Formul().e I Re1newe..r- for 1"" LE

.. l. = ( x,,X2., ... ,X11)' r.s. ot siie n ~ Fx (•;e) ~ 0 (0,,02., ... , 61,;)' : vec+or of k u116'own pa.r<:\lf\e.hs

Jl~ a p,.ro-.1<\e,·l-e.J' ~ : Set of all Ociff\1ss:;bj~ Vtllues ot 0

! : ( %, , 'e:i, . .. 1

~le. J' : vector- of edirt\o.tots .fot ~ t" (ft) o. °hAM"h 'o" of ~ u116'at... " poro.t11 e.-ler..s TC~) a sto:+,sfi'c.. used 01. OJI e.~iiN\a.-lor- tor T (6)

• De~n;+i 01'S

Poi11+ e.s-h'rt.o..tol' - (l s+c.tis+.·c used. \o estima:le QI\ Uf\bDl,jl\ pora,..e+e~ o,- a t11c.,1lof\ of Ul\tl\own po.ra((le;re("S .. 0 rule or 't4nc..ho1\

Poid e.dimo.+e. - u reali1..u:f value of a pcin+ ~-f1fl\a-h,r ~ a teo.l nutY1ber

• He.on sq_uared €./l-or- C HSE)

Fom-.ulc. c.t M~£ of TC5.) w;'"' res~ fo "t(~) :

t-1SE tt~/T) ~ E [ T - TC~) J" V(T) +(B,o.s/

VCT)+ [ECT)-'((~)] 2

• Col'\Sis~~

.&i.d ""t T'2., -.. , Tr,,... Ix a s-e'l,uenc.e of -e.stirt10.1'ors of -cl~), w~ T" = Tn(1) i& -Hie Safi\€ esfi'rne,:h>t k~d on o.. /'.<;.. of si1.e <'i • "the seq,ue.t\C.e {Tn 1 1r. :

b. weo.kl~ Cbf\s,s ~ .J.. iff: P{ITn-'t<ft)l<E}-">I as n-">oO , \J § e12ft

A suft'o~+ C=J.11\0I\. t1' {Tl\ 1 -fo be MS£ CoTISl'd«!.n+ 'h:>r re~) IS:

£CT.)->'!(~) °"d. VCT.'1-'>0 as n->oO

C\. A s+cJ,s-h'c S 'SC~) Is. d.eRned to be suth'e..l'el\1 1ft: fa (x ,,-x~1

• .• 1 Xn /S=s) ;~ i11depe11d.e,A of§., \/ S • [Su/;'~-J- .f,,- ft 1

b. A s+o.fi~h'c $:SC!) is detir,ed . +o be suffia·~+ iff: 91vCJ\ a.r.~ o-/l--er .shf,s,ic. T-:. Tc~) ,

fr ( t 1.5 = s ) is independe11t of ~ , Vs . [Suffic.re,.J fu...11

MEMendoza

1. X ~- BeCp), p ~ [0,11

~: (X,,X2,X3)1

I\

o.. S": ~ Xi fs suffic.,e.,J -for p 1:1

b. S(~), X112)h+X~ i's ~t suff.·c.fe.nf fur 'P .

Note.: ~e<o..ll!j fo.,. K ~"Be( p) ot s12e. n,

-<::' ." X ,· rs s l.lthc.i·er.+ for- -o L,~, ,- .

fur K of s~" ~ - PoO.), S" • Li"~, X; 1s

SL&ffideo+ for- IL .

Tue sin.-n.s·hc.s S,' S,C~), S-2 , Si~),·-·, Sr-, SrC~) are .so.,d. to be jo,ntllJ suftiae.-.t G>r.§ ,iff, f!J (x,,x2., ... ,x,., I S,:s,, S2=s,., ... ,S,-'Sr) is ,nd.epetd!!.f',t of e, l/s,,s2., ... ,Sr

hc..mp\es:

I · The r-.s. (! J ,t~f is joi"+I~ .suffi·cier.+

Le+ ! ~ - f X ( . j ~). § Ul\kl\t>t.01\ '. A ~+o.ti~tt'c, s ~ S(~) i~ sufr.·ci·el\1' for ~ ' itf,

fx,,x,,., ... ,x/x,,J<2,·-·/x .. ) ~ g[SC~);~] · h(~)

(}.)here. h (~h O , o tuftc.11'on of~ i"d€fle11de....+ c.+' .1 S ($(~); §] ~ 0 1 a func+.'o" i"volvi"<:f ~ but ck(lCf\ds on 1he t.s . ! onl~ fh~ SCK)

ii. h>t joi"+I~ suffi·e1e.nt sfo.fis+rc.s:

Le+ x.~s'fxC·;~).e. Ul\rl'\01.J.)f\• The.sh>.·ti'sfic.s S,:S,(~). ~2:S2(~), ... ,st,S'rC~) arejoi"H~ su-!t;-cien+ for ~ J++, [ · J '( ) fx,,'i,,. .. ,xJ"l<,,'X21 . •• ,x"): g 5:,C1),S"2C~), ... , SrC~); § • n ~

whe<e h(~) ~ 0 ' a fu11ct1on of r alo.,e independe,,+ of ~ g[S,C~),$2(.~), ... ,SrC~) i ©1 , o. fu,,c:hon ,'1111olv1f\9 Q butJ.epe"c.ls o" fl.ie r.s. ~ onl~-thru

-!he s-hi.+i:s+.cs , S,C!),.S2C!SJ, ... , SrC!) MEMendoza

No-re: 3[5(1);~], 3[Si<!),S2{n, ... , .$',-CJ);~] O.fld. f..cn o.re oJl t\Ol\f\e90-tive tof" the foHowi'n9

ex°"'pb:

I · K ~- BeCp) , Osps l

2.. 6 ~- PoO.), :>t->o

I\

~ (1) : JT Px; 6:;") 1=1

I\

, lJ Px:c1-p),-)(: Ifo11Cxi"} l-1 )

" f15. (!) ' p'i.-x; (/-p ') n· I:«; V, I {o,I.} (x;}

" o [~X;; p] .J 11.,

h c~) . : S11 ° Li~, X; is suffic.,e..+ fur -p

h fa) = lT.-~ px;(;x.-)

1T e-:\ i\°x; I fa·) j, , ;r; ! {0,1,- .. f I

- M 'l !.X, " I I ( ·) ;: e /1. !I ,_;! {0,1, ... } K,

h(J) s" : .ix, ;~ su.ttibe.,i+ for /1. '"

3. ~ ~- NCµ.,cr2 ) , µ. € fR , o-2 e (Rt

f1C! 1, lT,:, fxJx;)

: n,.:, o-~ e)(p{-~ ('x,· -µf] Ic ... ,dJ/x.-)

TI " -.!. f I '2 t I ,., (2no-2) 'l. exp L-2a2<-x.--,-,..) J I(-.o,o0)(~;)

~lrr%. e"'p {-~ 1,:. ('X; ·p..f-} 1Tic-c0,"')(x;) , .. f\

( 2ntr) -; exp {- 2~ [ Li~, ( x,·-x)2t 11 ( x·µ.l] J1f J,.ca Ol))x,) , .. , J J

9['f;µJ h(i)

MEMendoza

Ca~e 2. : /.L kr.oo>l"l I cr'l. Ul\(I\OUII\

" f5.(~): (1ncr1..f"hel(p{-~ r,:,('X;-µ)'l.} 1!, Ic- .. ,..,/xi )

J [ Li~. cx,-11-)2; a-'- ]

Cose 3 : both µ, a11d o- 'l Ul"lknoi.a o

f~_{!) = (2no-1fl'l exp f- 2~"' [ z.,:,rx,·-x)" + n(X-p.)2]} 'JJc- .. ,,,J:(;)

Theorem : If {S,,S2, .. . 1 Sr1 is c. set of joi"-+l'.1 suthcie"t .s-+a-h:d1cs , -thet\ °'"~ Set of at leo.s+ r 1-1 1n,.r,st'or"'a.-h'ons of { ~. , $21 ... , Sr} is also jo;"-\i~ suth·c,enf . 111 +he case of a sin3 le suffi·­Cie,J s+a:h:Sfic., if S : S l [ ') i~ sufl;-c.ie."+, ~ an~ 1-1 -fronsfur.,..a.tio" of S is o.lso .Suff;-cie.rJ .

'Ex Q.ty", p les :

I. !S. ~• Be.Cp) I 'P € (0,1]

Sn C Li~,)(,' ,~ su.ffic.;-en+ For -p - X: -f;-sn is o.lso suftic,ed for p

!l · X ~ - Po O.) , .11. > o

X is als,, sufr.oent ~r- A

3 . }i ~· N(p.,1r 2 ) , 1tef1

a-2 s IRt

" • L X; is suff:-c.iU\ + f;,,.. µ, , ..

Def:r.i·lior1 : A sh:.+1s+:-c. T•T(~) is soJd f-., be Cof<'lple+e iff'.

E[9CT)]=O il'l\phe.s P[9<n,o],I, V96.D. 9

1. Le+ lS. ~- 8eCp) . T~ T(~)= Li:,)(; 1s co"'ple.fe

:2. Ld· X, ,X'.I. ~ · &C-p'). T,T(n , X'l-X, is NOTc,,1'1\ple.+e.

r.s. T 3. Lei- X, r-., N( o,e') . • T(~) = l<'i 1s IJO"T c.omple-.fe

MEMendoza

2 . ~ ~ - 6eo(-p)

Px(x) • p C 1-p) -x I{o,,, ... />:) : p I<o. ,, .. 1 Cx) exp{ In Ul-p -yx]1

c p Iio.i, ... 1Cx)e>cp{ :xln(l-p)} 1 ~ t '-<--'

c.(p) b(>c) J.( x) cCp)

n n

~ d(Yj) : L x,- = Sn (:. J c.~,

r.s. 3. K'"" Pao,)

Pxf-x) = e->- 'A~ I

X' J{o,1, ... /x) = e-A ~! 1{0,1, ... l >c) e?Cpt ln(i\.")}

= e-A ~, Ifo,r, ... ) (x) exp{ x {nCll.)} t "'· t ......__,

a(,\) b(x) d(x) cU..)

C°"'Plefe S.A'.°icie."+ S+a.-iisht,: i c({ Xi) , t 'X ; ,: S,. l:1 (. t I

"I. K ~· fxpC~)

fxfa ): 11.e- .i\>c Ito,.a1Cx) ~ I\.Lo,.a)Cx)exp{(-).)('lC)} 1'-~ 1' f

a-C,\) b<'.x) c(>.) dh:)

" n Complete 5,..ffic,e..t .S+c..-frsk: L cl(,q), r x; = S,,

i=1 i: ,

r.s. ~ k 5 , l "- N(µ..,a'l), O' nov,n

fxtx) 0 O'~ e.xp{- ~ 2 (x-µ.)2

} Ic-00, 00) (:d

= o-f2nexp{- 2; 2 (?r2.-2p.x-+p.2 )}J(_14100/>c.)

I { I ii I I 2.}- ) C <r-l2n exp - 2o'l )( t ~ p..x - ')_~ µ. .L(-r,0,,o) (x

exp f- 2~7 x'l} o-~ 11.-.,,,.aiCx") exp {- 2~ µ,' J e)(p{ ; 2 µ x} .._.,-, t

a(µ.') cCµ,) d(x)

CO'l't'\plefe S1,1fficiwt S.fo:h:sfrc: I d(-i;) : i xi· : Sn (:1 (• 1

b · ! ~- B; Cn.p)

p~( ;{) :: (;) p )( (/-p)0 -"I{o,1, .. . ,n1 (x) • G )(1-p )" ( -r.%-Y' I,0,1, ... ,•}(x l

= (1-p)" (~)l~0•11 .. , 111 ('l< Je><p { ln[(,fy ')><]}: (l-p)"(~JI10,1, .. ,n)(x)exp{ X In(~)} t ~ f' ---.;-.,

a(p) b(xJ d(>c) dp)

COY'\pld·e St.rfficiel\+ S-to.f1~f1c: 1.;~. dCx.- ): I.:. t; = Sn MEMendoza

i. K ~· NCµ.,tr2 )

fx(x) = er~ exp {- 2~'L x?. +- ~'2 µ.x - 2: .. µ'l.} I(-.o, .o/>:)

= ~ exp {-~2 P,'2.} Ic .... , .. /Jc) exp { - 2~2 X 2

t ; 2 flX} '-r-'l, T~

a(l(.,a-'2.) bCx) c,(cr) d,(x) c2(f'.,112) d1 (>c)

g. X ~ · BefoCa,b)

I' ( ) re o.tb) a· I ( , b-1 I rxx: P(a)P(b)X l·xJ c0,11 {x)

rca+b) -1 c 1 { I [ a~'c )b·']} rco.)r( b) (0,1) l( exp I) X I-)(

rca+b) { } r(aW(b) Lo,nCx)exp ln(xa·,)+ln((f-x)b·I]

~(o.~b \ lc0,11 h:)exp { (a-() ln(l<) r (b-l)ln(I- )()} Ca)rcb L,-J ,,_, ~ 1 ~ a.(c.,b) bCx) C.(c) £1,(x) C2(b) d2(>c)

::bintl~ Compleie Sufficicn+ Stc.-b1.·ks: " J\ " I\ Ld;(X',' J' lln(x;), L d 2C>c;):: L ln(l·J<;) ~' .-~, ,,, i:,

r.r •. q · f "' (;\a Cr, i\.)

f () "A." r-1 -Axr () JC x = r(r-) x e co,.01 x

" r~ Ico,cj)i(x'l exp { ln(x'·')}exp{ ·J.)( 1

= A" Ico,.oi>c) exp{ 11"1 (xr·') - AX} rcr-) i\:'

Iio,11>) (x) exp { (r-1 )lnCx) - J\x J ~ I I I IL

a(~) I:,()() c,(~) d,(x) ~l~) d-/x)

Join~~ G.mplek Sutf}d~t .Sh1tisttc-s: " f\ " " L~ 1(x,) ~ i ln(x;)

1 L.d,.hiJ: ,I 'X;

l-:, 1t1 1:1 i1:,

• U"bio.sedness

Dehnd,on : An es1iN'lo.+o1' T: T(!) I~ d.ei';neJ +o be Ul\biosed for a. tl<t\e,i\'on of unknov.:>n Pofo.rne~Cs)

Sc.~ 't(~).,ff' : ECT), £(TC~)]• TC~)

DefZo;fi'on: I~ T,. T(K) ,'s o.n estirt\otor of T(~), tl-.e b,Qs elf T c..:,;+h resped tot(~) is defined os: brc~)Cr) ; r(~)- ECT) MEMendoza

Defi11iti'on : fi.11 es-n/1\o..+or T= T(f) of T(~) is. detine.d. to be -!he. 11\in,mum HS[;- es+i'«10.+or oP !'(~), it for o.11~ other '-S.t/'l\o+or S.o~ T4"(~) oF T(!)1

Msf(T) :s MsHT") , V .§. '= .o.~, o.r.d

I-ts£ (T") < t-1SE (T4') for o.t least 1 ~ E 11.§

Defi";ti'on : A!\ erlima:h,r r,,. T(~) o~ 1:rn) c's de.fined to be -the UH VUE ~ r(t) ;ff 'v !1. '= Jl~ :

(i.) HT)= r(~)

(ri) V(T) ~ V(T*) 1 tor M':j o+he.r Uflbi'c,sed es.f11Y10.tor Tlf of T'(§)

! . If T:. T(~) ,dhe Ut-lv'UE for r(e) , -the.(\ (o.T+b) 1\ -the lJf.NUE f0 , [o.t(~) +b]

1 . I~ T1 : T,C!) ,s+he lJMVUE fur (,(Ii) and T1 ° T2,(f) is The OMVUf fo,. -r,_(§),1,,}h T,a11d.T.2

,rde.p€.t'lde.('t, -then (011, +ei2.T2) is fue oµVU6 fur [ l\11:"1 r~) +Q2. T2(~)]

CRL8 ~ ['C'(~)]\l

(\ J(9)

TC 8 ) • Fi-sher; lnhif"'<Ahc,n HeQSure

E [ ( ~Ge lnf'x (:II',·; El) )2] = - E [ :a: In fx (x;; e) ]

Let ! ~- fx ( • ; 9) 1

Gt .O.e C. ~ (i .e ., e is. 0 ~c.o.lc..r) · furfk.r, let $" SC~) be O. s11.fFi'c-1ef\ t sta+is ­

tfc. iol' El o.rd let T=TCi) be. M Uf\Diased estirno.ibr- fur- T(6), with ~ni+e. vafii,.nte. V(T)< ~ ­

Define T' a.s : T' = EcTI S). 1k.n,

1. T' is o. hlflc.1'on of the sutti'6e11+ sto.-ns-t,-c.. S. II. E (T') : "C'Ca) • \J e € .Q9 ( T' is ul\b,'osed f;,r n.e) 1 n·, . V(T') ~ VCT) I V e E ne ' w;-1-h e.q,ua.hl:J ,ff' P( T'.: T) : I •

MEMendoza

• lehrno."" - Scheffe Theorem

let tS_ ~· fx(•;0), es llec.lR (r.e ., 9 isn seo.lnr-;. If f\..es+a.-tisf,c Se SC~) ,'s C,ot'Y\pleJe (m1(\1/IIC,.I)

a11J. sufficief\+ for- 0 0.11d. -1+-ei~ exi's.ts Oil ul\b£A~d e.J;(l\O.t,r- TC~) ~r rl0), -ttie,-. +hele c1<1'sfs o. ul\1~ue.

Uf..\VOt .for T(8_),-gi11C11 b!j: T* : t (TIS) .

1. Ar. ul\bic.sed estif11l>.+or- T, TC!) or T(e) is defi"eJ +o be e~Cl'ent -fur rce), ,H, .fhe vo.rrb."ce of T atb.il\S ~e C.RLB of -the Vo.ri'o.l\c.es of Ut\b1asecl esfi,oo:h,rs of T(&) ·

!l. . ~e. effi'a'e..-c'::l csf °" u"bio.sed esiirno-+or T: T~) I!. defined as +he ro:h'o ot the CDrresp:,t'd. i'11g

CRLB C ~ 'l'o.n'=c,e,s of ul\biosed es+i<M-br i. of Tte)) +o Vor(T), i.c.,

fff(T) = (RLB X roo 'l. Vor(T)

(ii) Eff'CT) _.,. 1 cts n -~ oe .

Detn;+,ol\: Let r : C X,, h, ... , X" Y be. o. r-.s. ~ fx C •; ~) , ~ e. .n.~ . Suppose T, TO~ ) rs on edi'mo.+o" of t<§) . Tl\eo T i~ def;r-eJ to be

( o) Loco.-h'on itwo.no.f\+ . i ft

T< ~ + c ~"): TC~) . 'v ~ t S , V c. e R

( b) Sc.ale i,wori'ont-, iff

T(c~)r:T(~), V~e.S, v'c,dR

( C') lo=-ti'ol'\ ~u1\/onoo+ I itt

T(K~c111): TC~)tc, V1tS, VcetR

( d) S.o:,le e.,_uivo.ri°M+, i" ft

T ( c .K) , c. T(1) , 'v 1 e S. \Jc. e IR

( f) loCA+,'on- scale eq,uiv-o..n'o.f'lt; iff

T(c.~ +c.21n J • c,T(~) + c.2 , V ~ e. S, Ve, B 11(+,

~~E,(R

MEMendoza

Robus+l\e.ss : Al\ es+imo.+ot Y pertotfl\s we.JI ur.de.r- l'f\Od;f.'c.a.fi'ol\!; of ~e UMeM1j1'n9 ossump-h'ons 1s

So.i'J +o be. robud

• Anc;ll~6+~

•

Dei·"i·hon : Le..+ l ~- fx ( • i ~) . Let -fhe Mi·sf1c. f: TC.K) be 5utti'ci€L)+ for ~, ond suppose. d;mCT) '>di'"'(~). If T co.n be w...;t\.eo as Ts (T,,T2 ) where T2: T<J.C~) ho.s o · tr10.r3i"al d,'s.J-r; btl'i,'on tohw:1-1 1.s il\dep:!"ded of §, fier. +he s+crhs-f,t T2 is define.cl +o be. al\ ctncillo.r:lj st<rt,'sti'c . Norecvef', ihe s+atis-1-ic Ti: T, (~) is co-lied a (.Ol\cl1 ·­

tioMll':l su-ffic.ut stt,.-hsfk .

(i) Tit is o. lil\e.or fu~d-1011 of X,,x?., ... , Xn (ii') f(Tll-) =Tle), ;.e., T~ is u"biosed .for Tl6) (ijj) Vor-CT*) ~ Var CT) I 'v e E; lle J £r ().t\~ other li'neo.resh'rT\c..1'of' Tot rce)

~esul+ Let b.. ~- fx (• i µ,o- 2), (.,,:)r+h 0-

2< oo. lhe" X is +~e BLOf I OHVULE for J,L, r~ordless of

-lhe .form of fx .

Let K ~- fxC·;~),~Ell1 w,+h PDF fx<·;~). leJ µ; de"o+e.tl-e.r-*'populo..-h'o11ro.w rr.ome..,-.+. Ger.e.ro..11'1 , µ.; w,\l be c.. 'h>J\e,hOI\ of ~ , so OJe. C.Ol\ ...,r1+e µ../' fl; C~) · Le+ H( d.e.tiote fl-.c. r-!h Sa.f"ple r llW ft\011\1!.f\ + :

> q,1..de +l\e. -Brs+ k ~pie ro.v.i mume,-.+s ( it ~ ho.s k CCO\Pone"h) to 1heil' correspand.1,\9 r-o..w

tno"'cnh, i.e.. 1, ' 1 lJ I I I J I t Mt : µ., 1 1'11 : µ.2_ I ' ' ' / f'1 k C Ilk

II, "" "' > Tke. Soluti'of\S 1

~o-kcl. b~ a,, Bi, --. , ek , are the t-H<t.s of-fhe pnro.me+us e,;92.,-,.,Gk_, respec.+ivel~ . ·

µ/= f(X.") 11 , r,:. x/' r , n

MEMendoza

"> Likel;l--ood fuf\Ghon ·. 1' f,.'l\cf1'0" of e,,e:i., ... , 91r: w;tt1 -+l--e vo.lues of 4e r.v. :; tflV\JJT\:

U~l~·!) ' LCe.,e:1., .. ,81c. /X,,'t,,X2;~2.,. ·· ,Xn=x~)

" = Jr Fx< X, j ~) , ..

> Let ~ ~ - tx( •; ~) , ~ €.12.~ Ofl6 le+ LC.§(5.) be th€ likel!-oo~ 'h.u1ef1ot-, of fhe r.s. 11.e esii'Mo ·

-tor : - " "' " "' § : ( e, ,e2

1 ... I 81c.)

is de~r.e.d as +1-.e /l'0./1,,""uf'Y\ l;\:e,1;~od e~hrr.o.1-or- of ~ iff', A A

L(§l~)=sup L(~f~), i.e, U§l~)>L(~l~),V@G 12~ ~t D.t

> 1-1~ : Mo.x,11-.;2.e L rn I~)

(i) Use ! LC~lf) = 0

(ii) Fi"d !n[L(~l~)J ~en u~e :~ /n[l(~I~)] • 0

" "' "' "" Let~• (e, , 02 ,. .. , ek )' be f+.-e MU: of .§ . Sv.ppcse r,C§.)1 rz(~), ... , rkC~) , tsr~k ,

are r funcfiol\s ot -fk k unlo-C>l.ur> poro.r11d-us . Thel\ 1 fhe klfs of 1hete (' tu11cti'of\S ore -fhe .some 1W\c:lions evolvo.fed c.+ +he Mlts of -f+-e k UT1k:t10LDr. pc.rorr.e ks I i.e., 1

the t--\LEs of? 't 1(§),T2(~) 1 .. .. , t'1::Cfl) c.re 'C,CV , !'2(§), . .. ,"ik(~) .

Theo~: If -J+.e)~=-~"t ~FI PMF so.tis-Res c~+o.;" r~v.lo.n~ COTldri,<JflS I tk tk µI.£ o-f TC~) .SC.!:j T(.!1) ' t'( ~) ts as!:J~'c.o..ll~ I\OTI'\o.11~ d1s+nbv\.c.d ;

TC§) ~ N ( µ. = T(~) i 0-'2.: CR.U3wn) Q5 I')-') o(l

2.. MLE s o.re ~"'phrfi'Cblllj \J.l\bio.sed o."d o.sH~pto-hca.llj etfic.te.r.'l- .for #.e .fo.~e-+ fX>-to..mele.r, <11\d c.re a.lso a.t~rnp~+iCAll~ I\Uf"'-o.ll~ J ;sinbu+ed.

4. If 011 Mlt k. ul\b1ased tor o. +o.~e+ pa.r-omefer Ol\d. ;F ;+ is o. .fu.l\cf.iol\ ot il,.e. CSS: 1 -lk." i~ 1s fhe. UMVlJE · MEMendoza