= 2. Statistic - function that doesn't depend on any of the known parameters; examples:

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1 of Samplg Theory amples - uemploymet househol cosumpto survey Raom sample - set of rv's... ; 's have ot strbuto [ ] f f s vector of parameters e.g. Statstc - fucto that oes't epe o ay of the ow parameters; eamples: Sample Mea: Sample ace: Sample Covarace: y y y Note for sample varace a covarace must replace the mea wth a fucto of the observatos; ca't have a fucto o the uow parameters Theorem: f raom sample s from a populato wth mea a varace the the sample mea s a raom varable wth mea a varace /. Sample Mea: X X Sample ace: assume ~ N X X ca sum varaces of epeet ormals Geeral case for varace: [ ]

2 [ ] X Cov stmato - wat estmators that are ubase a effcet small varace Ubase - the mea of the strbuto of the estmator equals the parameter we're tryg to estmate: f ample - Normal Dstrbuto ~ N / show last tme so s ubase estmate mea... ; for N / ; s ubase estmate 5 ~ N ; s ubase estmate / ; /[ ] /[ ] ; s ubase estmate Moe ample - poetal Dstrbuto / f e > / F e > Mea Mea ;... ubase / mea F e. 5 mea l... base 5 ;... ubase / ; /[ ] /[ ]... s ubase Bas - [ ] b ; ubase measb ffcecy - s more effcet tha f < ample - Normal Dstrbuto /... ths s the most effcet / Purposely Choose Base stmator? - maybe f you have great gas effcecy much lower varace; eample: mea may be base but bas may be small a mea has lower varace tha mea so the trae-off may be worth whle of

3 of Mea Square rror - [ ] [ ] b MS ; traes off bas a varace; ew staar for estmate s lowest MS although sometmes ubase s more mportat Ubase - MS Relatve ffcecy - s relatvely more effcet tha f [ ] [ ] < MS MS Ubase - relatvely more effcet f < UMVU - uformly mmum varace ubase estmate; ubase estmator such that for ay other ubase estmator * * < ; f we wat a ubase estmator ths wll be the best oe to use Sample ace: or s... has smaller MS but s s ubase Show s : s comes from assumpto that mea a varace Proof: [ ]

4 Proof: Cov f 's are epeet Cov so s... ubase s / - MSs s [bs ] / - s s b s MS MS MS s / / < s a relatvely more effcet estmator of tha s base... people stll ebate whch s better to use s but s s ubase a Asymptotc Theory What happes to our statstc as the sze of our sample creases? Does our statstc coverge to the correct value... o s a coverge to as? No-Stochastc Covergece - let {b } be a sequece of real umbers. f there ests a real umber b a f for every δ > there ests a teger Nδ such that for all Nδ b - b < δ the b s the lmt of the sequece b a wrte b b rea "b coverges to b" ample - {b} b b / / b / / 7 / / / / 8... of

5 Coverges to b Statstcal Covergece - fferet types; some stroger tha others; ; ;. Almost Sure Covergece. Covergece r th mea. Covergece Probablty. Covergece Dstrbuto et b be a statstc base o a raom sample of observatos e.g. b sample mea or b sample varace use observatos of to costruct the statstc b Almost Sure Covergece - {b } coverges almost surely to b ff there ests a real umber as b such that Pr[b b] ; wrtte b b ample - { } s sequece of rv's wth < strbuto type oes't matter as Covergece Probablty - {b } coverges probablty f there ests a real umber b P such that for every ε > Pr[ b - b < ε] as ; wrtte b b ample - { } s sequece of rv's wth < a Cov ; assumptos are much weaer easer to verfy tha almost sure covergece; o assumptos about etcal strbuto or epeece Cov epeet; oly lear epeece P Covergece r th Mea - use tme seres; {b } s sequece of real value rv's; f there ests a real umber b such that [ b - b r ] as for some r > the b coverges r th r mea; wrtte b b ; q m Quaratc Mea - use r whch we usually o; b.. b ; ote b - b b - b so [ b - b ] [b - b ] b ; coverges f varace of statstc coverges to zero r ower r - b b for some r the s b b for < s < r.e. quaratc mea covergece mea covergece r Covergece Dstrbuto - {b } s sequece of real value rv's wth strbuto fuctos {F }; f F F as for every cotuty pot where F s the strbuto fucto of a rv X the b coverges strbuto to the rv X; wrtte b X ample - t N 5 of

6 Cosstet stmator - s cosstet estmator of ff P amples: s cosstet estmator of mea mea; f Cov a a < a come from a symmetrc strbuto the mea mea a s cosstet estmator of poetal Dstrbuto - use mea/l 5 a are ot cosstet estmators; they o't chage at all as sample sze gets bgger Wea aw of arge Numbers WN - as sample sze creases the sample mea coverges probablty to populato mea; sets of cotos uer whch WN hols Khch - { } s a sequece of rv's wth fte mea P Chebychev - { } s a sequece of epeet rv's wth meas a varaces ; f varaces are boue above.e. < c < a P the sample mea mus mea of the meas Marov - { } s a sequece of rv's wth meas f as the P oes't assume epeece Kolmogorov - { } s a sequece of epeet rv's; z P f lm the z z z Strog aw of arge Numbers - tereste almost sure covergece; Theorem - { } s a sequece of rv's the ff [ ] <.e. fte mea Kolmogorov - { } s a sequece of epeet rv's wth fte varaces as f < the allows mea to chage over tme but mea of sample meas approaches mea of meas Cetral mt Theorem - for a large sample from ay strbuto we ca appromate the strbuto of the sample mea wth a ormal strbuto; { } be a sequece of rv's wth s s s a s ; the staarze mea z where s s a ; z ~ N as 6 of

7 N N De Movre - prove CT where 's are epeet Beroull rv's berg-evy - prove CT where 's are wth < berg-feller - prove CT where 's are epeet wth X a s S ; for some ε > lm f s > εs Oly loog at porto of varace that s far away from mea; f that s covergg to zero.e. tals are't too bg the CT hols Multvarate CT - { } s sequece of -varate rv's s vector wth mea a varace Σ the z N Chebychev's equalty - for costat > Pr - X X/ ; probablty that you're more tha away from the mea s less tha or equal to varace over ; proves a upper bou o Pr - X... usually a very geerous upper bou much hgher tha the actual probablty wll be... wll see o HW As ower Bou - Pr - X < > - X/ Marov's equalty - Pr λx /λ for a postve rv a λ > ; ample - loo at epoetal: f e -/ X... Pr /... yep elhoo Fuctos - how to estmate parameters; let... be a raom sample from a esty fucto f; where s vector of strbuto parameters; lelhoo fucto ; f... f ; loos ust le a ot pf ecept ow the 's are gve a the parameters are the uows ample -... are N so / ; e π f ; e π og-elhoo Fucto - atural logarthm of the lelhoo fucto; wll be egatve because you're tag l of probabltes < ample - l [ ; ] lπ l formato Matr - estmate parameters for strbuto from sample e.g. a for ormal strbuto; ee to estmate covarace matr for ubase estmators l ; l ; Sgle Parameter of

8 8 of Or More Parameters - - ; l ; l ; l formato Matr - ; so t's symmetrc matr ample -... are N so [ ] l l ; l π l l l l l l l [ ] l Regularty Cotos - l ests for all a l l.e. ca swtch orer of tegrato a fferetato l l

9 l < < for all l l 5 a est a are cotuous for all Cramer-Rao ower Bou CRB - Sgle Parameter - f regularty cotos hol f s a ubase estmator of the parameter the Multple Parameters - f s a ubase estmator of for... parameters. Assumg regularty cotos hol [ ].e. [ ] Cov s a postve sem-efte matr s a lower bou for the covarace matr of a ubase estmator Sgfcace - f you have a ubase estmator that attas the CRB of the varace the you ow that ths s the most effcet ubase estmator UMVU; t's ot always possble to f a ubase estmator that attas the CRB ample - coser a raom sample from a ormal strbuto N Ubase estmators: a s formato matr ths last tme Tae verse of ths [ ] so the CRB for the varace of s / s? ~ χ s s ~ χ... so s s NOT the CRB... so s the CRB Cramer-Rao equalty - let ~ f ; a T T... be a statstc such that T u some fucto of. Assume regularty cotos. The T [u'] / 9 of

10 Mamum elhoo stmator M - choose values of the parameters that mamze the lelhoo fucto or the log-lelhoo fucto. Tae all the partal ervatves of l; a set them equal to zero a solve for ~ Score - ervatve of l Score Vector S - comprse of all partal ervatves of l; ample - et... be a raom sample form a N strbuto / ; / π ep π ep l ; lπ l l s the M of ample - et... be a raom sample from a Pareto strbuto f ; for < ; l l ; l l l l l ample - et... be a raom sample from a strbuto wth esty f ; for otherwse ; for... otherwse l ; l for... otherwse l s ; l for... otherwse Problem - there s a scotuty at ; s ; s ot a fucto of ; ca't set s ; of

11 Soluto - wat to mae s ; as close to zero as possble so we wat to mae as large as possble; we ow that m possble value of s ; s at ma Multple Parameter ample - et... be a raom sample from N ; π / / ep / / π ep l ; lπ l l l s ; Substtute to the seco equato NOT: Ths s a base estmator Propertes of Ms varace - et be a M of. f g s a fucto of the the M of g ests a s gve by g Cosstecy a Uqueess - uer regularty cotos for Cramer-Rao lower bou there ests a soluto vector to the lelhoo equatos that s cosstet lm Pr S estece a cosstecy [ ] Asymptotc Normalty - lm / uer the regularty cotos of the CRB N ;.e. Ms have ormal strbuto at the lmt Asymptotc ffcecy - asymptotc varace of equals the lmt of the CRB verse of matr of

12 a b c a cb c b a of

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