Study of the Balance for the DHS Distribution

Transcription

1 Itatioal Mathmatical Foum, Vol. 7, 01, o. 3, Study of th Balac fo th Distibutio S. A. El-Shhawy Datmt of Mathmatics, Faculty of scic Moufiya Uivsity, Shbi El-Kom, Egyt Cut addss: Datmt of Mathmatics, Collg of Scic Qassim Uivsity, P.O. Box 66 Buaidah 515, Kigdom of Saudi Aabia Abstact This a dals with alyig th dfomatio tchiqu o th hybolic scat (HS) distibutio by itoducig two ositiv aamts ad q. A family of dfomd hybolic scat (-) distibutios is obtaid. Ou goal is to study som otis of this family ad also to fomulat its cosodig fuctios ad masus. W dtmi th maximum liklihood stimats (MLE) fo th dfomatio aamts ad q. A alicatio will b std. Fially, som scial cass of th costuctd family of distibutios will b viwd ad comad. Mathmatics Subjct Classificatio: 60E05, 60E10, 6E17, 6E0 Kywods: Galizd hybolic fuctios, Galizd hybolic scat distibutio, Hybolic scat distibutio, Skw galizd scat distibutios 1 Itoductio I ct yas, sval tchiqus alid to symmtic distibutio i od to galiz ad gat symmtic o asymmtic distibutio with ossibly light o havi tails. Basd o [8], som mthods alid to us a tasfomatio of th obability dsity fuctio (df) as multilicatio of df of th oigial symmtic distibutio by a aoiat wightig fuctio of th cosodig cumulativ distibutio fuctio (cdf) with aamt vcto o a fixd itval. Oth tchiqus w suggstd i som litatus [, 6, 7, 11, 13]. A dfomatio tchiqu fo hybolic fuctios has b viwd ad alid i sval filds [3,, 5, 10]. Moov, this tchiqu has b usd fo obability distibutios ad th autho costuctd th -dfomd hybolic scat (-)

2 155 S. A. El-Shhawy distibutio ad th q-dfomd hybolic scat (q-) distibutio [, 5]. Pst study dals with th dfomatio of th balac btw xotial gowth ad dcay ats of th HS distibutio by itoducig, q simultaously. Som otis of th costuctd - distibutio will b viwd. This a is ogaizd as follows: Th followig sctio sts a sot of th HS distibutio. Sctio 3 is dvotd to xlai a -dfomatio tchiqu ad also to costuct th - distibutio with a suvy of its itstig otis. Som cosodig fuctios ad masus of costuctd distibutio a discussd i Sctio. I Sctio 5 th ML mthod with sct to th - distibutio is xlaid. Som scial cass of th - distibutio with thi otis will b oosd i Sctio 6. A bif summay ad discussio a std. Saml of tabls a ad ad giv i Adix. Symmtic hybolic scat distibutio Basd o [1,, 5, 7, 1], a cotiuous adom vaiabl X has a HS distibutio if its df is giv by 1 1 f HS( x) sch( x /) ; x( x / ) + x( x / ) x (1) ad th cosodig cdf is FHS( x) = acta[ x ( x / ) ]. () Moov, th ivs cdf (o quatil fuctio) of th HS distibutio is giv by HS -1 xα = FHS( α) = l[ta((1 α) /)], (3) HS HS wh PX [ > xα ] = 1 FHS( xα ) = α, α (0,1). This distibutio shas som otis with th stadad omal distibutio: - it is symmtic with uit vaiac ad zo ma, mdia ad mod, - th skwss ad th xcss kutosis a qual to 0 ad sctivly, - th momt-gatig fuctio (mgf) is M HS( t) = sc t, t < /, - th chaactistic fuctio (cf) is HS () sch, t < / 3 A dfomatio tchiqu ad th - distibutio 3.1 Dfomd hybolic fuctios I od to obtai th - distibutio by mas of th -dfomatio fo th hybolic fuctios [3, 10], som otis of th dfomd hybolic fuctios (DHF) will fist b calld, that is

3 Study of th balac fo th distibutio 1555 x x x x q + q sih x 1 sih x =, cosh x =, tah x =, sch x =, () cosh x cosh x wh x ad th dfomatio aamts, q > 0. Not that if, q 1 th sih x is ot odd fuctio ad cosh x is ot v fuctio, i.. sih ( x ) = sih x, cosh ( x) = cosh x. (5) Moov, a family of th DHF satisfis th followig latios: (sih x) = cosh x, (cosh x) = sih x, (tah x) = sch x, (6) (sch x ) = sch x tah x, cosh x sih x =, tah x = 1 sch x. 3. Exlaatio of th - distibutio Th mai ida of th dfomatio tchiqu is to wit th symmtic HS distibutio i a simila fomula of distibutio with two ositiv aamts ad to study som otis of this costuctd distibutio. Lt us dfi th followig al valud fuctio: 1 g( x;, q) = sch ( x /) ;, q > 0, (7) which satisfis g( x;, q) > 0, x, ad g( x;, q) d x = 1/ q. Thus, w ca dfi th df of a - distibutio as: f x q q g x q x I additio, th cosodig cdf ca b divd ad giv by th followig fom: F ( x;, q) = + acta[ sih ( x / )], () ( ;, ) = ( ;, ) = sch ( /). (8) ad its ivs (citical valu) is -HS xα = F -HS( α;, q) = [acsih[ta ( ( α))] l ( / q) ], (10) wh PX [ > xα ] = 1 F ( xα ) = α (0,1). Th dfomatio aamts, q idicat th sha of th costuctd distibutio. -HS Fo som difft valus of, q, th citical valu x α ca b comutd ad tabulatd (s Tabls, 5, 6). 3.3 Potis of th - distibutio Usig a aoiat substitutio of x, th xctatio of th - vaiabl X ad X ca b divd ad giv sctivly by th followig foms:

4 1556 S. A. El-Shhawy μ = EX [ ] = l q/ ad μ = EX [ ] = 1 + ( l q/ ). (11) I this cas, w ca fid that th vaiac of X quals 1. Poositio 1 Th - distibutio with two ositiv al valud aamts ad q is symmtic about 0 fo = q. Moov, it skwd mo to th ight fo > q ad skwd mo to th lft fo < q.fo all ositiv al valus of th aamts ad q, th kutosis is always costat. Th followig figus (a) ad (b) illustat difft dsitis fo th - distibutio with < q ad thi cosodig dsitis with > q fo som ositiv al valus of ad q. Moov, th obability dsity fuctio fo th - distibutio with th cas of = q is lottd i th figu (c). (a) (b) (c) Figu 1: Pobability dsity fuctio of th - distibutio fo som valus of ad q Gahically, it bcoms obvious that th Poositio 1 is valid. Basd o Tabl 7, w ca fid th followig: 1- fo fixd valu of, it is cla that th ma of th - distibutio is ivsly ootioal with th valu of q ad th vaiac quals o. - fo fixd valu of q, it is cla that th ma of th - distibutio is ivsly ootioal with th valu of ad th vaiac quals o. Accodig to th fom S ( x ) = ( df ) /( df ) of th sco fuctio, w ca also div this fuctio fo th - distibutio. Poositio Th sco fuctio S ( ;, ) x q of X is giv by S x q x Sttig = q = 1, quatio (1) ducs to SHS ( x) = tah( x / ), wh HS S ( x ) is th sco fuctio of th HS distibutio. ( ;, ) = tah ( / ). (1) Poositio 3 Th - distibutio is uimodal fo >, 0.

5 Study of th balac fo th distibutio 1557 Poof: Basd o th giv df f ( ;, ) x q i (8) fo th - distibutio, w aim to show th uimodality of this fuctio fo all valus of ad q. Sic this df is a cotiuously difftiabl fuctio, th oly citical oits fo this fuctio satisfy th quatio f ( ;, ) 0 x q =. Thus w wat to ov that this quatio has xactly o oot yilds a lativ maximum. Sic lim f ( x;, q) = 0, th x ± if th is o citical oit, it must yild th absolut maximum, so fa w d to ov that th is xactly o oot to th divativ quatio. Aft simlificatio, (this ca b s to b quivalt to ovig) sch ( x / ) tah ( x / ) = 0 has xactly o oot. By takig x = [ y + l q / ], th last statmt is quivalt to showig sch( y) tah( y ) = 0 has xactly o al oot y = 0. This mas that th quatio ( ;, ) 0 f * x q = has oly th al oot x = l q /. Sic * f ( ;, ) /8 * x q = is gativ, th th oit x is th maximum valu of th - distibutio. It th also follows this yilds a lativ maximum (ad hc absolut maximum) sic f ( ;, ) * x q is ositiv to th lft of th oot x, ad gativ to th ight (s Figu ). Figu : Divativ of th Uimodal df of th - distibutios with >, 0 Not that, th mod fo th - distibutio has th valu l q / Poositio 3 ad it is qual to th ma fo th - distibutio. * of x i Poositio Th mod ad th mdia fo th - distibutio hav th sam valu of th ma, fo >, 0. Poof: Du to th uimodality of th giv distibutio ad th vious obtaid sults ad th fact that th mdia of th uimodal distibutio lis btw th ma ad th mod of th sam distibutio, w ca simly fid that th mod ( Mod ) ad th mdia ( Mdia - ) fo th - distibutio hav th sam valu of th ma fo th - distibutio fo >, 0. -

6 1558 S. A. El-Shhawy Momts of th - distibutio I this sctio, w will div th fomulas fo th mgf, th cf ad also th cgf of th - distibutio. Moov, th momts ad cumulats ca b dducd. Cosqutly, w dduc th cosodig skwss ad kutosis cofficits. Poositio 5 Th mgf, cf ad cgf of th vaiabl X with >, 0 a giv by th followig fomulas sctivly: t l q / M ( t;, q) = sc t, it l q/ Ψ (; t, q) = t K ( t;, q) = l sch t, q / + l(sc t), wh t < / ad i = 1. Poof: Aly th dfiitio of th mgf M( t) = E[ ] fo th giv vaiabl ad us th substitutios x = [ z + l q / ] ad B = t /, w gt o th followig: tx 1 l / M ( ;, ) ( ;, ) B q B z t q = f x q d x = sch z d z. (1) Accodig to [], th itgatio i th ight sid of (1) ca b obtaid as: B z tx (13) sch ( z) d z = sc( t) ; B < 1, (15) which ca b wokd out with th hl of Mal o Mathmatica. Fom (1) ad (15) th mgf of X is obtaid. Th cosodig cf of X ca b dictly dducd fom th gal latio Ψ () t = M( it). Fom th latio K( t) = l[m( t)] w ca also obtai th quid fomula of th cgf. Poositio 6 Th vaiabl X with >, 0 has th followig statmts: 1- th -th o-ctal momts μ, = 1,,3, K, a giv by d ( ) μ t= 0 t= 0 = [ M ( t;, q)] = [M ( t;, q)], = 1,,3, K, () dt ad th fist fou o-ctal momts a giv by

7 Study of th balac fo th distibutio 155 μ 1 = EX [ ] = l q/, μ = EX [ ] = 1 + (l q/ ), μ 3 = E[ X ] = l q / + (l q / ), (17) 3 μ = E[ X ] = 5 + (l q / ) + (l q / ). - th -th cumulats k, = 1,,3, K, of X ca b dtmid by d ( ) k = [ K ( t;, q)] t 0 [K ( t;, q)] t 0, 1,,3, = = = = K (18) dt wh (1) K ( t;, q) = l q / + ta t, () K ( t;, q) = 1+ ta t, (1) (3) K ( t;, q) = ta t (1+ ta t), () = K ( t;, q) (1 ta t) ta t (1 ta t), K ( t;, q) = (1 + ta t) tat + ta t (1 + ta t), K (5) 3 Not that th fomulas (17) ca b wokd out with th hl of Mal o Mathmatica. Usig (17) ad th latio btw th ctal momts ad th octal momts, w ca dtmi th fist fou ctal momts of X as μ1 = μ3 = 0, μ = σ = 1, μ = 5. This imlis that X has skwss 0 ad xcss kutosis. Th -th cumulats k of X fo som valus of ca b wokd out with th hl of Mal o Mathmatica ad th sults hav b tabulatd (s Tabl 8). Moov, th momts of X a latd with th cumulats, μ 1 = μ = k1, σ = μ = k, μ3 = k3, μ = k + 3( k ), K. (0) Comutatioally, it ca b s that k 1 = μ ad k is qual to 0, 3 wh is odd. Moov, k is dictly ootioal to, wh is v (s Tabl 8). 5 Maximum liklihood aamt stimatio To obtai th MLE fo th dfomatio aamts of th - distibutio, stat with th df of th - distibutio i th fom ( ;, ) f ; x q = x x( x / ) + q x( x / ). (1)

8 1560 S. A. El-Shhawy Suos that X 1, X, K, X a a iid adom saml wh ach X i, i = 1,,..., has th - distibutio. Th th log-liklihood fuctio fo th costuctd - distibutio is l (, q) = l( q) l[ x( x i / ) + qx( x i / )]. () i = 1 Takig th atial divativ of th log-liklihood fuctio with sct to ach o of ad q ad sttig ach o of th sults qual to zo. Th, w obtai x( xi / )sch ( xi / ) = 0, x( xi / )sch ( xi / ) = 0. (3) i= 1 q i= 1 Solv th systm (3) itativly, th MLE ˆ ad ˆq ca b dducd. 6 Alicatio Basd o [1], a datast "fusio.tim" is th tim to cogiz stogam of adom oits, ad th uos is to s th vval xlaatio shots th tim. W will us two samls wh th fist (with th vval xlaatio) is dotd by (V) ad th scod (without th vval xlaatio) is dotd by (N). Ths two samls a assumd to b fom th - distibutio. Th aamts of th - distibutio ad th iducd distibutios (th - distibutio ad th q- distibutio [, 5]) will b stimatd. Th ML mthod ca b alid to obtai th MLE fo th dfomatio aamts of th - distibutio ad its iducd distibutios. Usig EXCEL fo som choics valus of th mtiod aamts, w ca obtai MLE of ad q fo ach o of th two samls (V) ad (N). Th sults a moitod ad also tabulatd. Tabl 1: MLE fo ad q of - Distibutio, q- Distibutio ad - Distibutio to th saml (V) Data (V) Distibutio - q- - ML Fuctio.3778E E E-8 ( ) (, ) MLE ˆ = ˆq = 1500 ˆ, q ˆ = Tabl : MLE fo ad q of - Distibutio, q- Distibutio ad - Distibutio to th saml (N) Data (N) Distibutio - q- - ML Fuctio 3.705E E E-151 ( ) (, ) MLE ˆ = ˆq = 301 ˆ, q ˆ = It is obvious that, th balac is valid fo ach of th dfomd distibutio at th MLE of th cosodig aamt i Tabls 1 ad fo ach o of th samls (V) ad (N). Th omal cuvs a addd to th obtaid histogams of th two

9 Study of th balac fo th distibutio 1561 samls (Figu 3). This idicats th o-omality i both samls. I th followig figu, th diffc btw th cosidd samls (V) ad (N) is vidt Dsity Ma StDv.80 Saml Siz Dsity Ma StDv Saml Siz Nomal 0.03 Nomal (V) (a) (b) Figu 3: Th tim to cogiz histogam of th samls (V) ad (N) 7 Scial cass of th - distibutio Fist of all, fo = q = 1 th HS distibutio is covd. Sttig = 1 o q = 1, skw HS distibutios ca b obtaid [, 5]. Moov, th followig tabl cotais som scial cass of th - distibutio accodig to th aamts ad q. Tabl 3: Scial cass of th - distibutio a + 1 Cas q =, a, a, + + a q = = a q = a, a q =, a a f sch( x + l ) sch( x + a ) sch( x l a) sch( x + l ) a 1 1 x 1 1 F + acta[sih( + x l )] acta[sih( a 1 1 x x 1 a + + )] + acta[sih( l a)] + acta[sih( + l )] a x x x 1 x 1 a S tah( + l ) tah( + a) tah( l a) tah( + l ) a M t l at t la ( a 1) t a l sct Ψ sct it l iat a scht t K l + l(sc t ) a μ l a μ 1 + (l ) a scht 0 10 it la (N) sct 0 scht 30 0 ( a 1) i t l sct scht at t ( a 1) t + l(sc t ) l a+ l(sc t) l + l(sc t ) 1 + a a l a l a 1 + ( ) ( a 1) l ( a 1)l 1 + [ ] 7 Summay I this a, w alid th dfomatio tchiqu of th hybolic fuctios to obability distibutio by itoducig two ositiv al valud dfomatio aamts ad q. W costuctd th - distibutio which aiss as

10 156 S. A. El-Shhawy dfomatio fo th HS distibutio. Th aamts ad q hav b itoducd sctivly as scala factos of th xotial gowth ad dcay ats of th HS fuctio i th HS distibutio. Basd o th difft valus of th ai of aamts (, q ), a family of cotiuous skwd obability distibutios has b obtaid. Fo fixd valu of o of th dfomatio aamts, w foud that th ma of th - distibutio is ivsly ootioal with th valu of th oth dfomatio aamt ad th vaiac quals o. Th uimodality of th costuctd - distibutios is valid ad vidt. Moov, th ma, th mdia ad th mod hav th sam valu. Th cosodig mgf, cf, cgf ad th sco fuctio fo th - distibutio hav b divd i closd foms. W foud also that th skwss ad xcss kutosis of this distibutio a sctivly 0 ad. Futhmo, som scial cass of th - distibutio with thi chaactistics hav b illustatd. Samls of th citical valus of th - distibutio, valus of ma ad -th cumumats fo som valus of ad q hav b comutd ad tabulatd. Th MLE ˆ ad ˆq hav b dducd. Fially, a alicatio has b std to xlai th MLE ˆ ad ˆq. I th futu, w ho to ovid futh studis o th alicatio of th dfomatio tchiqu with oth hybolic o tigoomtic distibutios. Adix Th tabulatd valus ca asily b wokd out with th hl of Mal o Mathmatica. Tabl : Citical Valus x α of th - distibutio which satisfy th latio PX [ > x α ] = α fo = q α x α α x α

11 Study of th balac fo th distibutio 1563 Tabl 5: Citical Valus x α of th - distibutio which satisfy th latio PX [ > x α ] = α fo q q α Tabl 6: Citical Valus x α of th - distibutio which satisfy th latio PX [ > x α ] = α fo q q α

12 156 S. A. El-Shhawy Tabl 7: Som valus fo th ma of th - distibutio q> > 0 > q> 0 q Ma q Ma Tabl 8: Som valus fo th -th cumulat of th - distibutio with >, k l q / k Ackowldgmts. I would lik to off thaks to Pofsso D. Mohammd Hlmi Maha - Dt. of Math., Collg of Scic, Qassim Uivsity - fo his scitific discussios ad suggstios that cotibutd to th aatio of this wok. Rfcs 1. W. D. Bat, Th obability law fo th sum of iddt vaiabls, ach 1 subjcts to th law ( h) sch( x / h), Bullti of th Amica Mathmatical Socity, 0 (13), L. Dvoy, O adom vaiat gatio fo th galizd hybolic scat distibutios, Statistics ad Comutig, 3 (13), M. F. El-Sabbagh, M. M. Hassa, ad E. A-B. Abdl-Salam, Quasi-iodic wavs ad thi itactios i th (+1)-dimsioal modifid dissiv wat-wav systm, Phys. Sc., 80 (00), S. A. El-Shhawy ad E. A-B. Abdl-Salam, Th q-dfomd Hybolic Scat Family, It. J. Al. Math. Stat. (01), No. 5, 51-6.

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