TESTING HYPOTHESIS ON STABILITY OF EXPECTED VALUE AND VARIANCE
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1 B A D A N A O P E R A C Y J N E D E C Y Z J E N 6 Gzeoz KOŃCZAK* Jausz WYWAŁ* ENG HYPOHE ON ABLY O EXPECED VALUE AND VARANCE he smple samples ae depedetly tae om omal dstuto. he two utos o the sample meas ad sample aaes ae osdeed. he desty utos o these two statsts hae ee deed. hese statsts a e appled o ey the hypothess o stalty o expeted alue ad aae o omal dstuto osdeed e.. statstal poess otol. he tal alues o these statsts hae ee oud us umeal teato. he tales wth appoxmated tal alues o these statsts hae ee peseted. Keywods: desty uto sample aae test statst umeal teato statstal poess otol. toduto Oe o the polems o statstal poess otol s osdeed. t s a poedue so alled otol hats o moto stalty o the expeted alue ad the aae o daost aales. We assume that du the st peods the mea alues o a daost aale ae the same ut uow. he same stuato s wth the aae o the aale. he uased estmatos o the expeted alue µ ad the aae σ ealuated o the ass o data oseed the st peods ad the peod ae X Ŝ ad X ad espetely. Ou polem s the ollow: s the poess haatezed y the daost aale stale all o the peods? yes the dstaes: X X o should ot e sat. uh a polem s osdeed e.. []. Moe omally we hae the polem o test the hypothess * Depatmet o tatsts Katowe Uesty o Eooms ul. Boua -6 Katowe. oz@ae.atowe.pl wywal@ae.atowe.pl
2 7 G. KOŃCZAK J. WYWAŁ H : E X E X ad E E. We ae o to ostut a test statst o ths hypothess the ext paaaphs.. Bas detos ad popetes Let J a e a olum eto osst o a elemet eual oe ad let a e a ut matx o deee a. Moeoe let X [ X X... X... X ] whee X [ X X...X j...x ] j. We wll osde the ollow statsts: o X X j j XJ X X XJ whee j X j X j XMX M J J j whee M s the lo-daoal matx o deee ~ X X XMX M [ M ] XMX N M J J whee s the sample aae wth the -th oup Ŝ s the mea o oup aaes ad ~ s the aae etwee oups. Moeoe let us ote that M M M M ad N N. he patula ase o the theoem o depedee o the uadat o lea oms o omal etos oespod to ou polem s as ollows see [] pp. ad ext.
3 est hypothess o stalty o heoem. Let X hae a o-sula omal dstuto NµJ σ ad QA XAX QB XBX L Xa whee a s a olum o-adom eto A B ae symmet ad o-adom mates o deee eah. he the set o eessay ad suet odtos o Q A ad Q B to e depedetly dstuted s a AB O o ABJ O. he uadat om Q A ad the lea om L ae to e depedetly dstuted ad oly Aa O. O the ass o ths theoem we a show that the statsts the ollow pas ~ ae depedetly dstuted X ~ X. Moeoe the statsts X ae depedet ad they ae depedet o eah o the statsts X ~. ad he otaed esult ad the well ow detos let us dee the ollow dstutos: ad Moeoe X X Z : N. σ U U ~ : χ σ χ σ : U : χ σ. O the ass o these expessos we hae ~ Z σ : 5 Z σ : 6 : 7 ~ : 8
4 G. KOŃCZAK J. WYWAŁ 76 whee m deotes the well ow dstuto wth ad m deees o eedom ad the ollow desty uto: m m m m m. 9. he statsts Q ad Q Let us osde two statsts Q ad Q e as ollows ~ Z Q ~ Z Q whee ae deed y expessos 5 8 espetely. O the ass o the peous esults the dstutos o the adom aales Z U U ad U ae depedet. ally ths ad the at the samples ae depedet lead to the oluso that the dstutos o the statsts ad as well as the dstutos o ad ae depedet. he desty uto o adom aale s as ollows: whee A x A x x A.
5 est hypothess o stalty o he desty uto o adom aale s:. he desty utos o ad ae as ollows:. 5 whee ad. Let us dee the dstuto o adom aale. the we hae ad d d ad a. 6 Whe the we hae ± ad d. ] the ad o we hae.
6 G. KOŃCZAK J. WYWAŁ 78 [. 7 [ the. 8 We a wte the desty uto o the ollow way. 9 W: the the desty uto o W s e y expesso. Let V e the adom aale o dstuto wth ad - deees o eedom. he desty uto o V s e y 9. Now we ae o to ealuate the desty utos o the adom aale: Q W V. he desty uto o the statst Q s as ollows d h
7 est hypothess o stalty o h whee. mlaly the desty uto o Q s deed the ollow way. d h whee s the desty uto o [
8 G. KOŃCZAK J. WYWAŁ 8 ally the desty uto h o the statst Q s as ollows [ h whee. he dstuto utos o Q o Q ae ealuated y meas o the ollow teal < ds s h Q P H. 5 o the e sae leel α the uatl α s detemed o the ass o the teal: α α ds s h. 6. Numeal omputatos A exat soluto o euato 6 s ey dult. ths stuato the uatles o Q ad Q a e oud us umeal teato see e.. [] []. he ua-
9 est hypothess o stalty o... 8 tles wee oud o thee sae leels α..5.. ale pesets uatles o the statst Q ad ale pesets uatles o the statst Q. hese uatles wee ealuated o the ase 5. hese uatles hae ee ealuated o the umes o oups om to ad o 5 5 ad. Quatles o statst Q Nume o ae leel α oups ale he aph dsplay o uatles o statst Q o sae leel α..5 ad. he ue pesets uatles o the statst Q o the sae leels..5 ad.. hese uatles ae peseted o the same ases as ale. ue pesets the same esults as ue ut o the statst Q.
10 8 G. KOŃCZAK J. WYWAŁ ale Quatles o statst Q Nume o ae leel α oups he aph dsplay o uatles o statst Q o sae leel α..5 ad. Reeees [] BRAND. tatstal ad Computatoal Methods Data Aalyss pe Vela New Yo 997. [] DAHLQU G. BJORCK A. Numeal Methods Polsh PWN Waszawa 99. [] MAHA A.M. PROVO.B. Quadat oms adom aales Mael Dee. New Yo Basel Ho Ko 99. [] HOMPON J.R. KORONACK J. tatstal Poess Cotol: he Dem Paadm ad Beyod Chapma ad Hall/CRC New Yo Lodo.
11 est hypothess o stalty o... 8 estowae hpotezy o staloś watoś ozewaej waaj W pay jest ozważae zaadee jedozesej staloś watoś pzeętej waaj. Póy poste są poeae ezależe z populaj o ozładze omalym. Rozważa sę dwe uje śedej waaj z póy. Dla ozważayh statysty zostały wypowadzoe uje ęstoś. Popoowae statysty moą yć wyozystae do weyaj hpotezy o staloś watoś ozewaej waaj dla ozładu omaleo. Hpoteza taa może yć ozważaa p. w statystyzym steowau poesem pzy ostuj at otolyh. Badzo tude jest dołade wyzazee watyl ozważayh statysty. Dlateo watoś ytyze dla tyh statysty zostały wyzazoe dla tzeh zwyle używayh pozomów stotoś α 5 dla pó o lzeośah od do z wyozystaem ałowaa umeyzeo. Zapezetowao tale watoś ytyzyh dla tyh statysty. Zapopoowae statysty wyzazoe watoś ytyze moą yć óweż pzydate do wyywaa zma w poesah poduyjyh. łowa luzowe: uja ęstoś waaja z póy test umeyze ałowae statystyza otola poesu
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