A modified hyperbolic secant distribution
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- Lesley Chandler
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1 Songklnkrin J Sci Tchnol 39 ( Jn - Fb 2017 hp://wwwsjspsuch Originl Aricl A modifid hyprbolic scn disribuion Pnu Thongchn nd Wini Bodhisuwn * Dprmn of Sisics Fculy of Scinc Kssr Univrsiy Chuchk Bngkok Thilnd Rcivd: 17 Dcmbr 2015; Rvisd: 4 Mrch 2016; Accpd: 24 Mrch 2016 Absrc Th ims of his ppr r o inroduc nd vlid nw disribuion which is rld o h hyprbolic funcion Th diffrnil quion is pplid o obin h survivl nd probbilisic funcions Th momn gnring funcion is providd by using h ingrl rnsform Th proposd disribuion is pplid o rl d ss I is dmonsrd h i cn b usd s n lrniv modl in vrious disciplins such s lcronics finncil whr nd rrivl ims Kywords: hyprbolic funcion hyprbolic scn disribuion scn funcion 1 Inroducion Th hyprbolic funcion is n imporn mhmicl funcion in rlion o rigonomric funcions xponnil funcions nd complx numbrs Th hyprbolic scn is pr of s of hyprbolic funcions which is dfind s 2 sch( x In 1934 Bn inroducd probbiliy x x ( disribuion rld o h hyprbolic scn funcion which is clld h hyprbolic scn disribuion (HSD (Bn 1934 This work ws hn xpndd by Tlcko (1956 who proposd h disribuion for finncil rurn modls Consqunly h HSD provids n opiml fi nd xhibis mor lpokurosis hn boh h norml nd logisic disribuions Howvr i is limid in is uiliy; i cnno k on vrious shps du o lck of flxibiliy in is prmrs hus nurl bhviors cnno b sufficinly xplind wih his modl Mny brnchs of nurl scincs mphsiz h sudy of phnomn such s h sprding of diss or growh of populion by looking rs of chng Diffrnil quion is n imporn chniqu usd o solv hs yps of problm Diffrnil quion is usful chniqu * Corrsponding uhor Emil ddrss: fsciwnb@kuch in xplining sisicl propris nd vil ool in proving or disproving sisic-bsd issus This modl cn b pplid o survivl nlysis lifim d nlysis nd rlibiliy nlysis This ppr inroducs chniqus usd o gnr nw disribuion h is consrucd by modifying h hyprbolic funcions ino mor flxibl modl using diffrnil quions W implmn h chniqu wih s ( S( b s im dfind ( S( S( ( S( S( 0 n n n whr n blongs o {123 T} T / n nd S( S( S( k k k Th 1 procss cn b xprssd S( S( S( S( S S( S( S S( S( S k k 1 k Undr h ssumpion h is vry smll w obin S( S( S ds( k k 1 k Som propris of h survivl funcion S( r righ coninuous in Nvrhlss for coninuous survivl im T S( is coninuous wih non-incrsing funcion ( S( 1 0 nd lim S( 0 n n Th rs of h ppr is orgnizd s follows In Scion 2 nw fmily of disribuions is proposd from h hyprbolic funcions using diffrnil quions o dvlop his nw probbiliy funcion Also w will pply rl d ss o h proposd modl in Scion 3 bfor sring our conclusions in Scion 4
2 12 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( A Nw Disribuion This scion prsns n lrniv wy in which h nw probbiliy funcion could b dvlopd W pply h diffrnil quion o driv probbiliy dnsiy funcion by sing up h firs ordr diffrnil quion nd solv i o g survivl funcion W hn k h driviv of h survivl funcion Consqunly probbiliy dnsiy funcion ssocid wih survivl funcion will b obind Dfiniion 1: L T b rndom vribl on probbiliy spc ( F P wih probbiliy dnsiy funcion ( f ( ; disribuion funcion ( F ( ; nd survivl funcion ( S( ; 1 F ( ; nd ( f ( ; dfind by ( F ( ; or ( S ( ; Proposiion 1: L T b rndom vribl on probbiliy spc ( F P wih probbiliy dnsiy funcion f ( ; k nd T [0 which producs h quion blow: f ( k k k 2 2 (2 log[2 ] log[ ] 2 ( 2 whr is n iniil vlu nd 2 0 k 2 (2 1 log(2 log(1 Proof: W propos h nw disribuion by finding h soluion of h following diffrnil quion whn ( ( ki S S S ( 1 ( ( ( 1 [0 I 0 ohrwis Tking h firs driviv w will obin h probbiliy funcions s sn in Eq (1 As for h soluion funcion of h diffrnil quion w s U ( s n rbirry funcion hn muliplying Eq (2 by U ( w obin U ( ki U S ( ( ( U ( S( ( ( by driviv produc rul i is hn s o h following form U ( ki ( U ( S( ( ( ( whr U ( d Thrfor h rsul of diffrnil quion Eq (2 is (1 (2 (3 1 U ( ki S ( [ d C] ( ( U ( ( which is h survivl funcion In his cs U ( nd h soluion of h diffrnil quion wih iniil vlu problm S( 1 is ( (2 k log(2 k log(2 2 S( 2 Now king h firs driviv h pdf of T is ( ( (2 k log(2 k log(2 2 f ( 2 Corollry 1: k 2 2 ( L T b rndom vribl wih h pdf f ( ; b which could b xprssd s b 2 f ( ; b 2b (1 wih h locion prmr b nd iniil vlu c Proof: Similr o Eq (1 f ( 2 b 2b (1 solving h diffrnil quion xprssd s 2 ( S ( c 1 ( b ( b ( ' S b (4 obind by by ingring boh sids w obin h survivl funcion which is h rsul from solving Eq(5 cb b 2rcn( 2rcn( S( hn by king firs driviv h pdf is obind in h form b 2 f ( ; b 2b (1 whr T ( I is lso possibl o prsn h gnrl diffrnil quion s AkI 2(1 A ( AS ( ( ( ( b ( b ( ( ' S by imposing A 0 or 1 Th Eq(6 cn b rducd o Eq (2 nd Eq(5 rspcivly Following his logic S ( 1 nd 1 S ( c 1 r hrfor h iniil vlus of Eq (2 nd Eq (5 2 rspcivly (5 (6
3 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( Morovr h pdf drivd from h survivl funcion is probbiliy funcion which is ssifid h following propris; 1 Sing S( 1 nd S( c 1 r iniil vlu funcions 2 Th survivl funcion is monoonic dcrsing 3 lim S( 0 4 If h pdf corrspond o prmr spc hn i will b grr hn zro 21 Som propris of h nw disribuion Thr r mny mhods o solv h diffrnil quions An imporn mhod uss h Lplc rnsform which is rld o h momn gnring funcions Th Lplc rnsform of Eq (1 is givn by L( s s f ( d ; s s k (2 k log[2 ] k log[ ] ( d( 2 ( 2 ( s 1 ( s 1 ( s1 2 k 2 k log[2 ] [ ( ( ( 2 (1 2 2 ( s 1 ( s 1 2 ( s 1 k (log[2 ] k ( d ( ( 2 (1 ( s 1 2 ( s 1 ( s1 2 k log[ ] ( ] d 2 ( s1 2 k log[ ] ( d 2 ( s 1 s 2 s ( s 1 2 k (log[2 ] k k log[ ] ( d ( ( ( d (7 2 2 (1 ( s 1 2 ( s 1 Focusing on h ls rm nd using h by pr ingrion chniqu w hv ( s ( s1 ( s 1 k log[ ] k log[ ] k (8 ( d [ ( d] ( s 1 ( s 1 (1 subsiu h ls rm of Eq (7 wih Eq (8 bcoms ( s 1 s 2 s k (log[2 ] k L( s ( d ( ( 2 2 (1 ( s 1 2 ( s ( s 1 ( s 1 k log[ ] k [ ] ( d 2 2 ( s 1 ( s 1 (1 ( s1 s 2 s s 2 1 k (log[2 ] k k log[2 ] ( 1 ( d ( ( ( 2 ( s 1 (1 ( s 1 2 ( s 1 (( s 12 Th convrgnc of h ingrl pr of Eq (9 will b shown in Appndix Undr h condiion of 0 wih powr sris w rrrng Eq (9 o obin h following Lplc rnsformion k 1 L S d s ( s1 n 2( n ( ( 1 (( 1 ( ( s 1 n 0 ( s 1 s In ddiion h momn gnring funcion (mgf of is givn by M ( E( L( s for ll momns if h Lplc rnsform xiss (s shown in Appndix nd whr h rh momn is E( r ( 1 r L r (0 In priculr Th momn gnring funcion of Eq (4 is givn by (9
4 14 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( Figur 1 Som plos of h survivl funcion of h nw disribuion whr k = from h boom lin rspcivly s M ( s f ( d s 2 d ( b ( b ( ( s 1 b ( s 1 b b 2 2 d d ; s 0 2 ( b b 2( b (1 ( 1 (1 ( 1 n sb 4 ( 1 (10 ( s 1 2n n 0 In h cs of Eq (4 whr symmric propris dfin h s of domin s ( hn M ( s cn b xprssd s Eq(10 Som plos of h survivl nd rlibiliy funcions rld o h proposd disribuion r prsnd in Figur 1 Th survivl funcion of Eq (4 r xhibid in Figur 2 In survivl nlysis h hzrd funcion is dfind s f ( h( whr f ( nd S( r h pdf nd survivl S( Figur 2 Som plos of h survivl funcion for h nw disribuion wih som prmr vlus of Eq (4 funcion rspcivly Th hzrd funcion of h proposd disribuion (Eq (1 cn b wrin s 2 k h( 1 ( (2 k log[2 ] k log[ ] Som shps of h hzrd funcion of h proposd disribuion (Eq (1 wih som prmrs r shown in Figur 3 Figur 3 Som plos of hzrd funcions for h proposd disribuion wih prmr = 0 of Eq(1
5 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( Figur 4 Som plos of h hzrd funcions for h proposd disribuion wih som prmr vlus of Eq(4 Th hzrd funcion of Eq (4 cn b wrin s b 2 h( 2( b ( cb ( b (1 ( 2 rcn( 2rcn( whr svrl funcion wih som prmrs r illusrd in Figur 4 Figur 5 shows som shps of h pdf bsd on som slcd prmr vlus I illusrs h h nw proposd disribuion consiss of vrious shps Th symmric bhviors of Eq (4 r illusrd in Figur 6 3 Applicions Four d ss r fid wih h proposd disribuion Th firs xmpl dls wih h r of chng of filur ims of lcronic dvics rpord by Domm (2014 Th scond xmpl is h r of chng of soybn prics Chicgo Bord of Trd (CBOT from Jun 09 - Jun 14 fid wih h proposd disribuion Th hird nd fourh d ss r US July prcipiion (Top 8 soybns producion s from nd h inr-rrivl ims (10 minus d s for crs (Lw 2015 rspcivly Figur 5 Som pdf plos of nw disribuion wih som prmr vlus of Eq (1
6 16 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( Figur 6 Som pdf plos of h nw disribuion wih som prmr vlus of Eq (4 Th simd prmrs could b crrid ou by mximum liklihood simion (MLE In his sudy w us bbml (Bolkr & R Tm 2014 pckg of R progrmming lngug (R Cor Tm 2014 o obin h prmr sims Th fiing disribuions for h four xmpls r vrifid s shown in Figur 7 In ddiion h simd prmrs using MLE r shown in Tbl 1 4 Conclusions Th im of his ppr is o propos n lrniv mhod o gnr nw survivl funcion W hv don so using h rlionship bwn diffrnil quion nd hyprbolic funcion nd obind h soluion of h mhod s nonincrsing funcion; w obin h survivl nd h probbiliy dnsiy funcions from hs chniqus Morovr Figur 7 Som fid disribuions wih min(
7 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( Tbl 1 Th simd prmrs using MLE D s Esimd prmrs 1 Th r of chng of filur ims of lcronic dvics Th r of chng of soybn prics Chicgo Bord of Trd (CBOT from Jun 09 - Jun July prcipiion from Th inr-rrivl ims (10 minus d s for crs 0 2 k w orgniz h gnrl form of h modl Eq(6 which producs wo survivl nd h probbilisic funcions Th vrious grphicl syls of h rsul cn b usd o dmonsr is flxibiliy in nlyzing bhvior in rl d Acknowldgmns Th uhors would lik o hnk o Dprmn of Sisics Fculy of Scinc nd h Grdu School of Kssr Univrsiy Also w hnk o Chron Pokphnd Foods (CPF for supporing h firs uhor Rfrncs Bn W D (1934 Th probbiliy lw for h sum of n indpndn vribls Bullin of Amricn Mhmicl Sociy Bolkr B & Tm R D C (2014 Tools for Gnrl Mximum Liklihood Esimion (R pckg vrsion 1017 Domm F & Condino F (2014 A nw clss of disribuion funcions for lifim d Rlibiliy Enginring nd Sysm Sfy Fishr M J (2014 Gnrlizd hyprbolic scn disribuion Brlin Grmny: Springr Jffry A & Zwillingr D (2007 Tbl of ingrl sris nd produc Nw York NY: Acdmic Prss Jing R (2013 A nw bhub curv modl wih fini suppor Rlibiliy Enginring nd Sysm Sfy Klbflisch J D & Prnic R L (1980 Th sisicl nlysis of filur d Nw York NY: John Wily & Son Lw A (2015 Simulion modling nd nlysis (McGrw- Hill sris in indusril nginring nd mngmn Nw York NY: McGrw-Hill Lwlss J F (1982 Sisicl modls nd mhods for lifim d Nw York NY: John Wily & Son Mnoukin E B & Ndu P (1988 A no on h hyprbolic scn disribuion Th Amricn Sisicin R Cor Tm (2014 R: A lngug nd nvironmn for sisicl compuing Vinn Ausri: R Foundion for Sisicl Compuing Tlcko J (1956 Prk disribuions nd hir rol in h hory of winr s sochsic vribls Trbjos d Esisic Wng F K (2000 A nw modl wih bhub-shpd filur r using n ddiiv burr xii disribuion Rlibiliy Enginring nd Sysm Sfy
8 18 P Thongchn & W Bodhisuwn / Songklnkrin J Sci Tchnol 39 ( APPENDIX A convrgnc will b provd ( s 1 s 2 s 1 k (log[2 ] k d 2 2 ( s 1 (1 ( s 1 2 ( s 1 ( 1 ( ( ( Th ingrl is impropr ingrl w us h comprison s o prov convrgnc by h following horm A comprison s horm Suppos h f nd g r coninuous funcions wih 0 f ( g( for hn 1 If g( d is convrgn hn f ( d is convrgn 2 If g( d is divrgn hn f ( d is divrgn Espcilly his impropr ingrl s shown nd ( s1 k ( d 2 (1 W s ( s 1 g( d k ( s1 g( d k = k ( s 1 ( s 1 k f ( d ( d 2 2 (1 whrs ( ( s 1 [ ] ; s 0 k ( s1 I is vrifid h g( d convrg o [ ] hrrfor ( s 1 g is grr hn f ( for ll s 0 ( s 1 k f ( d ( d 2 (1 is convrgn oo
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