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PureMathematicsnØêÆ,2023,13(3),533-540
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.133057
Vä‘ÅCþCox-CzannerÑÝ
ëë뀀€www
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2023c216F¶¹^Fϵ2023c316F¶uÙFϵ2023c327F
Á‡
CoxÚCzanner(2016)JÑ)•ÑÝVg¿ïÄ˜©Ù) •ÑÝ"Š•KLÑÝí
2§)•ÑÝ3ÚOÆ!)Æ+•¼2•A^"©JÑCox-CzannerÑÝ3Vä
‘ÅCþe©ÙɧÏL2”Ç•{ïÄVä‘ÅCþ)•ÑÝk.5ÚüN
5§¿?ØüNC†é)•ÑÝK•§•ÏL©ÙC†òVä‘ÅCþ)•ÑÝA
^'~`³.?1¢~u"
'…c
Cox-CzannerÑݧ2”ǧA§'~`³.
Cox-CzannerDivergenceof
aDoublyTruncatedRandom
Variable
SuyuanZhao
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Feb.16
th
,2023;accepted:Mar.16
th
,2023;published:Mar.27
th
,2023
©ÙÚ^:ë€w.Vä‘ÅCþCox-CzannerÑÝ[J].nØêÆ,2023,13(3):533-540.
DOI:10.12677/pm.2023.133057
ë€w
Abstract
CoxandCzanner(2016)putforwardtheconceptofsurvivaldivergenceandstudied
thesurvivaldivergenceofsomedistributions.AsageneralizationofKLdivergence,
survivaldivergencehasbeenwidelyusedinstatistics,ecologyandotherfields.This
paperproposesthedistributiondifferenceofCox-Czannerdivergenceunderdouble-
truncated random variables, studies the boundedness and monotonicity of the survival
divergenceofdouble-truncatedrandomvariablesbymeansofgeneralizedfailurerate
method,anddiscussestheinfluenceofmonotonictransformationonthesurvivaldi-
vergence.Finally,thesurvivaldivergenceofdoubletruncatedrandomvariablesis
appliedtotheproportionaldominancemodelbythetransformationofdistribution.
Keywords
Cox-CzannerDivergence,GeneralizedFailureRate,Characteristic,Proportional
OddModel
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
ÑÝÝþnØ•£3CA›c5éÐïÄ,¿2•/A^3ó§ÚšÆˆ‡+
•,•~„ÝþkKullback-leiblerÑÝ(KLdivergence)[1]!RenyiÑÝ[2]!KagansÑÝ(k•
ål)[3]!Cox-CzannerÑÝ.'uÑÝÝþnØÚA^•õ[!,Œ±ë•Basseville,M.[4],
Vonta, F.ÚKaragrigoriou,A. [5]©z.Ù¥Cox-Czanner[6]ÑÝÏÝþü|)•¼êƒmÉ
2•A^,~XŒ±)º•´˜|‡ö3žmtk,˜|‡ö3žmt•¹ýéV
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(x)=f(x)/
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Ý½Â•,
I(F,G) =
Z
∞
0


f(x)
¯
G(x)−g(x)
¯
F(x)


dx.
(1)
DOI:10.12677/pm.2023.133057534nØêÆ
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2022c,MansourzarÚAsadi[7]?˜Ú*ÐCox-CzannerÑÝ, ?Øü«•{Æ·X
t
ÚY
t
ƒ
mÑÝ,Ù¥X
t
=(X−t|X>t),Y
t
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t
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t
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X
t
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G(t),e
3D(F,G;t) =
R
∞
t


f(x)
¯
G(x)−g(x)
¯
F(x)


dx
¯
F(t)
¯
G(t)
.
(2)
K¡D(F,G;t)•Cox-Czanner•{Æ·ÑÝ.
2022c,Mansourzar[8]ïÄ•{Æ·éóÝþ,=>Žžm
t
X=(t−X|X≤t)Ú
t
Y=
(t−Y|Y≤t)ÑÝ,ÙVǗݼê©O•f
t
X
(x)=f(x)/F(t)Úg
t
Y
(x)=g(x)/G(t),\È©Ù
¼ê©O•F
t
X
(x) = F(x)/F(t)ÚG
t
Y
(x) = G(x)/G(t),e
¯
D(F,G;t) =
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t
0
|f(x)G(x)−g(x)F(x)|dx
F(t)G(t)
.
(3)
K¡
¯
D(F,G;t)•Cox-Czanner>ŽžmÑÝ.
Cc5,'uVä‘ÅCþÝþ¯K5É•H[4,9–11],3)•©ÛÚŒ‚5ó§
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)ž,I‡ïÄT«mSÆ·&E.••
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‘ÅCþCox-CzannerÑÝ,e¡‰ÑÙ½Â.
½Â1bXÚYL«ü‡ëY‘ÅCþ,©Ù¼ê©O•F(x)ÚG(x),—ݼê©O•f(x)Úg(x),
X
t
1
,t
2
=[X|t
1
<X<t
2
]ÚY
t
1
,t
2
=[Y|t
1
<Y<t
2
]©O´XÚYƒ'Väžm,
(t
1
,t
2
) ∈D= {(x,y)|F(x) <F(y)ÚG(x) <G(y)},e
ID(X,Y;t
1
,t
2
) =
Z
∞
0
|f
t
1
,t
2
(x)G
t
1
,t
2
(x)−g
t
1
,t
2
(x)F
t
1
,t
2
(x)|dx
=
R
t
2
t
1
|f(x)G(x)−g(x)F(x)|dx
∆F∆G
+




F(t
1
)
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−
G(t
1
)
∆G




,
(4)
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1
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1
),K¡ID(X,Y;t
1
,t
2
)•VäCox-CzannerÑÝ.
©SüXeµ312!¥,·‚‰Ñ'uVÑÝ̇(J.äN/,·‚¼ÑÝ
..313!¥ïÄVäÑÝüN1•±9üNC†éVäÑÝK•.314!¥,é,
a.=†.?1ÑÝÝþµ.
2.VäCox-CzannerÑÝ5Ÿ
2.1.k.5
½n1bXÚY´ü‡‘ÅCþ,X
t
1
,t
2
=[X|t
1
<X<t
2
]ÚY
t
1
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2
=[Y|t
1
<Y<t
2
]©O´
†XÚYƒ'Väžm,é(4)ª¥L«ÑÝÝþID(X,Y;t
1
,t
2
),k0 ≤ID(X,Y;t
1
,t
2
) ≤1.
DOI:10.12677/pm.2023.133057535nØêÆ
ë€w
y²w,,†žmƒ'ÑÝÝþID(X,Y;t
1
,t
2
)´šK,=ID(X,Y;t
1
t
2
)≥0,…=
F(x) = G(x)žÒ¤á.Ïd•Iy²ID(X,Y;t
1
t
2
) ≤1.
ID(X,Y;t
1
,t
2
)
=
R
t
2
t
1
|f(x)(G(t
2
)−G(x))−g(x)(F(t
2
)−F(x))|dx
∆F∆G
=





R
t
2
t
1
f(x)G(t
2
)dx−
R
t
2
t
1
f(x)G(x)
∆F∆G
−
R
t
2
t
1
g(x)F(t
2
)dx−
R
t
2
t
1
g(x)f(x)dx
∆F∆G





≤




G(t
2
)∆F−G(t
2
)∆F−F(t
2
)∆G+F(t
2
)G(t
2
)−F(t
1
)G(t
1
)−G(t
2
)∆F
∆F∆G




= 1.
(5)
y²¤.
2.2.üN5
½n2‘ÅCþXÚY©Ok—ݼêf(x)Úg(x),©Ù¼ê©O•F(x)ÚG(x).
(i)éu?¿t
1
≤t
2
,t
2
½ž,eID(X,Y;t
1
,t
2
)'u´t
1
4O(4~),KID(X,Y;t
1
,t
2
) ≥(≤)



h
X
1
(t
1
,t
2
)
G(t
1
)
∆G
−h
Y
1
(t
1
,t
2
)
F(t
1
)
∆F



−



h
X
1
(t
1
,t
2
)
F(t
2
)
∆F
−h
Y
1
(t
1
,t
2
)
G(t
2
)
∆G



h
X
1
(t
1
,t
2
)+h
Y
1
(t
1
,t
2
)
;(6)
(ii)éu?¿t
1
≤t
2
,t
1
½ž,et
2
'uID(X,Y;t
1
,t
2
)´4O(4~),KID(X,Y;t
1
,t
2
)≤
(≥)



h
X
2
(t
1
,t
2
)
G(t
2
)
∆G
−h
Y
2
(t
1
,t
2
)
F(t
2
)
∆F



+



h
Y
2
(t
1
,t
2
)
G(t
1
)
∆G
−h
X
2
(t
1
,t
2
)
F(t
1
)
∆F



h
X
2
(t
1
,t
2
)+h
Y
2
(t
1
,t
2
)
.(7)
y²(4)©O'ut
1
Út
2
¦,
∂
∂t
1
ID(X,Y;t
1
,t
2
)
= (h
X
1
(t
1
,t
2
)+h
Y
1
(t
1
,t
2
))ID(X,Y;t
1
,t
2
)−




h
Y
1
(t
1
,t
2
)
F(t
1
)
∆F
−h
X
1
(t
1
,t
2
)
G(t
1
)
∆G




+




h
Y
1
(t
1
,t
2
)
G(t
2
)
∆G
−h
X
1
(t
1
,t
2
)
F(t
2
)
∆F




,
(8)
DOI:10.12677/pm.2023.133057536nØêÆ
ë€w
Ú
∂
∂t
2
ID(X,Y;t
1
,t
2
)
= −(h
X
1
(t
1
,t
2
)+h
Y
1
(t
1
,t
2
))ID(X,Y;t
1
,t
2
)+




h
Y
2
(t
1
,t
2
)
F(t
2
)
∆F
−h
X
2
(t
1
,t
2
)
G(t
2
)
∆G




+




h
X
2
(t
1
,t
2
)
F(t
1
)
∆F
−h
Y
2
(t
1
,t
2
)
G(t
1
)
∆G




.
(9)
?-
∂ID(X,Y;t
1
t
2
)
∂t
1
≥(≤)0,
∂ID(X,Y;t
1
t
2
)
∂t
2
≥(≤)0,²L{z§(6)Ú(7),=½ny²¤.
3.üNC†
y3?ØüNC†éVäCox-CzannerÑÝID(X,Y;t
1
t
2
)K•.
½n3XÚY´ü‡ ýéëYšK‘ÅCþ,VǗݼê©O•f(x)Úg(x),©Ù¼ê©O
•F(x)ÚG(x).eV¼êφ
1
Úφ
2
´î‚üN…Œ‡,Kéu¤k0 ≤t
1
<t
2
<+∞,k
ID(φ
1
(X),φ
2
(Y);t
1
,t
2
) =























ID
φ
1
(X,φ
−1
1
(φ
2
(Y));φ
−1
1
(t
1
),φ
−1
1
(t
2
)),
XJφ
1
Úφ
2
´î‚üN4O;
ID
φ
1
(X,φ
−1
1
(φ
2
(Y));φ
−1
1
(t
2
),φ
−1
1
(t
1
)),
XJφ
1
Úφ
2
´î‚üN4~.
y²XJφ
1
(x)Úφ
2
(x)´î‚4O¼ê,@oφ
1
(X)Úφ
2
(Y)VǗݼêÚ©Ù¼ê©OŒ±
L«•
f
φ
1
(x) =
f(φ
−1
1
(x))
φ
1
0
(φ
−1
(x))
F
φ
(x) = F(φ
−1
1
(x)),(10)
Ú
g
φ
2
(x) =
g(φ
−1
2
(x))
φ
1
0
(φ
−1
(x))
G
φ
(x) = G(φ
−1
2
(x)),(11)
d,Œ±φ
−1
1
(φ
2
(x)VǗݼêÚ©Ù¼ê©O•
g
φ
−1
1
(φ
2
)
(x) =
g(φ
−1
2
(φ
1
(x))φ
1
0
(x)
φ
2
0
(φ
−1
2
(φ
1
(x))
,G
φ
−1
1
(φ
2
)
(x) = G(φ
−1
2
(φ
1
(x))).(12)
DOI:10.12677/pm.2023.133057537nØêÆ
ë€w
ò(10)Ú(11)“\(4)
ID(φ
1
(X),φ
2
(Y);t
1
,t
2
)
=





R
t
2
t
1
(F(φ
−1
1
(x))g(φ
−1
2
(x))/φ
2
0
(φ
−1
2
(x))−f(φ
−1
1
(x))G(φ
−1
2
(x))/φ
1
0
(φ
−1
1
(x)))dx
∆F
φ
1
∆G
φ
2





+




G(φ
−1
2
(t
2
))
∆G
φ
2
−
F(φ
−1
1
(t
1
))
∆F
φ
1




,
(13)
Ù¥∆F
φ
1
=F(φ
−1
1
(t
2
)) −F(φ
−1
1
(t
1
))Ú∆G
φ
1
=G(φ
−1
1
(t
2
)) −G(φ
−1
1
(t
1
)),é(13)¦^C†u=
φ
−1
1
(x)k
ID(φ
1
(X),φ
2
(Y);t
1
,t
2
)
=






R
φ
−1
1
(t
2
)
φ
−1
1
(t
1
)
(F(u)g(φ
−1
2
(φ
1
(u)))φ
0
(u)/φ
2
0
(φ
−1
2
(φ
1
(u)))−f(u)G(φ
−1
2
(φ
1
(u)))/)du
∆F
φ
1
∆G
φ
2






+




G(φ
−1
2
(t
2
))
∆G
φ
2
−
F(φ
−1
1
(t
1
))
∆F
φ
1




,
= ID
φ
1
(X,φ
−1
1
(φ
2
(Y));φ
−1
1
(t
1
),φ
−1
1
(t
2
)).
(14)
½n3y²Ò¤.
4.'~`³.
y3·‚ïÄ©ÙC†.eÑÝÝþID(X,Y;t
1
,t
2
),¿$^VäCox-CzannerÑÝé
'~`³.?1Ýþ.
½Â2(©ÙC†.)H´ëY©Ù¼ê,ÙVǗݼêh∈[0,1],Š•=†ª½ó¼ê,
FÄ©Ù¼ê,H•©ÙC†¼êG,eé¤kx>0,k
G(x) = H(F(x)),(15)
K¡FÚG÷v©ÙC†..
4.1.'~`³.
½Â3('~`³.)b‘ÅCþXÚY©Oäk)•¼ê
¯
F(x)Ú
¯
G(x),©Ù¼ê©O•F(x)
ÚG(x),e•3'~~êθ>0,é¤kx>0,Ñk
¯
G(x)
G(x)
= θ
¯
F(x)
F(x)
,(16)
K¡XÚY÷v'~`³..
½n4b‘ÅCþXÚY©Ok©Ù¼êF(x)ÚG(x)±9)•¼ê
¯
F(x)Ú
¯
G(x),©ÙC
DOI:10.12677/pm.2023.133057538nØêÆ
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†H(x) = x/(θ+(1−θ)x),Ù¥θ<x<1,θ>0,Kk
ID(X,Y;t
1
,t
2
) =
F(t
2
)+F(t
1
)
F(t
2
)−F(t
1
)
+
2θ
(1−θ)(F(t
2
)−F(t
1
))
−
2(θ+(1−θ)F(t
2
))(θ+(1−θ)F(t
1
))log
θ+(1−θ)F(t
2
)
θ+(1−θ)F(t
1
)
(1−θ)
2
(F(t
2
)−F(t
1
))
2
.
(17)
y²ò©ÙC†H(x)“\(15)ª,KFÚGáu'~`³.a(16).Ïd(16)ªŒ±-#U•
G(x) =
F(x)
θ+(1−θ)F(x)
,x>0,(18)
dž
g(x) =
θf(x)
[θ+(1−θ)F(x)]
2
,(19)
ò(18)ªÚ(19)ª“\(4)ª¥§Xe(J
ID(X,Y;t
1
,t
2
)
=
R
t
2
t
1
(1−θ)f(x)F
2
(x)
[θ+(1−θ)F(x)]
2
dx+
F(t
1
)F(t
2
)
θ(1−θ)F(t
2
)
−
F(t
1
)F(t
2
)
θ(1−θ)F(t
1
)
θ(F(t
2
)−F(t
1
)
2
)
(θ+(1−θ)F(t
2
))(θ+(1−θ)F(t
1
))
=
(θ+(1−θ)F(t
2
))(θ+(1−θ)F(t
1
))
θ(1−θ)(F(t
2
)−F(t
1
))
−
(1−θ)F(t
1
)F(t
2
)
θ(F(t
2
)−F(t
1
))
+
θ
(1−θ)(F(t
2
)−F(t
1
))
−
2(θ+(1−θ)F(t
2
))(θ+(1−θ)F(t
1
))log
θ+(1−θ)F(t
2
)
θ+(1−θ)F(t
1
)
(1−θ)
2
(F(t
2
)−F(t
1
)
2
)
=
F(t
2
)+F(t
1
)
F(t
2
)−F(t
1
)
+
2θ
(1−θ)(F(t
2
)−F(t
1
))
−
2(θ+(1−θ)F(t
2
))(θ+(1−θ)F(t
1
))log
θ+(1−θ)F(t
2
)
θ+(1−θ)F(t
1
)
(1−θ)
2
(F(t
2
)−F(t
1
))
2
.
(20)
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ý¢êâ§édØ2Þ¢~`².
DOI:10.12677/pm.2023.133057539nØêÆ
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ë•©z
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DOI:10.12677/pm.2023.133057540nØêÆ

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