A Class of Deformed Hyperbolic Secant Distributions Using Two Parametric Functions. S. A. El-Shehawy

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1 A Class o Deormed Hyperbolc Secat Dstrbutos Usg Two Parametrc Fuctos S. A. El-Shehawy Departmet o Mathematcs Faculty o Scece Meoua Uversty Sheb El-om Egypt shshehawy6@yahoo.com Abstract: Ths paper presets a ovel class o deormed hyperbolc secat dstrbutos. We apply the deormato techque by troducg two parametrc uctos uder some certa approprate assumptos. We dscuss some mportat propertes o ths deed class o dstrbutos. Some measures ad uctos o ths ew class o dstrbutos are derved. A smple llustratve example s gve. [S. A. El-Shehawy. A Class o Deormed Hyperbolc Secat Dstrbutos Usg Two Parametrc Fuctos] Le Sc J 0; 0 ():897-90] (ISSN:097-85).. 6 ey words: Geeralzed hyperbolc secat dstrbuto Hyperbolc secat dstrbuto q(t)-hyperbolc ucto p- DHS dstrbuto p-dhs dstrbuto pq-dhs dstrbuto. Mathematcs Subject Classcato: 60E05 60E0 6E7 6E0. Itroducto The cocept o deormato techque has bee exploted to a great extet several elds o sceces [- 6 ]. The deormato techque s appled or the hyperbolc ad trgoometrc uctos. Recetly ths techque has bee used especally or the cotuous hyperbolc secat dstrbuto "HS-Dstrbuto" whch s symmetrc about zero wth ut varace [8 9]. Ths dstrbuto has probablty desty ucto "pd" the orm HS( x ) sech( x /) ; x R () ad t has some closed orms or some correspodg uctos (the momets-geeratg ucto "mg" characterstc ucto "c" cumulats-geeratg ucto "cg" ad score ucto "s") ad measures [5 0 ]. The amly o the cotuous pqdeormed hyperbolc secat dstrbutos "{pq-dhs dstrbuto}" has bee costructed ad studed [7]. Each pq-dhs dstrbuto has bee obtaed by troducg two postve scalar deormato parameters p ad q respectvely as two actors o the expoetal growth ad decay parts o the hyperbolc secat ucto "HS ucto" the HS dstrbuto. The pq-dhs dstrbuto s umodal wth ut varace. Its pd s gve by p q pq DHS ( x ; p q) sech pq ( x / ) ; x R. () The correspodg mg c cg ad s o ths dstrbuto have closed orms whch deped o the troduced scalar parameters p ad q. All momets o ths dstrbuto exst ad the mea the meda ad the mode have equal o-zero values as a ucto o the troduced real valued postve parameters p ad q [7]. The ma o ths paper s to dee ad study a class o p(w)q(w)-deormed hyperbolc secat dstrbutos whch s deoted by "p(w) q ( w ) -DHS dstrbutos" by troducg two real valued postve parametrc uctos p(w) ad q(w) (the deormato parametrc uctos). We cosder a lear ucto o the metoed radom varable wth coecets as uctos o a scalar parameter w. I ths study we wll cosder some approprate assumptos wth respect to the troduced para-metrc uctos as well as the used coecets the metoed lear ucto o the radom varable.. The p(w)q(w)-deormed hyperbolc secat dstrbuto Frstly we cosder the deormato techque or whch two real valued postve parametrc uctos p( w ) ad q ( w ) are troduced respectvely as two actors o the expoetal growth ad decay parts o the HS ucto the HS dstrbuto. The p( w ) q ( w ) -DHS dstrbuto s deed by meas o the p( w ) q ( w ) -deormato or the hyperbolc uctos. Now we dee the deormed hyperbolc uctos by troducg two arbtrary deormato parametrc uctos p( w ) ad q ( w ) ad we expla ther propertes. Deto. Let p( w ) ad q ( w ) are two arbtrary real postve deormato parametrc deretable uctos o w w R. We dee the p( w ) q ( w ) -deormed hyperbolc uctos to be a amly o uctos 897

2 cosh tah sech coth ad csch as pw ( ) e qw ( ) e pw ( ) e qw ( ) e p( w) q( w) cosh p( w) q( w) p( w) q( w) cosh p( w) q( w) tah p( w) q( w) coth q( w) cosh p( w) q( w) p( w) q( w) sech csch ; p( w) q( w) p( w) q( w) cosh p( w) q( w) p( w) q( w) () where ( x w ) s a real deretable ucto o x ad w ad t s a lear ucto x wth postve partal dervatve wth respect to x.e. x D ( w ) (0 ) as a dervatve o wth respect to x ad D ( w ) R. Lemma. A amly o the p( w ) q ( w ) -deormed hyperbolc uctos satses the ollowg relatos o the rst dervatves o tah cosh sech p w q w wth respect to x : ( ) ( ) ( p( w ) q( w ) ) C( w) cosh p( w) q( w ) (tah p( w ) q( w) ) C( w) p( w) q( w)sech p( w ) q( w) () (cosh p( w ) q( w ) ) C( w) p( w ) q( w ) (sech ) C ( w) sech tah. p( w ) q( w ) p( w ) q( w ) p( w ) q( w ) Furthermore p( w ) or q ( w ) the s ot odd ucto wth respect to ad cosh s ot eve ucto wth respect to.e. ( ) p( w) q( w) p( w ) q( w ) p( w) q( w ) cosh ( ) p( w) q( w) cosh. p( w) q( w ) p( w) q( w ) Moreover the ollowg relatos are satsed: cosh p( w ) q( w ) p( w ) q( w ) p( w ) q( w ) tah p( w ) q( w ) sech p( w ) q( w ) p( w ) q( w ) coth p( w ) q( w ) csch. p( w ) q( w ) p( w ) q( w ) Proo: Based o [ 7 ] ad Deto we ca drectly prove ths lemma. The ma dea o the suggested deormato techque s to geeralze the HS-dstrbuto a alteratve ormula whch depeds o two real postve parametrc uctos ad also to study ts mportat correspodg characterstcs. Here we exted the radom varable by ( w ) where w R. As a mmedate cosequece o prevous deto ad lemma we ca dee the pd o the costructed p( w ) q( w ) -DHS dstrbuto as the ollowg. Deto. Let DHS be a cotuous radom varable. Ths varable has a p( w ) q( w ) - DHS dstrbuto wth two postve real deormato parametrc uctos p( w ) ad q ( w ) ts pd gve by DHS( ; p ( w ) q ( w )) sech ( ) ( )( p w q w ); x w R (5) where p( w ) q( w ) (0 ) ad ( x w) R. I ths case DHS s sad to be a p( w ) q( w ) -DHS radom varable wth two parametrc uctos p( w ) ad q ( w ) deed over R. Furthermore the correspodg real valued cd amed Fp ( w ) q( w ) DHS ( ; p( w ) q( w )) s deed as F DHS( ; p( w ) q ( w )) arcta [ ( ) ( ) ( ) ] (6) p w q w wth the verse cd (crtcal value) p( w ) q ( w ) DHS x [arc[ta ( ( ))]. p( w ) D( w ) l ] q( w ) (7) where p( w) q( w) DHS p( w) q( w) DHS p( w) q( w) DHS P[ x ] F ( x ) (0). Wthout loss o geeralty let D ( w ) 0 ad ths DHS case the values x or some deret values o w ad or each xed par o the parametrc uctos p( w ) ad q ( w ) usg (7) ca be computed. 898

3 Now we wll ext preset some mportat propertes o ths costructed p( w ) q( w ) -DHS dstrbuto. Based o [7-9] ad the graphcal explaato ad uder some approprate assumptos the expoetal tal behavor o the p( w ) q( w ) - DHS dstrbuto guaratees the exstece o the expectato o DHS ad geerally all momets. I partcular the expectato o the varables DHS ad also DHS ca be derved ad gve respectvely by E[ p( w ) q( w ) DHS ] l[ q( w)/ p( w)] C( w) (8) E[ p( w ) q( w ) DHS ] (l[ q( w)/ p( w)]). C ( w) C ( w) Moreover the varace o DHS s / C ( w). Proposto. The -DHS dstrbuto wth two postve real deormato parametrc uctos p( w ) ad q ( w ) s symmetrc about 0 or p( w ) q( w ). Moreover t skewed more to the rght or p( w ) q( w ) ad skewed more to the let or p( w ) q( w ). For all postve real values o p( w ) ad q( w ) the kurtoss s always costat. Based o [7-9] deret pd's or the p( w ) q( w )- DHS dstrbuto wth pw ( ) q( w) (or pw ( ) qw ( )) or each xed par ( p( w ) q( w )) or p( w) q( w) or some real values o w ca be plotted ad llustrated whch t obvous graphcally that the Proposto s vald. Computatoally we ca also d the ollowg results: - or xed value o the parametrc ucto p( w ) t s clear that the mea o p( w ) q( w ) -DHS dstrbuto s versely proportoal wth the value o the parametrc ucto q ( w ). - or xed value o the parametrc ucto q ( w ) t s clear that the mea o p( w ) q( w )-DHS dstrbuto s versely proportoal wth the value o the parametrc ucto p( w ). Accordg to the orm S ( x ) ( pd ) /( pd ) o the score ucto "s" we ca derve ths ucto or the p( w ) q( w ) -DHS dstrbuto. Proposto. The score ucto o the varable DHS wth p( w ) q( w ) (0 ) s gve by S DHS ( ; p( w ) q( w )) tah. (9) Settg p( w ) q( w ) ad the last equato reduces to ( ) tah x S HS x where S ( ) HS x s the s o HS dstrbuto. Moreover whe p( w ) p ad q ( w ) q (.e. parameters) ad equato (9) reduces to ( ) tah x S p q DHS x p q whch s the s o the varable p q DHS wth p q (0 ). Proo: By usg (5) the orm (9) ca be obtaed wth the reduced cases S ( x ) HS ad S ( ) pq DHS x or p( w ) q( w ) C( w) ad p( w ) p q ( w ) q C( w) respectvely. Proposto. The p( w ) q( w ) -DHS dstrbuto s umodal or p( w ) q( w ) (0 ). Proo: Based o the pd o DHS (5) we am to show that ths ucto s umodal or all par o p( w ) ad q ( w ). Sce ths pd s a cotuously deretable ucto the oly crtcal pots or ths ucto satsy p( w ) q( w ) DHS ( ; p( w ) q( w )) 0 (the dervatve wth respect to x ). Now we wat to prove that the last equato has exactly oe root ad that ths yelds a relatve maxmum. Sce lm ( ; p( w ) q( w )) 0 DHS the there s oe crtcal pot t must yeld the absolute maxmum so we eed to prove there s exactly oe root to the dervatve equato. Ater smplcato ths ca be see to be equvalet to provg (sech ).(tah ) 0 has exactly oe root. Set 899

4 q ( w ) ( x ; p( w ) q( w )) ( y l[ ] ) the p( w ) last statemet s equvalet to showg that the equato sech( y ) tah( y ) 0 has exactly oe root y 0 R. Ths meas that the equato DHS ( ; p( w ) q( w )) 0 has oly the q ( w ) root ( x w ) l[ ] (.e. p( w ) q( w ) x l[ ] ) R. Sce the d p( w ) dervatve p( w ) q( w ) DHS ( ; p( w ) q( w )) wth respect to x s less tha 0 wth ( x w ) the the pot x s the maxmum value o the p( w ) q( w ) -DHS dstrbuto. Based o [7-9] ths yelds a relatve maxmum (ad hece absolute maxmum) sce the st dervatve p( w ) q( w ) DHS ( ; p( w) q( w)) s postve to the let o the root x ad egatve to the rght. Note that the mode or the p( w ) q( w ) -DHS dstrbuto has the above value o the root x whch equals the obtaed mea. Proposto. The mode " Mode -DHS " ad the meda " Meda -DHS " or the pw ( ) qw ( )- DHS dstrbuto wth pw ( ) qw ( ) (0 ) have the same value o the mea (8). Proo: Due to the umodalty o the dstrbuto the prevous obtaed results ad the act that the meda o the umodal dstrbuto les betwee the mea ad the mode o the same dstrbuto the gve statemet the proposto s vald. Note that the case whe p( w) p ad q( w) q (where p q 0) Cw ( ) the pq-dhs dstrbuto s recovered ad also the case o p( w) q( w) C( w ) gves the orgal HS dstrbuto. Now we wll derve some closed orms or the correspodg mg cg ad c o the p( w ) q( w ) - DHS dstrbuto. Moreover we wll deduce the correspodg momets skewess ad kurtoss coecets o ths costructed dstrbuto. Proposto 5. The mg o the varable wth p( w ) q( w ) (0 ) s gve by M ( ; ( ) ( )) sec DHS t p w q w t p( w) q( w) DHS t l[ q( w )/ p( w ) ] e (0) where t. I partcular all momets o DHS exst. Proo: By usg the substtutos ( x ; p( w ) q( w )) [ y l ( q( w ) / p( w ))] t ad B we d that C( w ) B y e sech y dy sec t where B. The the mg (0) o DHS ca be drectly obtaed rom the ollowg: B l[ q( w )/ p( w )] M ( t; p( w) q( w)) e p( w ) q( w ) DHS B y e sech y dy B. Proposto 6. The rst our o-cetral momets o DHS wth p( w ) q( w ) (0 ) are gve by l[ q( w )/ p( w )] C( w ) (l[ q( w )/ p( w )]) l[ ( )/ ( )] (l[ ( )/ ( )]) ( ) q w p w C w C ( w) q w p w 5 6 (l [ q( w )/ p( w )]) (l [ q( w )/ p( w )]). Proo: The prevous orms ths proposto ca be drectly derved by applyg the deto o ocetral momet where the obtaed tegra-to ca be easly worked out wth the help o some mathematcal packages. 900

5 From the prevous results Proposto 6 we ca d that the rst our cetral momets o are p( w ) q( w ) DHS Cosequetly the skewess ad the excess kurtoss are 0 ad respectvely. Usg the relato betwee the c ad mg we ca obta the c o the p( w ) q( w ) -DHS dstrbuto the ollowg closed orm: DHS ( t ; p ( w ) q ( w )) secht t l[ q( w )/ p( w ) ] e () where t. The ext proposto gves the closed orm o the cg ad the used closed orm to calculate the r-th cumulat k o p( w ) q( w ) -DHS dstrbuto. r Proposto 7. The correspodg cg o the varable wth p( w) q( w) (0 ) s gve by p( w ) q ( w ) DHS ( ; ( ) ( )) l[sec ] p( w ) q( w ) DHS t p w q w t t q( w ) l[ ] C( w ) p( w ) () C( w) where t. Moreover the r-th cumulat k r o q ( w ) DHS s determed by r k [ ( t ; p ( w ) q ( w ))] ( r ) r DH S t 0 r () where the deretato wth respect to t ad () q( w) p( w ) q( w ) DHS ( t; p( w) q( w)) l[ ] C( w) p( w) ta t p w q w ( t; p( w ) q( w )) ta t () ( ) ( ) DHS p w q w ( t; p( w ) q( w )) ta t ( ta t) () ( ) ( ) DHS () p( w ) q( w ) DHS ( t; p( w) q( w)) ( ta t) ta t ( ta t)... Proo: The orm () ca be derved by applyg the deto o cg where the obtaed tegrato ca be worked out wth the help o some mathe-matcal packages. Smlarly the r-th cumulats k o p( w) q( w) DHS or each value o r ca be drectly determed. From the prevous results we d that the momets are related wth the cumulats o DHS k k k C ( w).e. [ ( ) ] k k ad so o.. Maxmum Lkelhood Parameter Estmato I ths secto we wll llustrate the ML Method to determe a certa value o the parametrc ucto that maxmzes the probablty o the sample data rom the -DHS dstrbuto. To obta the MLE's or the para-metrc uctos pw ( ) ad q( w ) or the pw ( ) qw ( )-DHS dstrbuto we start wth the pd o the p( w) q( w) -DHS dstrbuto whch s gve (5). Suppose that are a d radom sample rom the p( w) q( w) -DHS dstrbuto the the lkelhood ucto s gve by / L( x x x ( pw ( ) q( w))) C ( w)( pw ( ) q( w)) [ p( w)exp( ) q( w)exp( )] () wth ( x ; w). The log-lkelhood ucto s ( w ) l( C ( w) p( w) q( w )) l[ p( w) exp( ) q( w)exp( )]. (5) Takg the dervatve o ( w ) ( p( w ) q( w )) wth respect to w ad settg t equals 0 yelds p( w ) q( w ) p( w ) q( w ) [ p( w ) q( w ) p( w ) q( w )] [ tah p( w ) q( w )( ) { p( w )exp( ) q( w)exp( )}sech p( w ) q( w )( )] (6) r 90

6 wth C ( w) x D ( w). Solvg (6) teratvely the the MLE's pˆ ( w) p( wˆ ) ad qˆ ( w ) q( wˆ ) ca be obtaed.. Illustratve Example We gve a llustratve example o the deormed dstrbuto ad expla some results. Let DHS be a cotuous radom varable whch ollows the -DHS dstrbuto wth p ( w ) ad q( w) exp( w). We cosder cosh( w ) x. I ths case we ca d that pd o DHS ca be deed by exp( w / ) ( ;exp( w )) sech( w ) sech ( ); x w R DHS ad the correspodg cd o DHS s F DHS( ;exp( w )) arcta[ exp( w / ) ( )] wth the crtcal value p( w ) q( w ) DHS x sech( w) [ (arc[ta( ( )) ] w ) ] where p( w ) q( w) DHS p( w ) q( w) DHS p( w ) q( w) DHS P[ x ] F ( x ) (0). We ca d that the st ad d o-cetral momet are gve respectvely by o DHS s.ech( w ) w w ad sech ( w ) [ ]. Moreover the varace s =sech ( w ). We ca also d that mg s gve by tw sech( w) M p( w) q( w) DHS ( t;exp( w)) sec( t)exp[ ] wth t. sech( w ) Moreover the rd ad th o-cetral momets o ca be obtaed as DHS w w sech ( w ) [ ] 6w w sech ( w ) [5 ]. Thus the rst our cetral momets o the varable are 0 sech ( w ) 0 p( w ) q ( w ) DHS 5 sech ( w ) ad also 0 ad. Drectly we ca d that c ad cg are gve respectvely by: t w.sech( w) p ( w) q( w) DHS ( t; exp( w)) sech( t)exp[ ] t w sech( w) p( w ) q( w ) DHS ( t; exp( w)) l[sec( t)] wth t. From ths orm o cg we sech( w ) ca d that: () w sech( w ) p( w ) q( w ) DHS ( t; exp( w )) ta( t) ( k ) ad DHS ( t;exp( w )) k... are deed Proposto 7. Moreover the momets are related wth the cumulats k sech ( w ) k k sech ( w ) [ k ( k ) ]... ad so o. Fally by solvg the ollowg olear system w teratvely [ x ( w ) tah ( ) tah( w ) exp( w )sech ( )] wth cosh( w ) x ad ( w ) x... oe ca obta w ˆ ad thus the MLE o q ( w ) cosh( w ) s qˆ ( w ) cosh( wˆ ). Deret destes or the p( w ) q( w ) -DHS dstrbuto wth q ( w ) (.e exp( w ) ) ad ther correspodg destes wth q ( w ) (.e exp( w ) ) or some values o w ca be graphcally llustrated. Moreover the dervatve o the umodal pd o p( w) q( w) -DHS dstrbutos s explaed the ollowg gure: 90

7 example o the obtaed results has bee preseted ud dscussed. Ackowledgemet I am thakul to Pro. M. F. El-Sayed A Shams Uversty (Caro) or hs ecouragemet sghtul commets ad scetc dscussos that cotrbuted to the preparato o ths work. Fgure : Dervatve o the umodal pd o p( w ) q( w ) -DHS dstrbutos wth ( p( w ) q ( w )) ( exp( w )) 5. Coclusos Ths paper dscussed the costructo o the class o p( w ) q( w )-DHS dstrbutos whch ca be cosdered as a correspodg exteso o the class o pq -DHS dstrbutos. Frstly we deed the pw ( ) qw ( )-deormed hyperbolc uctos whch have bee mplemeted by troducg two postve real valued parametrc uctos pw ( ) ad q( w ) as two actors o the expoetal growth ad decay parts o the HS dstrbuto. We studed the eect o these deormato parametrc uctos comparg wth other prevous studes o the HS-dstrbuto. We cosdered a deretable real valued ucto ( w ) stead o. We assumed that ths ucto s lear ucto x wth postve partal dervatve wth respect to x. We oud that each -DHS dstrbuto o the costructed class s umodal. I geeral t has varace wth value deret tha uty. We oted also that the derved closed orms o the correspodg mg c cg ad s or the p( w ) q( w ) -DHS dstrbuto deped o p( w ) q ( w ) ad the partal dervatve o wth respect to x. Furthermore some mportat propertes o the costructed class o deormed dstrbutos were dscussed. We oted that ther momets exst. There s uque value o ther mea meda ad mode whch stll also as a ucto o p ( w ) q( w ) ad C( w ). The skewess ad excess kurtoss o these costructed dstrbutos are stll res-pectvely equal to 0 ad. By applyg the ML method to determe the MLE or the parameters p( w ) ad q ( w ) we obtaed a olear system whch ca be solved teratvely by usg hgh processg systems o computers. A llustratve Reereces. Abdel-Salam E. A-B. (0). Q(t)-hyperbolc ucto method ad o-travellg wave solutos Nolear Sc. Lett. A No Al-Muhameed Z. I. A. Abdel-Salam E. A-B. (00) Geeralzed hyperbolc ad tragular ucto solutos o olear evoluto equatos Far East J. Math. Sc. 6 No Ara A. (99). Exactly solvable super-symmetrc quatum mechacs J. Math. Aal. Appl. 58 No Ara A. (00). Exact solutos o mult-compoet olear Schrödger ad le-gordo equatos two-dmesoal space-tme J. Phys. A: Math. Ge. No Bate W. D. (9). The probablty law or the sum o depedet varables each subjects to the law ( h) sech( x / h) Bull. Amer. Math. Soc. 0 No El-Sabbagh M. F. Hassa M. M. ad Abdel-Salam E. A-B. (009). Quas-perodc waves ad ther teractos the (+)-dmesoal moded dspersve water-wave system Phys. Scr. 80 No El-Shehawy S. A. (0). Study o the balace or the DHS dstrbuto It. Math. Forum 7 No El-Shehawy S. A. Abdel-Salam E. A-B. (0). The q- deormed hyperbolc secat amly It. J. Appl. Math. Stat. 9 No El-Shehawy S. A. Abdel-Salam E. A-B. (0). O deormato techque o the hyperbolc secat dstrbuto Far East J. Math. Sc. 6 No Fscher M. (006). The skew geeralzed secat hyperbolc amly Aust. J. o Stat. 5 No Hassa M. M. Abdel-Salam E. A-B. (00). New exact solutos o a class o hgher-order olear schroedger equatos J. Egypt Math. Soc. 8 No Lu. Q. Jag S. (00). The sec q -tah q method ad ts applcatos Phys. Lett. A 98 No Vaugha D. C. (00). The geeralzed hyper-bolc secat dstrbuto ad ts applcato Commu. stat. Theory ad Methods No //0 90