Mathematical Statistics

Transcription

1 7 75 Ode Sttistics The ode sttistics e the items o the dom smple ed o odeed i mitude om the smllest to the lest Recetl the impotce o ode sttistics hs icesed owi to the moe equet use o opmetic ieeces d obust pocedues Howeve ode sttistics hve lws bee pomiet becuse mo othe this the e eeded to detemie the simple sttistics such s the smple medi the smple e d the empiicl distibutio uctio I most o ou discussio we will ssume tht the dom smple ises om cotiuous-tpe distibutio This mes mo othe thi tht the pobbilit o two smple items bei equl is zeo Tht is the pobbilit is oe tht the items c be odeed om smllest to lest without hvi two equl vlues O couse i pctice we do equetl obseve ties; but i the pobbilit o this is smll the ollowi distibutio theo will hold ppoximtel Thus i the discussio hee we e ssumi tht the pobbilit o tie is zeo I X X L X e obsevtios o dom smple o size om cotiuous-tpe distibutio with distibutio uctio x d pd x we let the dom vibles L deote the ode sttistics o tht smple Tht is smllest o X X L X secod smllest o M lest o X X X X L X L X The pobbilit desit uctios o d c be oud usi the method o distibutio uctios Becuse is the lest o X X L X the evet will occu i d ol i the evet X occu o eve i Tht is i X X X L To detemie the distibutio o the th ode sttistic depeds o the biomil distibutio Suppose tht x o x b d b It is possible tht d/o b The evet tht the th ode sttistic c occu i d ol i t lest o the obsevtios e less th o equl to Tht is hee the pobbilit o success o ech til is x d we must hve t lest successes Thus [ ] [ ] Thus the pd o is

2 7 Hece we hve tht the pd o is b It is woth oti tht the pd o the smllest ode sttistic is b d the pd o the lest ode sttistic is b REMARK: Thee is oe ve stiscto w bsed o the multiomil pobbilit to costuct heuisticll the expessio o the pd o Accodi to the deiitio o deivtive we hve Aothe iteesti heuistic umet c be ive bsed o the otio tht the lielihood o obsevtio is ssied b the pd To hve oe must hve obsevtios less th oe t d obsevtios ete th whee d the lielihood o obsevtio t is Thee e

3 7 [ ] possible odeis o the idepedet obsevtios d is ive b the bove multiomil expessio This is illustted i iue } L L iue 7 The th ode obsevtio A simil umet c be used to esil ive the joit pd o set o ode sttistics o exmple coside pi o ode sttistics i d j whee i j To hve i i d j j oe must hve i obsevtios less th i oe t j oe t o i d j d j s i j i betwee i d j ete th j Appli the multiomil om ives the joit pd i ji j ij i j [ i ] i [ j i ] [ j ] j i j i j i i j b d zeo othewise This is illustted b iue 7 uthemoe the joit pd o L is ive b i L L L b d zeo othewise i ji } } L L i i i j j j j L iue 7 The ith d jth ode obsevtios Exmple 75-: Coside dom smple o size om distibutio with pd d CD ive b x x d x x ; x The smllest d lest ode sttistics e d E d Let u d du u u u Deie the e o the smple s R The joit pd o d is du

4 74 Mi the tsomtio R S ields the ivese tsomtio s s d J Thus the joit pd o R d S is h s 4s s s s The mil desit o the e the is ive b h h s ds o exmple o the cse we hve h 8s s ds 4 Exmple 75-: Let L 7 be the ode sttistics o dom smple o size 7 om distibutio with pd x x x Compute the pobbilit tht the smple medi is less th 6 We could id the pd o ; tht is id Howeve ote tht the pobbilit o sile obsevtio bei less th 6 is Thus 6 [ x ] x dx Exmple 75-: I X hs distibutio uctio x o the cotiuous tpe the x hs uiom distibutio o the itevl zeo to oe I L e the ode sttistics o dom smple X X L X o size the L sice is odecesi uctio d the pobbilit o equlit is i zeo Note tht the lst displ could be looed o s odei o the mutull idepedet dom vibles L U Tht is ech o which is W W L W c be thouht o s the ode sttistics o dom smple o size om tht uiom distibutio Sice the distibutio uctio o sttistic W is U is w w w the pd o the th ode h w w w w The me E W E[ ] o E W W is ive b the itel w w w dw w w w dw

5 75 L is the cumulted pobbilit up to d icludi o equivletl the e ude x x but less th Hece c be teted s dom e Sice is lso dom e is the dom e ude x betwee d The expected vlue o the dom e betwee two djcet ode sttistics is the E[ ] Also it is es to show tht E [ ] d E[ ] Tht is the ode sttistics L ptitio the suppot o X ito pts d thus cete e ude x d bove the x-xis O the vee ech o the es Thee is extemel iteesti itepettio o W Note tht equl I we ecll tht the pth pecetile π is such tht the e ude x to the let o p π p is p the pecedi discussio suests tht we let be estimto o π p whee p o this eso we deie the pth pecetile o the smple s whee p I cse p is ot itee; we use weihted vee o vee o the two djcet ode sttistics d whee is the etest itee i p I pticul the smple medi is whe is odd M whe is eve