ) and o f v e r y s tro n g au t o c o r r e l a t i o n ( f > 0 . 7 ) .

4 . The t e s t S G i s s t r o n g e r than the t e s t SQ o n ly f o r ;p c lo s e to 0 .5 and f o r n o t v e r y s tro n g a u t o c o r r e l a t i o n a t a r e l a t i v e l y s m a ll s iz e of sa m p le s.

5. W ith th e in c r e a s e o f p the d i f f e r e n c e betw een the power o f t e a t s SA and Sq, and S 0 and SQ d e c re a s e s r a p i d l y (e x c lu d in g th e c a s e e o f v e r y s m a ll n (n < 1 5 ) ) . These d i f f e r e n c e s r e main s i g n i f i c a n t o n ly in th e c a s e o f s tro n g a u t o c o r r e l a t i o n >

(6)

Power o f t e s t » o f runs S^ f o r soma p a i r s ( p . £ ) and a ■ 0 .0 5 n p - 0 ,5 p ■ C>.7 p - 0 .9 e e 6 0 .1 0 .3 0 .5 0 .7 0 .9 0 .1 0 .3 0 .5 0 .7 0 .9 0 .1 0 .3 0 .5 0 .7 0 .9 5 68 118 190 289 420 59 81 109 148 184 52 57 62 67 73 10 74 145 243 360 467 69 124 214 353 562 55 67 81 98 118 15 78 165 292 436 524 71 134 236 386 586 58 79 107 143 190 20 02 184 336 505 583 73 146 265 432 625 62 93 140 207 306 25 84 197 371 560 634 75 157 294 485 668 65 110 183 302 492 30 87 212 406 613 682 77 168 320 532 712 68 123 221 391 683 40 90 232 455 686 756 80 183 363 605 785 69 127 229 404 694 50 93 250 500 746 814 82 197 400 664 839 70 133 243 428 714 60 95 268 542 795 859 84 208 431 711 879 71 139 260 458 741 60 98 290 593 853 916 87 227 482 781 932 73 152 295 526 807 100 101 313 644 898 950 89 242 522 830 961 75 163 327 588 872 N o t e s A l l p r o b a b i l i t i e s have been m u l t i p l i e d by 1000. T a b l e 2 Pow er o f t e s t a o f run s S B f o r some p a l r a (p ,£ > ) and a «■ 0 .0 5 p - 0 .5 p - 0 .7 p - 0 .9 n e 4* 0 .1 0 .3 0 .5 0 .7 0 .9 0 .1 o • 0 .5 o • Nl 0 .9 0 .1 0 .3 0 .5 0 .7 0 .9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5 68 118 190 289 420 73 127 185 241 285 76 114 133 134 116 10 74 145 243 360 467 82 167 263 338 344 90 154 191 193 146 15 78 165 292 436 524 88 194 319 415 401 96 180 238 246 178 20 82 184 335 505 583 91 214 365 482 455 102 207 284 299 209

(7)

T a b le 2 (c o n t d .)

1

2

3 1 4 5 1

6

7

8

9

10

11

[ 1 ] B a t e m a n G, ( 1 9 4 8 ) ; On th e Power F u n c t io n o f the Long e s t Run as a T e s t f o r Randomness in a Sequence o f A l t e r n a t i v e s " B io r a e t r ik a " 35, p . 97-112.

[ 2 ] 0 o n a r t 3 k l C . , T o i o a s z e w i o z A , ( i 9 6 0 ) , Va r i a n t s o f T e s ts Based on th e Len g th o f Rune. P a p e r p re s e n te d a t the c o n fe re n c e "P ro b le m s o f B u il d in g and E s t im a t io n o f L a rg e E c o n o m e tric M o d e ls“ , P o l a n lc a Z d r 6 J .

[ 3 ] M o o d A. M. (1 9 4 0 ), The D i s t r i b u t i o n T h e o ry o f R u n s. Ann*, o f M ath. S t a t i s t . 11, p . 367-392.

[ 4 ] 0 » a t o d P . S . (1 9 5 8 ); Runs D e term in e d in a Sam ple by an A r b i t r a r y C u t, B e l l System Tech n. O our. 37, p . 55-58.

(9)

C z e sła w Dom ański, A n d rz e j Tom aszewlcz

MOC TESTÓW OPARTYCH NA DŁUGOŚCI S E R I I

A r t y k u ł d o ty c z y a n a l i z y mocy te s tó w o p a r ty c h na m aksym alnej d łu g o ś c i a e r l l z je d n e j s tr o n y m ediany ( S . ) , m n ie js z e j z maksy m alnych d łu g o ś c i s e r i i z k a ż d ej s tr o n y m eolany ( S ) , w ię k s z e j z maksymalnych d łu g o ś c i s e r i i z k a ż d e j s tr o n y m ediany w e r y f i k u ją c y c h h ip o te z ę o n ie z a le ż n o ś c i k o le jn y c h elem ontów w p r ó b ie . W ś w i e t l e przeprow adzonych badań uzyskano m iędzy in n ym i n a s tę p u ją c e w n io s k it t e s t S G o k a z a ł s ię m o c n ie js z y od te s tó w S A i Sr, w yjąw szy p rz y p a d k i d u że j a s y m e t r i i i a u t o k o r e l a c j i ; t e s t Sq J e s t m o c n ie js z y od t e s t u S D t y l k o w przypadku p b l i s k i c h 0 ,5 i m a łs j a u t o k o r e l a c j i ; moc rozw ażanych te s tó w J e s t n a jw ię k s z a d la p ■ 0 ,5 .

The Power of Tests Based on the Length of Rune

100

0

0

«0

*

*0 * V

1

2

6

8

10

11

12

— —1

212

100

686

360

110

100

122

0

0 .1

0 .1

0 .1

66

21

102

10

120

202

211 122

100

20

2 11

102

101

1 1 1

020

100

S

0 .1

73

133

100

0 V