Avalable ole wwwjsaercom Joural o Scec ad Egeerg Research, 0, ():0-9 Research Arcle ISSN: 39-630 CODEN(USA): JSERBR NEW INFORMATION INEUALITIES ON DIFFERENCE OF GENERALIZED DIVERGENCES AND ITS APPLICATION KC Ja, Praphull Chhabra Deparme o Mahemacs, Malavya Naoal Isue o Techology, Japur- 3007 (Rajasha), INDIA Absrac Dvergece measures are useul or comparg wo probably dsrbuos Depedg o he aure o he problem, he dere dvergeces are suable So s always desrable o creae a ew dvergece measure I hs work, ew ormao equales, correspodg o derece o wo geeralzed - dvergeces, are obaed ad characerzed Secodly, we oba ew dvergece measure correspodg o ew covex uco ad dee he properes Furher, bouds o ew dvergece erms o oher sadard dvergeces are evaluaed Comparso o hs dvergece wh ohers s doe as well Idex erms: New Covex ad ormalzed uco, New dvergece measure, Comparso graph o dvergeces, New ormao equales, Bouds o ew dvergece Mahemacs Subjec Classcao: Prmary 9A7, Secodary 6D5 Iroduco P p, p, p3, p : p 0, p, Le be he se o all complee e dscree probably dsrbuos I we ake p 0 or some,, 3,,, he we have o suppose 0 ha 0 0 0 0 0 Csszar s - dvergece [] s a geeralzed ormao dvergece measure, whch s gve by (), e, p C P, q q () P p E C PC p q q Ad,,, () Smlarly (Ja ad Saraswa [5]) roduced a geeralzed measure o ormao gve by p q S P, q q (3) Joural o Scec ad Egeerg Research 0
Joural o Scec ad Egeerg Research, 0, ():0-9 Where : (0,) R (se o real o) s real, couous ad covex uco ad,,,,,,, P p p p p q q q q 3 3 Γ, where p ad q are probably mass ucos May kow dvergeces ca be obaed rom hese geeralzed measures by suably deg he covex uco Some o hose are as ollows Ch- square dvergece [8] = P, Relave JS dvergece [9] = F p q () q p, p log p q Relave J- dvergece [3] = J p q Relave AG dvergece [0] = G (5) p q, log q R Tragular dscrmao [] = P, J- dvergece [6,7] = J p q (6) p q p q, log p (7) p q p, log (8) p q (9) q We ca see ha JR P, F, P G, P, P, W P,,,,,whereW J J J P R R, pq p q ad s Harmoc mea dvergece Dvergeces rom () o (7) are o- symmerc ad (8), (9) are symmerc, wh respec o probably dsrbuo P, Γ () ad (9) are also kow as Pearso dvergece ad Jereys- Kullback- Lebler dvergece, respecvely Besde hese, Symmerc Ch- square dvergece [] ca be wre as he sum o Ch- square dvergece ad s adjo, e, P,, P P, p q p q (0) pq New Iormao Iequales I hs seco, we roduce ew ormao equales o derece o wo geeralzed - dvergeces Such equales are or sace eeded order o calculae he relave ececy o wo dvergeces Theorem Le : I R R, 0 ad suppose he assumpos: a be wo covex ad ormalzed ucos, e ad are wce dereable o (α, β) where 0, b There exss he real cosas m, M such ha m < M ad, 0, m M, () Joural o Scec ad Egeerg Research
I P,, he we have he equales, Joural o Scec ad Egeerg Research, 0, ():0-9,,,,,, me S E S M E S () Where E P,, S P, Proo: Le us cosder wo ucos ad m are gve by () ad (3) respecvely F m, (3) M F M () Where m ad M are he mmum ad maxmum values o he uco Sce Fm FM ad he ucos ad ad,, 0 0, (5) are wce dereable The vew o (), we have 0, (6) Fm m m 0 (7) FM M M I vew (5), (6) ad (7), we ca say ha he ucos β) Now, wh he help o leary propery, we have, Fm ad FM are ormalzed ad covex o (α, E,,,, F m SF E m m S m E P, me,,, S ms ad E,,,, F M SF E M M S M ME P, E,,, MS S Sce E P, S P,, (8) (9) rom [5], hereore (8) ad (9) ca be wre as he ollowgs E P, S P, me P, S P, 0,,,,, 0 ad M E S E S Or E P, S P, m E P, S P, (0), ad M E P, S P, E P, S P, () (0) ad (), ogeher gve he resul () New Dvergece Measure ad Properes Joural o Scec ad Egeerg Research
Joural o Scec ad Egeerg Research, 0, ():0-9 I hs seco, we oba ew dvergece measure or ew covex uco; urher dee he properes o ew covex uco ad ew dvergece Frsly, Le : 0, ad R be a uco deed as 3, 0,, 0,, (3) 3 (3) 3 Properes o uco deed by (3), are as ollows a Sce 0 s a ormalzed uco s a covex uco as well b Sce 0 0, c Sce d a 0 mooocally creasg 0, ad a, s mooocally decreasg 0, ad 00 000 0, 8 0 0, ad 800 600 00 00 6 8 0 Fgure 3: Covex uco Now pu ad (3) ad () respecvely, we ge he ollowg ew dvergece measure,,,, S E S 3 3 p q 3p 3p q p q 6 pq 8q (33) 8p q p q Properes o ew dvergece measure deed (33), are as ollows a S P, s covex ad o- egave he par o probably dsrbuo S P, 0 P or p q (Aas s mmum value) b P, Joural o Scec ad Egeerg Research 3
c Sce S P, S, P S, Joural o Scec ad Egeerg Research, 0, ():0-9 s o- symmerc dvergece measure Applcao o New Iormao Iequales I hs seco, we oba bouds o ew dvergece measure (33) by usg ew equales deed (), erms o sadard dvergeces Proposo Le P,, J P,, J P, ad S, (33) respecvely For P, Γ, we have a I 0 065, he 863 J P,, P J P, S P, R R be deed as (), (6), (9) ad 3 3 max, J P,, P J P, R () b I 065, he 3 J P,, P J P, S P, R 3 J P,, P J P, R () Proo: Le us cosder log, 0,, 0, log ad (3) Sce 0 0 ad 0, so s covex ad ormalzed uco respecvely Now, Pu (3) ad Now, le g Ad (), we ge he ollowgs p q p q S J R q (), log, p p q, log,, (5) E p q J P q p 3 are gve by (3) ad (3) respecvely, where ad 5 6 9 g, g 3 3 3 Joural o Scec ad Egeerg Research
I g 0 069793 065 I s clear ha g () s decreasg (0, 065] ad creasg (065,) Joural o Scec ad Egeerg Research, 0, ():0-9 Also g () has a mmum value a =065, because g 065 30035 0 Now, a I 0 065, he, m g g 065 863 (6) 3 3 M sup g max g, g max,, b I 065, he (7) 3 m g g (8), 3 M sup g g (9), The resuls () ad () are obaed by usg (33), (), (5), (6), (7), (8) ad (9) () Proposo Le P,, F P, ad S, P, Γ, we have a I 0 058, he P F PS 68,,, 3 3 max,, P F, P b I 058, he 3,,, P F PS P F P 3,, Proo: Le us cosder Sce Pu be deed as (), (5) ad (33) respecvely For (0) () log, 0,, 0, ad () 0 0 ad 0, so s covex ad ormalzed uco respecvely Now, (3) ad (), we ge he ollowgs q S q F P p q (3), log, Joural o Scec ad Egeerg Research 5
E Now, le g Ad P, Joural o Scec ad Egeerg Research, 0, ():0-9 p q q q pq q p q p p q p p p p p q q p, P () p 3 9 g, g 9 3 g 0 05773 058 0 I are gve by (3) ad () respecvely, where ad I s clear ha g () s decreasg (0, 058] ad creasg (058,) Also g () has a mmum value a =058, because g 058 38 0 Now, a I 0 058, he, m g g 058 68 (5) 3 3 M sup g max g, g max,, b I 058, he (6) 3 m g g (7), 3 M sup g g (8), The resuls (0) ad () are obaed by usg (33), (3), (), (5), (6), (7) ad (8) () Proposo 3 Le,,, P, Γ, we have a I 0 076, he G J ad S, 698 J P, G, P S P, max,,, b I 076, he 3 3 J G P 3 J P, G, P S P,,, Proo: Le us cosder 3 J G P be deed as (7), (9) ad (33) respecvely For (9) (0) Joural o Scec ad Egeerg Research 6
Sce Pu log, 0,, 0, log ad Joural o Scec ad Egeerg Research, 0, ():0-9 () 0 0 ad 0, so s covex ad ormalzed uco respecvely Now, (3) ad Now, le g Ad (), we ge he ollowgs p q p q S G P q (), log, p E p q J q (3), log, 3 3 g, g 3 I g 0 07598 076 0 are gve by (3) ad () respecvely, where ad I s clear ha g () s decreasg (0, 076] ad creasg (076,) Also g () has a mmum value a =076, because g 076 796 0 Now, a I 0 076, he, m g g 076 698 () 3 3 sup max, max, M g g g, b I 076, he (5) 3 m g g, (6) 3 M sup g g (7), The resuls (9) ad (0) are obaed by usg (33), (), (3), (), (5), (6) ad (7) () Proposo Le P,, P, ad S, P, Γ, we have 3, P U P, P, S P, 3, P U P, P, be deed as (), (8) ad (33) respecvely For (8) Joural o Scec ad Egeerg Research 7
Proo: Le us cosder Sce Pu, 0,, 0, Joural o Scec ad Egeerg Research, 0, ():0-9 ad (9) 3 0 0 ad 0, so s covex ad ormalzed uco respecvely Now, (3) ad (), we ge he ollowgs p q (30) S P, P, p q E P, where U P, p q p q p q p q q p p p =, P U P, (3) p q q p 3 Now, le g 3 ad g 0 0 ad (9), respecvely I s clear ha g () s creasg 0,, so, are gve by (3), where ad m g g 3 (3), M sup g g 3 (33) The resul (8) s obaed by usg (33), (30), (3), (3) ad (33) () Fgure shows he behavor o S P,, P,, J P,, P, ad P, cosdered p a, a, q a, a, where 0, dvergece S P, has a seeper slope ha P,, J P,, P, ad P, We have a I s clear rom gure ha he ew Joural o Scec ad Egeerg Research 8
Joural o Scec ad Egeerg Research, 0, ():0-9 Reereces Csszar, I, Iormao ype measures o dereces o probably dsrbuo ad drec observaos, Suda Mah Hugarca, Vol, pp 99-38, 967 Dacuha- Caselle D, Ecole d Ee de Probables de, Sa-Flour VII-977, Berl, Hedelberg, New York: Sprger, 978 3 Dragomr SS, Gluscevc V, Pearce CEM, Approxmao or he Csszar -dvergece va mdpo equales, equaly heory ad applcaos - YJ Cho, JK Km ad SS Dragomr (Eds), Nova Scece Publshers, Ic, Hugo, New York, Vol, 00, pp 39-5 Dragomr SS, Sude J ad Buse C, New equales or Jereys dvergece measure, Tamus Oxord Joural o Mahemacal Sceces, 6() (000), 95-309 5 Ja K C, Saraswa RN, Some ew ormao equales ad s applcaos ormao heory, Ieraoal joural o mahemacs research, Volume, Number 3 (0), pp 95-307 6 Jereys: A vara orm or he pror probably esmao problem Proc Roy Soc Lo Ser A, 86(96), 53-6 7 Kullback S ad Lebler RA, O Iormao ad Sucecy, A Mah Sas, (95), 79-86 8 Pearso K, O he Crero ha a gve sysem o devaos rom he probable he case o correlaed sysem o varables s such ha ca be reasoable supposed o have arse rom radom samplg, Phl Mag, 50(900), 57-7 9 Sbso R, Iormao radus, Z Wahrs Udverw Geb, () (969),9-60 0 Taeja IJ, New developmes geeralzed ormao measures, Chaper : Advaces Imagg ad Elecro Physcs, Ed PW Hawkes, 9(995), 37-35 Joural o Scec ad Egeerg Research 9