The Interplay between l-max, l-min, p-max and p-min Stable Distributions

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1 DOI: 0.545/mjis Th Itrplay btw lma lmi pma ad pmi Stabl Distributios S Ravi ad TS Mavitha Dpartmt of Studis i Statistics Uivrsity of Mysor Maasagagotri Mysuru Idia. ravi@statistics.uimysor.ac.i Rcivd: July 0 05 Rvisd: Sptmbr 9 05 Accptd: Sptmbr 7 05 Publishd oli: Sptmbr Th Author(s 05. This articl is publishd with op accss at Abstract: Etrm valu laws ar limit laws of liarly ormalizd partial maima of idpdt ad idtically distributd (iid radom variabls (rvs also calld as lma stabl laws. Similar to lma stabl laws w hav th lmi stabl laws which ar th limit laws of ctrd ad scald partial miima pma ad pmi stabl laws which ar rspctivly th limit laws of ormalizd maima ad miima udr powr ormalizatio. I this articl w look at trasformatios btw lma lmi pma ad pmi stabl distributios ad thir domais. Th trasformatios i this articl ar usful i simulatio studis. Mathmatics Subjct Classificatio: Primary 60G70 scodary 60E05. Kywords ad Phrass: lma stabl laws lmi stabl laws pma stabl laws pmi stabl laws domais of attractio.. Itroductio Etrm valu thory is a classical topic i probability thory ad mathmatical statistics. Th fild of trms has attractd th atttio of girs scitists actuaris ad statisticias for may yars. Th fudamtal rsult i trm valu thory is th form of limit distributios for ctrd ad scald maima/miima. Lt b a squc of idpdt idtically distributd (iid radom variabls (rvs with distributio fuctio (df F ad M = ma { }. Suppos that thr ists ormig costats a > 0 ad b R R th ral li such that lim P M b = lim F ( a + b = G( CG ( th st of a Mathmatical Joural of Itrdiscipliary Scics Vol. 4 No. Sptmbr 05 pp

2 Ravi S Mavitha TS all cotiuity poits of a odgrat df G. W th say that th df F blogs to th lma domai of attractio of G ad dot this by F Dl ma ( G. Th limit dfs G ar th wll kow trm valu laws ad G ca b oly o of thr typs of trm valu dfs amly (s for ampl [5]: { } { } < 0 { }} th Frècht law Φ ( = p 0; th Wibull law Ψ ( = p ( ; th Gumbl law Λ( = p p( R whr > 0 is a paramtr ad dfs ar giv hr ad lswhr i this articl oly for valus for which thy blog to (0. Th trm valu dfs G satisfy th stability proprty G ( a+ b = G ( R for costats a > 0 b R ad wr calld lma stabl laws i [7] l stadig for liar maig that ormalizatio is liar. Hr two dfs F ad G ar said to b of th sam typ if F ( = GA ( + B for all for costats A > 0 ad B R. W say that F blogs to th lmi domai of attractio of th odgrat df L udr liar ormalizatio ad dot it by F Dl mi ( L if thr ist ormig costats c > 0 ad d R such that limp m d lim( F( c d L( c = + = C ( L whr m mi Th df L is calld lmi stabl df ad ca b oly o of th followig thr typs of dfs (s for ampl [5]: = { } { } < { } { } gativ Frècht law L ( = p ( 0; gativ Wibull law L ( = p 0 ; gativ Gumbl law L ( = p R. 3 Thr ar svral rfrcs for trm valu distributios udr liar ormalizatio. W am a fw [ 36 9]. Similar to lma ad lmi stabl laws w hav th pma ad pmi stabl laws which ar rspctivly th limit laws of ormalizd partial maima ad partial miima udr powr ormalizatio. A oliar ormalizatio calld th powr ormalizatio was itroducd i [8]. A df F is said to blog to th pma domai of attractio of a odgrat df H udr powr ormalizatio dotd by F Dp ma ( H if thr ist ormig costats > 0 ad > 0 such that lim P M ( M = lim F sig ( sig( = H( CH ( sig ( = 0 or 50

3 accordig as < 0 = 0or > 0 rspctivly. Th df H is calld pma stabl df (s for ampl [7]. It is kow that th pma stabl dfs ca b a ptyp of oly o of th si dfs amly logfrècht law H ( = p {(log } ; logwibull law H ( = p { ( log } 0 < ; ivrs logfrècht law H3 ( = p { ( log( } < 0; ivrs logwibull law H4 ( = p {(log( } <; stadard Frècht law Φ( = Φ( R; stadard Wibull law Ψ( = Ψ ( R whr > 0 is a paramtr. Hr two dfs F ad G ar said to b of th sam ptyp if F( = GA ( B sig( for all for costats A> 0 B> 0. W say that F blogs to th pmi domai of attractio of a odgrat df K udr powr ormalizatio ad dot it by F Dp mi ( K if thr ist ormig costats γ > 0 ad δ > 0 such that lim P m δ sig( = lim F( γ δ { sig( } = K( C(K. γ Th pmi stabl dfs ca b ptyps of th followig si dfs: gativ logfrècht law K ( = p log( ; gativ logwibull law K ( = p log( 0; ivrs gativ logfrècht law K3 ( = p log 0 ; ivrs gativ logwibull law K4 ( p log ; { ( } < { ( } < { ( } < { } = ( stadard gativ Frècht law K ( = L ( R; 5 stadard potial law K ( = L ( R. 6 I this articl w look at trasformatios btw lma lmi pma ad pmi stabl distributios ad thir domais. Eight familis of trmal stabl laws ar cosidrd for study. Th mappig that maps a rv withi o family to a rv withi aothr family is costructd for all pairs of familis. Ad th trasformatios that map a ma/mi stabl rv to a ma/mi stabl rv of a diffrt family ar w. Sctio 3 cotais th rlatioship amog lma lmi pma ad pmi stabl distributios. I Sctio 4 ampls for dfs i th domai of attractio of pma ar giv. For asy udrstadig th itrrlatios ar tabulatd i Tabls through 7. W dot ma( ab = a b for a R b R. Th Itrplay btw lma lmi pma ad pmi Stabl Distributios 5

4 Ravi S Mavitha TS. Itrplay btw lma lmi pma ad pmi stabl distributios I this sctio th rlatioship amog domais of attractio of lma lmi pma ad pmi stabl distributios ar giv as thorms ad th rsults ar tabulatd for asy udrstadig i Tabl. Lt rvs ad Y hav rspctiv dfs F ad G ad a > 0 b a costat clos to th right trmity of th corrspodig df whrvr applicabl. Thorm. (i F Dl ma ( Φ Y = G Dp ma ( H ad Y G D ( H = log( a Y F D (. pma a lma Φa (ii F Dl ma ( Ψ Y = G Dp ma ( H ad Y G D ( H = log( a Y F D (. pma a lma Ψ a (iii F Dl ma ( Λ Y = G Dp ma ( Φ ad Y G D ( Φ = log( a Y F D ( Λ. pma lma (iv F Dl mi ( L Y = G Dp mi ( K3 ad Y G D ( K = log( a Y F D ( L. pmi 3 a lmi a (v F D ( L Y = G D ( K ad lmi pmi 4 Y G D ( K = log( a Y F D ( L. pmi 4 a lmi a (vi F Dl mi ( L3 Y = G Dp mi ( K6 ad Y G D ( K = log( a Y F D ( L. pmi 6 lmi 3. Rmark: Statmts (i (ii ad (iii i th abov thorm ca b provd as i []. W prov (iv ad proofs of (v ad (vi follow o similar lis ad ar omittd. Proof of (iv. Lt F Dl mi ( L with ormig costats c > 0 ad d R. Th df of Y = is giv by G PY P d ( = ( = ( = F(log 0. So with γ = ad δ = c δ δ G γ sig( F(log( γ ( Fc ( log( d ( ( = ( = + δ ( γ = = > 0. Thus lim ( G sig( L (log( K3 ( provig that G Dp mi ( K3. Lt Y G Dp mi ( K3 with ormig costats γ > 0 ad δ > 0. Th = log( a Y has df F ( = P ( = P(log( a Y = G( loga< so that with c = δ c + d δ ad d = log γ ( F( c+ d = ( G( = ( G( γ (. 5

5 Thuslim ( Fc ( + d = K3 ( = L ( provig that F D ( L l mi Thorm.. (i F Dl ma ( Φ Y = G Dp mi ( K. Th Itrplay btw lma lmi pma ad pmi Stabl Distributios (ii F Dl ma ( Ψ Y = G D mi ( K. (iii F Dl ma ( Λ Y = G Dp mi ( K 5. (iv F D ( L Y = G D ( H. lmi pma 3 (v F D ( L Y = G D ( H. lmi pma 4 (vi F D ( L Y = G D (. lmi 3 pma Ψ Rmark: W prov (i ad proofs of (ii ad (iii follow o similar lis ad ar omittd. Ad (iv (v ad (vi ca b provd as i [] ad th dtails ar omittd. Proof of (i: Lt F D lma ( Φ with ormig costats a > 0 ad b R. Th df of Y = is giv by a G ( = PY ( = P( = F(log( < 0. So with γ = ad δ = b lim δ G γ = F ( δ sig( log( γ ( F δ = ( a log( + b < 0. Th lim G γ sig( p ( ( = (log( = K ( < provig that G D ( K Φ. p mi Lt Y G Dp mi ( K with ormig costats γ > 0 ad δ > 0. Th df of = log( Y is giv by F ( = P ( = P(log( Y = G( R so that with a = δ ad a + b δ b = log γ F ( a+ b = G( G( γ (. ( = Φ ( Hc lim F ( a+ b = K ( = ( > 0 provig that F ma ( Φ. D l 53

6 Ravi S Mavitha TS Thorm.3. (i F Dl ma( Φ Y = G Dp mi ( K 3. ad Y G D ( K =log( a Y F D ( Φ. pmi 3 lma (ii F Dl ma( Ψ Y = G Dp mi ( K 4. ad Y G D ( K =log( a Y F D ( Ψ. pmi 4 lma (iii F D ( Λ Y = G D ( K6 ad lma pmi Y G Dp mi( K6 =log( a Y F Dl ma ( Λ. (iv F Dl mi ( L Y = G Dp ma ( H ad Y G D ( H =log( a Y F D ( L. pma lmi (v F Dl mi ( L Y = G Dp ma ( H ad Y G D ( H =log( a Y F D ( L. pma lmi (vi F Dl mi ( L3 Y = G Dp ma ( Φ ad Y G D ( Φ =log( a Y F D ( L3. pma lmi Rmark: W prov (i ad (iv th proofs of (ii ad (iii follow o lis similar to th proof of (i ad proofs of (v ad (vi follow o lis similar to th proof of (iv ad ar omittd. Proof of (i: Lt F D lma ( Φ with ormig costats a > 0 ad b R. Th df of Y = is giv by G ( = PY ( = P( = F( log 0 (. a so that with γ = adδ = b ( = + > ( ( = ( ( G δ F δ F γ log γ ( a ( log b 0. Th lim ( G( γ δ = Φ ( log = K3 ( 0< < provig that G D ( K p mi 3 LtY G Dp mi ( K3. Th df of =log( a Y is giv by F ( = P ( = P( log( a Y = G( log a (. so that with a = log γ b = δ 54

7 ( a b F ( a+ b = ( G ( = ( G( ( < log + γ δ γ a Th lim F ( a b K ( + = = ( 3 Φ > 0 provig that F ma ( D l Φ Proof of (iv: Lt F Dl mi ( L with ormig costats c > 0 ad d R. Th from (. with δ = cad d δ δ γ = G γ F γ F c d ( = = + ( ( log( ( ( ( log > 0. So lim G( = L ( log = H ( > provig that γ δ F Dl ma ( H. LtY G Dp ma ( H with ormig costats > 0 ad > 0. Th from (. with d c Fc d G c d = = ( ( + = ( ( + = G ( log (. So lim Fc ( + d a /δ. ( Th Itrplay btw lma lmi pma ad pmi Stabl Distributios = H ( = L ( < 0 provig (iv. Thorm.4. (i F D ( Y = G D ( H. lma Φ pma 3 (ii F D ( Y = G D ( H. lma Ψ pma 4 (iii F D ( Λ Y = G D ( Ψ. lma pma (iv F D ( L Y = G D ( K. lmi pmi (v F D ( L Y = G D ( K. lmi pmi (vi F D ( L Y = G D ( K. lmi 3 pmi S Rmark: W prov (i ad (iv th proofs of (ii ad (iii follow o lis similar to th proof of (i ad th proofs of (v ad (vi follow o lis similar to th proof of (iv ad ar omittd. 55

8 Ravi S Mavitha TS Proof of (i. Lt F D lma ( Φ with ormig costats a > 0 ad b R. Th df of Y = is giv by G ( = PY ( = P( = F( log( < 0 (.3 b so that with = a = G ( sig( = F ( log( ( = F ( a( log( + b < 0. So lim G ( sig( = Φ( log( = H3 ( < < 0 provig that G D ( H p ma 3. LtY G Dp ma ( H3 with ormig costats > 0 ad > 0. Th df of =log( Y is giv by F ( = P ( = P( log( Y = G( R (.4 so that with a = ad b =log ( a + b F ( a + b = G ( = G ( ( ad lim F ( a b H ( + = = ( Φ > 3 0 provig (i. Proof of (iv: Lt F Dl mi ( L with ormig costats c > 0 ad d R. ( ( = ( ( = δ δ Th from (.3 G γ sig( F log( γ ( d ( Fc ( ( log( + d < 0 so that with γ = δ = c δ lim G( γ sig( L ( log( K ( ( = = < provig that G D ( K. p mi LtY G Dp mi ( K with ormig costats γ > 0 ad δ > 0. + = = ( ( ( c + d Th from (.4 ( Fc ( d ( G( G γ δ so that with c = δ ad d =log γ lim δ( F( c + d = K ( = L ( provig (iv. Th followig tabl summarizs th rlatioship btw domais of attractio of lma lmi pma ad pmi stabl distributios. 56

9 Tabl : Rlatioship btw domais of attractio of lma lmi pma ad pmi stabl distributios. D lma ( Φ Dpma ( H Dpmi ( K3 Dpmi ( K Dpma ( H3 Th Itrplay btw lma lmi pma ad pmi Stabl Distributios D lma ( Ψ Dpma ( H Dpmi ( K4 Dpmi ( K Dpma ( H4 D lma ( Λ D pma ( Φ D ( pmi K 6 D ( pmi K 5 D pma ( Ψ D ( L lmi D pmi (K 3 D pma (H D pma (H 3 D pmi (K D lmi (L D pmi (K 4 D pma (H D pma (H 4 D pmi (K D lmi (L 3 D pmi (K 6 D pma (Φ D pma (Ψ D pmi (K 5 Th tabl is rad as follows: Th try say i row is rad as : If D lma ( Φ th Dp ma ( H Dp mi ( K3 Dp mi ( K ad D ( H pma Trasformatios Th followig tabls giv th itrrlatioship btw lma lmi pma ad pmi stabl distributios. Th tabls may b rad as follows: for ampl i Tabl blow th try say i row 3 ad colum is rad as: If Ψ th Y = Φ ad so o. Tabl. Rlatioship btw lma ad lmi stabl distributios. Φ Ψ Λ L L L 3 Φ Ψ Λ p log log( p p log log( p L log( log( 57

10 Ravi S Mavitha TS L log log L 3 p p p p Tabl 3. Rlatioship amog pma stabl distributios. H H H 3 H 4 Φ Ψ H (log (log (log (log H (log (log (log (log H 3 (log( (log( (log( ( log( H 4 Φ Ψ (log( / / log( / / ( / / ( / (log( (log( / Tabl 4. Rlatioship btw lma/lmi ad pma stabl distributios. H H H 3 H 4 Φ Ψ Φ Ψ ( ( Λ p ( p ( p ( p ( L ( ( 58

11 L L 3 p ( p ( p ( p ( Th Itrplay btw lma lmi pma ad pmi Stabl Distributios Tabl 5. Rlatioship amog pmi stabl distributios. K K K 3 K 4 K 5 K 6 K ( log( (log( (log( (log( K (log( (log( ( log( (log( K 3 (log (log ( log (log K 4 (log log (log (log K 5 ( / / ( / K 6 / ( / / / Tabl 6. Rlatioship btw lma/lmi ad pmi stabl distributios. K K K 3 K 4 K 5 K 6 Φ Ψ ( ( Λ p ( p ( p ( p ( L ( ( 59

12 Ravi S Mavitha TS L L 3 p ( p ( p ( p ( Tabl 7. Rlatioship btw pma ad pmi stabl distributios. K K K 3 K 4 K 5 K 6 H (log (log (log (log H (log (log ( log (log H 3 (log( (log( ( log( (log( H 4 (log( log (log( (log( Φ / / Ψ ( / ( / 4. Som Eampls for pma ad pmi domais. This sctio provids ampls for dfs blogig to domais of attractio of pma stabl laws ad pmi stabl laws. Som ampls of dfs i domai of attractio of lma stabl laws alog with ormig costats ar giv i [4]. O ca gt ampls for lmi by usig th rsult: if F Dl ma ( G with ormig costats a ad b th F Dl mi ( L with ormig costats c = a ad d = b ad with L ( = G(. Eampls for pma domais: Pdfs ar giv with ormig costats ad valu of th paramtr of th limit law whrvr applicabl. 60

13 . Dfs i D ( H : pma a. LogCauchy with pdf f( = ( (log 0< < + = = π π with =. b. LogParto with pdf f( = > = = with + (log =.. Dfs i Dpma ( H : a. Uiform with pdf f( = 0 < < = = with =. b. Bta with pdf f Bab a b a b ( = ( < < > = = 0 0 ( bbab ( with = b. c. Logbta with pdf f a b ( = (log ( log 0< < ab > 0 Bab ( b Th Itrplay btw lma lmi pma ad pmi Stabl Distributios = = bb( ab with = b. 3. Dfs i D ( H pma 3 b a. Ivrs logcauchy with pdf f( = + log < 0 π = = π with = b. Ivrs logparto with pdf f( = < < 0 = = + log with = 6

14 Ravi S Mavitha TS 4. Dfs i D ( H : pma 4 a. Ivrs logbta with pdf a f( = log log Bab ( b < ab > 0 = = ε b with = b. bbab ( 5. Dfs i D pma ( Φ : a. Cauchy with pdf f( = R ( + = = π π. b. Normal with pdf f( = π R { log4π + loglog } { log4π + loglog } log = log. log c. Gamma with pdf f( = > 0 > 0 = (log + ( log Γ( log logγ( = (log + ( logloglog Γ(. d. Loggamma with pdf f( = ( (log > Γ > 0 = = Γ( (log. 6. Dfs i D pma ( Ψ : a. Ivrs loggamma with pdf f( = log ( Γ < < > = 0 0 = ( (log Γ b. Ivrs gamma with pdf f( = Γ( < 0 6

15 = (log + ( log log log Γ( ( = log + ( loglog log Γ(. Rmark: W kow that if F Dp ma ( H with ormig costats > 0 ad > 0 th F Dp mi ( K with ormig costats γ = adδ = ad with K ( = H(. I particular Ki( = Hi( i = 34 ad K5( = Φ( K6( = Ψ (. O ca gt ampls for pmi domai by usig this rsult. Th Itrplay btw lma lmi pma ad pmi Stabl Distributios REFERENCES [] Castillo E. (988. Etrm Valu Thory i Egirig. Acadmic Prss Sa Digo Califoria. [] Christoph G. ad Falk M. (996. A ot o domais of attractio of pma stabl laws. Stat. Probab. Ltt [3] d Haa L. ad Frrira A. (006. Etrm Valu Thory A itroductio. Sprigr. [4] Embrchts P. Klüpplbrg C. ad Mikosch T. (997. Modllig Etrmal Evts. SprigrVrlag Brli. [5] Galambos J. (978. Th Asymptotic Thory of Etrm Ordr Statistics Wily Nw York. [6] Kotz S ad Nadarajah S. (000. Etrm valu distributiosthory ad Applicatios Imprial Collg Prss Lodo. [7] Moha N.R ad Ravi S. (993. Ma domais of attractio of uivariat ad multivariat pma stabl laws. Thory Probab. Appl. 37 No.4 Traslatd from Russia Joural [8] Pachva E. (984. Limit thorms for trm ordr statistics udr oliar ormalizatio I Stability Problms for Stochastic Modls (Lctur Nots i Math. Sprigr Brli [9] Rsick S.I. (987. Etrm Valus Rgular Variatio ad Poit Procsss SprigrVrlag Nw York. 63