MonotonicBehaviourofRelativeIncrementsofPearsonDistributions

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1 Globl Journl o Science Frontier Reserch: F Mthemtics nd Decision Sciences Volume 8 Issue 5 Version.0 Yer 208 Type : Double lind Peer Reviewed Interntionl Reserch Journl Publisher: Globl Journls Online ISSN: & Print ISSN: Monotonic ehviour o Reltive Increments o Person Distributions y University o otswn bstrct- Theory hs been developed in order to clssiy distributions c-cording to monotonic behviour o their reltive increment unctions. We pply the results to Person distributions. GJSFR-F Clssiiction: FOR Code: MSC 200: 2648 MonotonicehviouroReltiveIncrementsoPersonDistributions Strictly s per the complince nd regultions o: This is reserch/review pper, distributed under the terms o the Cretive Commons ttribution-noncommercil 3.0 Unported License permitting ll non commercil use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.

2 Re 2. Szbo, Z. : Investigtion o Reltive Increments o Distribution Functions. Publ. Mth. De-brecen ), pp Monotonic ehviour o Reltive Increments o Person Distributions Let x) be the probbility density unction o continuous rndom vrible X with open support I. The corresponding distribution unction o X, Xx), is deined by F x) = x t)dt The reltive increment unction, h, o distribution unction, F, is deined by where c is positive constnt. bstrct- Theory hs been developed in order to clssiy distributions c-cording to monotonic behviour o their reltive increment unctions. We pply the results to Person distributions. I. Introduction hx) = F x + c) F x) F x) Lemm. Let F be twice dierentible distribution unction with F x) <, F x) = x) > 0 or ll x in I. We deine the unction Ψ s ollows Ψx) = F x) ). x) 2 x) I Ψ < Ψ > ), then it hs been proven tht the unction h strictly increses strictly decreses)[2] probbility distribution with probbility density unction x) is sid to be Person distribution i x) x) = Qx) qx) where Qx) = x + nd qx) = x 2 + bx + c nd,,, b, c re rel constnts with 2 + b 2 + c 2 > 0, > 0 [] Globl Journl o Science Frontier Reserch Volume XVIII Issue V V ersion I Yer 208 F ) The ollowing our theorems hve been proven beore nd we will use them to ormulte results bout Person distributions. uthor: Deprtment o Mthemtics, University o otswn. e-mil: reikeletsengsn@mopipi.ub.bw 208 Globl Journls

3 Monotonic ehviour o Reltive Increments o Person Distributions II. Theorem. I the probbility density unction hs the ollowing properties..) I = r, s) R is the lrgest inite or ininite open intervl in which > 0..2) There exists m in I t which is continuous nd m) = 0.3) > 0 in r, m) nd < 0 in m, s).4) is twice dierentible in m, s) ) ).5) = d dx > 0 in m, s), Then the corresponding continuous reltive increment unction, h, behves s ollows: I Ψs ) = lim Ψx) exists, then x s h strictly increses in I i Ψs ) h strictly increses in r, y) nd strictly decreses in y, s) or some y in I i Ψs ) >.[2] Re Globl Journl o Science Frontier Reserch Volume XVIII Issue V V ersion I Yer 208 2F ) III. Theorem 2. I the probbility density unction hs the ollowing properties. 2.) I = r, s) R is the lrgest inite or ininite open intervl in which > ) m = r 2.3) < 0 in r, s) 2.4) is twice dierentible in r, s) 2.5) ) = d dx ) < 0 in r, s), then r is inite. i) I Ψr + ) < or Ψr + ) = nd Ψ < in some right neighbourhood o r), then Ψ < in I nd the corresponding reltive increment unction strictly increses in I. ii) I Ψr + ) >, then I Ψs ), then Ψ > nd the reltive increment unction strictly decreses in I. I Ψs ) <, then Ψ > in r, y) nd Ψ < in y, s) or some y I, so the reltive increment unction strictly decreses irst nd then strictly increses.[2] Monotonic ehviour o Reltive Increments o Person Distributions Theorem 3. I the probbility density unction hs the ollowing properties. 3.) I = r, s) R is the lrgest inite or ininite open intervl in which > ) m = r 3.3) < 0 in r, s) 3.4) is twice dierentible in r, s) 3.5) ) = d dx ) > 0 in r, s), then r is inite i) I Ψm + ) > or Ψm + ) = nd Ψ > in some right neighbourhood o m), then Ψ > in I nd the corresponding reltive increment unction strictly decreses in I. ii) I Ψm + ) <, then I Ψs ) <, then Ψ < nd the reltive increment unction strictly increses in I. I Ψs ) >, then Ψ < in m, x) nd Ψ > in x, s) or some x I, so the reltive increment unction strictly increses irst nd then strictly decreses.[4] Theorem 4. Let x) be probbility density unction o U type distribution where.).4) re ulilled. Suppose ) > 0 in m, y) nd ) < 0 in y, s) or some y m, s). Then the corresponding continuous reltive increment unction, h, behves s ollows: I Ψy) < nd Ψs ) < then strictly increses in I or 2. Szbo, Z. : Investigtion o Reltive Increments o Distribution Functions. Publ. Mth. De-brecen ), pp Globl Journls

4 Monotonic ehviour o Reltive Increments o Person Distributions Re 4. Szbo Z. : On Monotonic ehviour o Reltive Increments o Unimodl Distributions. I Ψy) then i Ψs ) > h strictly increses in r, x 0 ) or some x 0 m, y) nd strictly decreses in x 0, s). i Ψs ) < then h strictly increses in r, x ) or some x m, y) nd strictly decreses in x, x 2 ) or some x 2 y, s) then inlly strictly increses in x 2, s).[4] Theorem 5. Let x) be probbility density unction o J type distribution. ssume m = r,.),.3),.4) re ulilled. Suppose ) < 0 in m, Y ) nd ) > 0 in Y, s) or some Y m, s). Then the corresponding continuous reltive increment unction, h, behves s ollows: i Ψm + ) = lim Ψx) exists, then x m + I Ψm + ) < nd Ψs ) < then h strictly increses in I or Ψs ) > then h strictly increses in m, x 3 ) or some x 3 y, s) nd strictly decreses in x 3, s) I Ψm + ) nd Ψy) > then h strictly decreses in I Ψy) < then i i Ψs ) < then h strictly decreses in m, x 4 ) or some x 4 m, y) nd strictly increses in x 4, s) IV. i Ψs ) > then h strictly decreses in m, x 5 ) or some x 5 m, y) nd strictly increses in x 5, x 6 ) or some x 6 y, s) then inlly strictly decreses in x 6, s) [4] V. MIN RESULTS For the next two theorems, we use the nottion used in [3] Theorem 6. Let x) be density unction o Person distribution where.).4) re ulilled Let M = b..c, L =. 2.M, D =.L, D = D nd b = b 2 or 0. ) I = 0 nd < 0 then s + b 0. 2) I. > 0 nd q ) 0 nd D 0 then Y m) 0 nd 0 Y s) 2D 3) I. < 0 nd q ) 0 nd D 0 then 0 Y m) 2D nd Y s) 0 where Y v) = v + ) + D. Then ll the ssertions o Theorem 4 hold. Proo. y theorem 4, it is suicient to prove tht or some y in m, s) We hve where nd ) > 0 in m, y) nd ) < 0 in y, s) ) = q Q ) = px) Q 2 px) = x 2 + 2x + M ) > 0 i px) > 0 nd ) < 0 i px) < 0 I 0 nd D 0 then the roots o px) re Globl Journl o Science Frontier Reserch Volume XVIII Issue V V ersion I Yer 208 3F ) x = D) ) nd x 2 = +D) ) where D = D Globl Journls

5 Monotonic ehviour o Reltive Increments o Person Distributions Cse : 0, 0 nd q ) 0, so q nd Q hve no common zero... I. > 0, then px) is convex. I D 0 then there will be two rel roots x nd x 2 where x x 2. i nd only i px) > 0 in m, y) nd px) < 0 in y, s) which mens tht m x, x = y nd x < s < x 2 Y m) < 0 nd 0 Y s) 2D Globl Journl o Science Frontier Reserch Volume XVIII Issue V V ersion I Yer 208 4F ) MONOTONIC EHVIOUR OF RELTIVE INCREMENTS OF PERSON DISTRIUTIONS 5.2. I. < 0, then px) is concve. i nd only i which mens tht Cse 2 = 0 nd 0 thereore nd I > 0 then q Q ) > 0 i x > I < 0 then q Q ) > 0 i x < px) > 0 in m, y) nd px) < 0 in y, s) x m x 2 nd s x 2 = y 0 Y m) 2D nd Y s) 0 q Q = x2 + bx + c q Q ) = 2x + b nd q Q ) < 0 i x <. I < 0 then s > 2 nd q Q ) < 0 i x >. I > 0 then s > 2 Cse 3 I = 0, then px) = b. I b > 0 then Theorem pplies. I b < 0 then there is no cse like it s seen in remrk 2. in [3]. Theorem 7 Let x) be the density unction o Person distribution with m = r,.),.3),.4) re ulilled nd M, L, D deined s in theorem 5. ) I = 0 nd < 0 then m + b 0. 2) I. > 0 nd q ) 0 nd D 0 nd Y m) 0 nd 0 Y s) 2D 3) I. < 0 nd q ) 0 nd D 0 nd 0 Y m) 2D nd Y s) 0 where Y v) = v + ) D. Then ll the ssertions o Theorem 5 hold. Proo. y theorem 5, it is suicient to prove tht or some y in m, s). Cse : ) < 0 in m, y) nd ) > 0 in y, s) 0 nd q ) 0, so q nd Q hve no common zero... I. > 0, then px) is convex. I D 0 then there will be two rel roots x nd x 2 where x < x 2. i nd only i px) < 0 m, y) nd px) > 0 y, s) m = x, x 2 = y nd s > x 2 Re 3. Szbo, Z. : Reltive Increments o Person Distributions. ct Mth. c. Ped. Nyiregyh ), pp [Electronic Journl, websites Globl Journls

6 Monotonic ehviour o Reltive Increments o Person Distributions which mens tht Y s) > 0 VI. Monotonic ehviour o Reltive Increments o Person Distributions Re 3. Szbo, Z. : Reltive Increments o Person Distributions. ct Mth. c. Ped. Nyiregyh ), pp [Electronic Journl, websites I. < 0, then px) is concve. i nd only i Cse 2 = 0 thereore nd px) < 0 m, y) nd px) > 0 y, s) m < x which mens tht Y m) > 0, y = x nd s = x 2 I > 0 then q Q ) < 0 i x < I < 0 then q Q ) < 0 i x > q Q = x2 + bx + c q Q ) = 2x + b nd q Q ) > 0 i x >. I > 0 then m < 2 nd q Q ) > 0 i x <. I < 0 then m < 2 Cse 3 I = 0, then px) = b. I b < 0 then Theorem 2 pplies. > 0 then Theorem 3 pplies. I b We stte the ollowing Lemm s it helps in clssiying distributions. These re outlined nd proven in [3]. Lemm 2. Let x) be the density unction o Person distribution, then then: I Let s = nd = lim x x.x) I. I = = = 0 nd b + 0 then Ψ ) = b+) I.2 I 0 nd = 0 then Ψ ) = I.3 I.. + ) 0 nd = 0 then Ψ ) = +) I.4 I = = 0, b. = 0) or = 0, 0) or.. 0) then Ψ ) = 0 II Let s be inite number nd let lim x s x) = 0 II. I [.Qs).qs) 0] or [.q s) 0, Qs) = qs) = 0] or [ = 0, qs) 0] or [.Qs) 0, qs) = q s) = 0], then Ψs ) = Exmple We hve II.2 I 0, q ).Qs).[Qs) + q s)] 0, qs) = 0 then Ψs ) = Qs) [Qs)+q s)] II.3 I 0, q ).qs) 0, Qs) = 0 then Ψs = 0) II.4 I 0, q ) = qs) = 0, + 0 nd either Qs) = q s) = 0 or Qs) 0 then Ψs ) = +) II.5 I = = qs) = 0, q s).b + ) 0 then Ψs ) = VII. x) = Γn ) x n exp k x ), n = 2, 3, 4,..., I = 0, ) = nx+k x 2 so = n, = k, =, b = 0, c = 0 b+ Globl Journl o Science Frontier Reserch Volume XVIII Issue V V ersion I Yer 208 5F ) 208 Globl Journls

7 Globl Journl o Science Frontier Reserch ) Volume XVIII Issue V V ersion I 6F Yer 208 Monotonic ehviour o Reltive Increments o Person Distributions Theorem 6 prt 3 pplies since q ) = k2 n 0 nd. = n < 0. 2 mode = k n < y = 2k n, so it s U-type distribution with px) = x2k nx) > 0 or x < 2k n nd px) < 0 or x > 2k n lim xx) = lim x x Γn ) x n+) exp k/x) = 0 so I.3 in Lemm 2 pplies giving ψ ) = + = n n+ > This mens tht ψy) > since i it ws less thn then ccording to theory ψx) would decrese nd remin less thn in y, s). So the reltive increment unction increses nd then decreses. Exmple 2 x) = K + x 2 ) n exp rctnx)), n > 2 We hve, K is constnt nd I =, ) = 2nx x 2 + so = 2n, =, =, b = 0, c = Theorem 6 prt 3 pplies since q ) = 0 nd. = 2n < 0. mode = 2n < y = + 4n 2 + 2n, so it s U-type distribution with px) = 2 nx 2 + x n) > 0 or x < + 4n 2 + 2n nd px) < 0 or x > + 4n 2 + 2n lim xx) = 0 x so I.3 in Lemm 2 pplies giving ψ ) = + = 2n 2n+ > This mens tht ψy) > since i it ws less thn then ccording to theory ψx) would decrese nd remin less thn in y, s). So the reltive increment unction increses nd then decreses. VIII. Reerences Réérences Reerencis. Johnson N. nd Kotz S. : Distributions in Sttistics. Continuous Univrite Distributions, vol.,2, Houghton Mi_in, oston - New York, Szbo, Z. : Investigtion o Reltive Increments o Distribution Functions. Publ. Mth. De-brecen ), pp Szbo, Z. : Reltive Increments o Person Distributions. ct Mth. c. Ped. Nyiregyh ), pp [Electronic Journl, websites Szbo Z. : On Monotonic ehviour o Reltive Increments o Unimodl Distributions. 5. ct Mth. cd. Ped. Nyiregyh ), pp [Electronic Journl, website Notes 208 Globl Journls