Mathematical Statistics STAT 709
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Date Created: 09/17/15
Lecture 20 Minimal suf ciency There are many suf cient statistics for a given family P In fact X the whole data set is suf cient If T is a suf cient statistic and T 39LL S where U is i neasurable and S is another statistic then S is suf cient This is obvious from Theorem 22 if the population has a pdf but it can be proved directly from Definition 24 Exercise For instance if X1Xn are iid with PXi 1 E and PXi 0 1 9 then 2 X7 2 1 X7 is suf cient for E where m is any fixed integer between 1 and m If T is suf cient and T 39LL S with a i neasurable U that is not one to one then 0T C 0S and T is more useful than S since T provides a further reduction of the data or a field without loss of infori nation Is there a suf cient statistic that provides i naximal reduction of the data39 If a statement holds except for outcoi nes in an event satisfying PA 0 for all P 6 73 then we say that the statement holds P De nition 25 Minimal suf ciency Let T be a suf cient statistic for P 6 73 T is called a minimal suf cient statistic if and only if for any other statistic S suf cient for P 6 73 there is a i neasurable function U such that T 39LL S P If both T and S are minimal suf cient statistics then by definition there is a one to one i neasurable function U such that T 39LL S P Hence the minimal suf cient statistic is unique in the sense that two statistics that are one to one measurable functions of each other can be treated one statistic Example 213 Let X1 X7 be iid random variables from P9 the uniform distribution U9 E 1 E 6 R Suppose that n gt 1 The joint Lebesgue pdf of X1 Xn is n fob H 1ltoo1xi luminand x x1 mxn 6 73quot i1 where x denotes the ith smallest value of 331 xn By Theorem 22 T X1Xn is suf cient for 9 Note that 461 sup9 fg l gt 0 and arm 1 inf9 fg l gt 0 If S X is a statistic suf cient for 9 then by Theorem 22 there are Borel functions 1 and 9 such that fg l ggSxhx For x with 142 gt 0 461 sup9 ggSx gt 0 and arm 1 inf9 ggSx gt 0 Hence there is a i neasurable function U such that T 39LL S when 1142 gt 0 Since 11 gt 0 P we conclude that T is minimal suf cient Minii nal suf cient statistics exist under weak assui nptions eg 73 contains distributions on Rk doi ninated by a a finite i neasure Balladur 1957 lseful tools for finding minimal suf cient statistics Theorem 23 Let P be a family of distributions on 73quot i Suppose that 730 C P and 7 implies P If T is suf cient for P 6 P and minimal suf cient for P 6 730 then T is minimal suf cient for P 6 P ii Suppose that 73 contains pdf s f0f1f2 wrt a a finite measure Let Ziocyf x where 3 gt 0 for all i and 20201 1 and let when gt 0 i 012 Then TX T0T1T2 is minimal suf cient for P 6 P Furthermore if ff gt 0 C gt 0 for all i then we may replace f00 by y in which case T X T1T2 is minimal suf cient for P 6 P iii Suppose that P contains pdf s fP wrt a a finite measure and that there exists a suf cient statistic TX such that for any possible values 42 and y ofX fpx fpygbx y for all P implies T T y where 9 is a i neasurable function Then T X is minimal suf cient for P 6 P Proof If S is suf cient for P 6 P then it is also suf cient for P 6 730 and therefore T 105 7 holds for a i neasurable function 1 The result follows from the assumption that 7 implies P ii Note that f00 gt 0 P Let MT Ti 239 012 Then P By Theorem 22 T is suf cient for P 6 73 Suppose that SX is another suf cient statistic By Theorem 22 there are Borel functions 1 and if such that W iSxlwx i 012 Then mx iSxZ 0cj j5x for satisfying foox gt 0 By Definition 25 T is minimal suf cient for P 6 P The proof for the case where f00 is replaced by f0 is the same iii From Bahadur 1957 there exists a minimal suf cient statistic SX The result follows if we can show that T X 1SX P for a i neasurable function 1 By Theorem 22 there are Borel functions 1 and 1 such that fP gPSx1x for all P Let Mgr 0 Then 0 for all P For x and 3 such that 5y 41 and y g A fpfx gpfsfxllhfxl 9p SCIlMxlhylhy fpfylhxlhy for all P Hence T T This shows that there is a function 1 such that 39LL S except for x 6 It remains to show that 1 is i neasurable Since S is minimal suf cient gTX SX P for a i neasurable function 1 Hence 1 is one to one and O g l The i neasurability of 1 follows from Theorem 39 in Parthasarathy 1967 39Xdxa X sg as twp umaggns mngugm sg X 3g axdumxa 101 39 JQSQRN mangth maxdmgs am 51qu 301 8 398mm mngugm V 3331103 8 apom ngm Spanxa moug 301x op am mq smpom asaq 30 Gun 8 apom arm am Iqu uopmn guy Kim asq mu mop uopnzgnzmmns pm Imanan mpp 103 33939238 asaq Sugsn anl mpgth aw 6392 axdtmzxg u 339391238 mpm am pm 111 axdumxg u 0X 1gtX ZS mtmgmx axdums am X Imam axdums am 3th sanspms sxapom 30 ma ax apgm p 103 31219 pug um auo 939g u saspwxa mums pmz uogms Sm u saxdtmzxa am 1101 39apom pasodmd am 30 ssmnaaxxm am moqp s3qtmp amos Spq mm 3SRG 31gt xo Summ sg apom mpsnms pasodmd am 3g Magma mast p aq mu 51211 31 39sppom magmas suopmndod 30 CL 5mm pawxmsod up no spuadap Abuapggns mngugm pm Abuapggns aql 39 g39g umxoaql Sugsn Sq pamxd aq asp um 392 pm 317 saxdxmzxg ug smxsm aql 399 9 g m umpggns mngugm 8 LL g392 umxoaql Sq anl d sagdm 0d 31219 33 03 KSRG s 31 3909 9 g m umpggns mngugm 8 LL aauaH 39OCL XS71 XL 3219 3th 71 uomutg ammnsvmn auo m auo p s mam umpuadapug Spmaug aw S t mugs 07 Q9 Q amqm 0 9 g m maggth mngugm sg d xmuwxw w mum my 3mg If 00f mm Mg392 umxoaql mag mono 31 399 no puadap mu mop 0 lt f 338 am 3mg MON 06 9 g If 0d 331 399 9 g m umpggns mngugm 3312 u 8 LL 3219 moqs mou 3M 399 9 g m umpggns s Xx 3mg umoqs wzq 3M 31mm in 30 sg 5mm am 3g arm sg sgql 39dQL ug umpuadapug Spmaug aw dquotquot 1 g OQM mu th mamax am 3mg 3th 9 9 47 quotquot 19 09 0 83mm mam 3mg asoddng MT63 WLKQWHCW WW s j39p39d mm 5mm Rntmtmdxa UP aq 9 9 g If CL 331 39prz aldmexg