CC BY
19
WEIGHTED GENERALIZED ENTROPY: PROPRTIES
AND APPLICATION
Bilal Ahmad Bhat1*, M. Sultan Shah2 and M.A.K Baig3
Department of Statistics, University of Kashmir, Srinagar, J&K-190006, India
bhatbilal3819@gmail.com msultanshahhO 1@gmail.com baigmak@gmail.com 'Corresponding Author
Abstract
Recently, the measurement of uncertainty has attracted the attention of researchers. In this article, we introduce a new weighted uncertainty measure known as weighted generalized entropy. We also study its dynamic (residual) version which is known as weighted generalized residual entropy. These are length-biased shift-dependent uncertainty measures. It is shown that the proposed dynamic uncertainty measure uniquely determines the survival function. The various significant properties and the relationship with other well-known reliability measures of the proposed dynamic uncertainty measure are also studied. Finally, a real life data set is used to illustrate the usefulness of our proposed uncertainty measures.
Keywords: Weighted entropy, weighted residual entropy, hazard rate function and characterization results.
1. Introduction
The notion of entropy that was introduced by Shannon [1] is a very important and well known concept in the area of information theory. For an absolutely continuous non-negative r.v U having p.d.f g(u), the Shannon's entropy (SE) is defined as
Hu(g) = - £g(u) \ogg(u)du = -E[\og(U)]. (1)
Throughout this article, the notations r.v and p.d.f stands for an absolutely continuous nonnegative random variable and the probability density function respectively.
If a lifetime component has survived up to an age t, then the SE is not useful for measuring the uncertainty about its remaining life. To overcome this problem, Ebrahimi [2] has introduced the concept of residual entropy and is defined as
Hu(g;t) = -r^\og^du, (2)
/ Jt G(t) & G(t) ' v '
where, G(t) = 1 — G(t) is the survival function (s.f) of the r.v U.
It is clear that the SE is well-known by means of its applications in the area of information theory, but it is a shift-independent uncertainty measure (UM) because it remains unchanged, if for instance U is uniformly distributed in (c, d) or (c + h,d + h) for any h E'R. However, in some applied contexts, such as reliability or mathematical neurobiology, the shift-dependent UM's are
desirable. To fulfill this requirement, Belis and Guiasu [3] have introduced the concept of weighted entropy (a shift-independent UM) and is defined as
H(u,w)(g) = - f™w(u)g(u)\ogg(u)du
— -ug(u)\ogg(u)du, (3)
where, the coefficient u (i.e the length of the system or component under consideration) represents the weight function of the elementary events.
Similarly, Di Crescenzo and Longobardi [4] have introduced the weighted version of residual entropy (2) and is given by
Huw)(g;t) = -Cu^\og^du. (4)
In the recent literature, it is seen that the study of weighted UM's have attracted the attention of researchers for introducing the new flexible weighted UM's. For more details see Misagh et al. [5], Misagh and Yari [6], Nourbakhsh and Yari [7], Mirali and Baratpour [8], Kayal [9], Nair et al. [10], Rajesh et al. [11], Khammar and Jahanshahi [12], Bhat and Baig [13] and Bhat et al. [14] etc. Motivated with this research literature, here in this article, our objective is to introduce a new weighted UM and its dynamic (residual) version on the basis of the following new generalization of SE
where,
JjTiH^^ig) - -f? g(u)\ogg(u)du, which is the SE given in (1).
y.=l
Analogous to (2) and on the basis of (5), the generalized residual entropy can be defined as
^m-^^S?^*1*)-(6)
The rest of the article is organized as follows: In section 2, we discuss the weighted generalized entropy (WGE) of order t] and type i in the form of its definition and some properties. The section 3 presents the weighted generalized residual entropy (WGRE) and also some of its significant characterization results. In section 4, we study the various important properties of WGRE and also its relationship with other well-known reliability measures. In section 5, an application of the WGE and WGRE by using a real life data set is presented. Finally, we illustrate some concluding remarks in section 6.
2. Weighted Generalized Entropy (WGE)
In this section, we introduce the weighted version of (5) which is known as weighted generalized entropy (WGE) of order and type i.
Definition 2.1 For a r.v U having p.d.f g(u), the WGE of order t] and type i denoted by H^^Jg) is
defined as
= l^l0g(i:^(ug(u))2^-1du)^2 <V<H>1, (7)
where, the coefficient u in the integrand denotes the weight function as in (3). In the following example, we illustrate the importance of WGE. Exumple 2.1. Let U and V be two r.v's distributed as
g (u) = (2u, 0<u<1 g (v) = (2(1 - v), 0 < v < 1 guW = \0, otherwise, gv(v) =( 0 otherwise. Here, we can see that
But, the WGE's of U and V are different with each other as follows
/ 4-1
H^Ql) =
log
and
H
(V.H)
№W) ->g( 22» 1B(2-,2-
(v,w) 2r(ß-rj) \ V V V
where,
B(c,d) = Cyc-1(l -y)d-1, C'd > 0 =
T(c)T(d) r( c+d) '
Thus, even though H^^) = H^Q,), but * H^QjW
Example 2.2. Let g(u) be the p.d.f of a r.v U distributed as: (a) Exponentially with g(u) = pe-^u,u >0,p>0, then
H
(r,v)
(U,w)
(g) =
2r(v-r)
log
№
ß(
2v
(b) Gamma with g(u) Uu^ 1,0 < u < <x,p > 0, then
H
(rrv) (a) _ (U,w) (a)
2r(v-r)
log
r(ß(2
n ß(2^-1)+1)
(rW)fl-1(2l-1p2ï-1>+1
(c) Lomax with g(u) =
H
(1+u)1+m (rr'v) (a) =_1_
(U'W)^J 2V(V-V)
,u> 0,m> 0, then
log
m2»-1r(2^(m(2l-1)-1)
v[(2T1)(m+1)
•m(2i-l)
l) > l
2
(d) Rayleigh with g(u) = fiue 2U ,u> o,fi > 0, then
H
(rl,Ñ (a) _ (U,w) (g)
2v(p-v)
log
v 3
22~3<2l-2)
M2J¡r1Yr~2.
Lemma 2.1. If Z = mU, with m > 0, then
fives' ^ 1
H(
\(g) =
logm + H(rWW) (a).
Theorem 2.1. For a r.v U having SE Hu(g), we obtain
^^WW)(g)>^^[HlJ(g) — (^)E(iogu)]. Proof. By applying the log-sum inequality, we obtain
XTg(u)log
g(u)
(ug(u))2l* 1
— du>f0 g (u) du log-
J0°g(u)du
f (ug(u)) P du oo 2—1
= -logf0OO(ug(u)) P du.
Due to (7), the desired result is satisfied.
1
1
m
1
3. Weighted Generalized Residual Entropy (WGRE)
In this section, we introduce the dynamic (residual) version of (7) which is known as weighted generalized residual entropy (WGRE) of order q and type /j.. Some important characterization results of this UM are also discussed.
Definition 3.1 Let U be a r.v with p.d.f g(u) and s.f G(t), then the WGRE of order q and type ^ is defined as
KU'WW)(g;t) =
1
2v(n-v)
log
r(u^)5- du J t V G(t)J
Here, we evaluate the WGRE of some lifetime distributions.
Example 3.1. Let a r.v U be distributed as:
(a) Exponentially with p.d.f g(u) — fe-^u,uf > 0,f > 0, then
H
(U,w)
(b) Gamma with p.d.f g(u) =
2V(n-V)L-r- ■ ■ ßRR+1
1
rßß)
HlfX (g; t) = 1
e-uuß-1, 0 <u < ™,ß > 0,
T(Rß + 1,Rt)
(U,w)y
2V(P-V)
log
(c) Weibull with p.d.f g(u) = —e~
1
2v(n-v)
MRß+i(rmy ■n ), u > n,m > 0,n > 0, then
t i —t(r + 1,r—
R-+\°gl—^-m
2
(d) Rayleigh with p.d.f g(u) — flue 2u , u>0,f > 0, then
KU'WW^g^) =
2 "I JßRR*i
where, T(n,mz) — mn f? e mxxn 1dx, m, n > 0 is an upper incomplete gamma function and R — 2^-1 respectively.
Theorem 3.1 If H^^ig) and H^^ig; t) denotes the GRE and WGRE of a r.v U, then for all t > 0, we have
H&W)(g;t) = -
:l°g|t
exp
, X Jx=t
Proof. From (8), we have
S? (u aMir1 du — f? [C (21-1) mf-1 du
- - 2^-1 g})" exp(2i(ß-i)H™(g;x)dx)
= H-l) JT [f:y2("-1)dy + f/^dy] -1 du
= t2ß-1 fT (3W)2r
J t \G(t)J
d u + (2
H-iVT^HfzMî^d»
From (6), we have
and
JT (W " du = exp [2V(ß - i)H^(g; t)j. f t°°g2ï-1du = G2l-1(t)exp [2V(ß - ^H^ig; t)j
dy. (9)
(10) (11)
Using (9), (10) and (11) in (8), we obtain the required result. Here, we show that G(t) is uniquely determined by H^^ig; t).
Theorem 3.2. Let U be a r.v having p.d.f g(u), s.f G(t) and WGRE H^^ig; t) < <] < i,i > 1 respectively. If H^fy (g; t) is increasing in t, then H^fy (g; t) uniquely determines G(t). Proof. Rewriting (8) as
2v_1
exp [2t](i - riH^ig; t)] — f? (ua-^) " du. Differentiating (12) w.r.t , we have
(12)
~texp
21 (ß- riHUW)(g;t) = (2l-l)Ac(t) J" (uß du - (tXG(t))
2—-1 ß
where, Ac (t) — represents the hazard rate of U. Using (12), we can rewrite (13) as
( tAc(t))2»-1 -(2l-l) exp [2i(ß - l)HtUW)(g: Mc(t)]
1
1
ß
2rj(ji — V)exp [2rj(M — v)H^n^^)(g;t)]^H^'u-WW.)(g;t) = 0. (14)
I dt
Hence for fixed t > 0, Ac(t) is a solution of *p(ut) = 0, where
^(ut) = t2l-1 — (2^ — 1) exv [2r(M — ri)H(UW-) (a; 0] ut
+2rj(n — r)exp [2VQi — ^H^g; t^jH^g; t). Differentiating both sides w.r.t ut, we have
s
Also,
¿■KM = (21-1) t^u2-1' -(2l-l)exp[lv(M- v)H%%tg; t)].
Now, -^K(Ut) = 0 gives
u =
exp(2n(ii-r,))H{^)(.g-,t)
4-1 t V
= uo (say).
U
For -<r<p, ^ > 1, ^¡2^(uo) < 0. Thus, ip(ut) attains maximum at u0. Also, ip(0) > 0 and
^(m) = —m. Further it can be easily observed that ^(ut) first increases for 0 < ut < u0 and then decreases for ut>u0. So, the unique solution to ^(ut) = 0 is given by ut = Ac(t). Thus, H(U'U) (3 '; t) uniquely determines Ac (t) which in turns determines G (t).
4. Properties and Inequalities of WGRE
This section presents some interesting properties and inequalities of weighted generalized residual entropy .
Definition 4.1. Let U and V be two r.v's having WGRE's H^'U^g; 0 and H^te; 0, then U is said
wgre , ,
to be smaller than V in WGRE of order rj and type ^ (denoted by U < V), if H(U'w) (3; ^ —
H^U'WWM.t), v t > 0.
Definition 4.2. A r.v U or a s.f G will be said to have increasing (decreasing) WGE for residual life of order r and type ^ IWGERL (DWGERL), if H^ (g; t) is increasing (decreasing) in t, t > 0. Lemma 4.1. If Y = aU, with a > 0 is a constant, then
Proof.
Setting Y = aU, a strictly increasing function of U, we have
a \t (u—-) du
„ \ G(t)
By using (8), the desired result is obtained.
Theorem 4.1. For two r.v's U and V, let us define Y1 = a1u and Y2 = a2V with a1,a2 > 0. Let U WfE V and ai — a2. Then Yi WfE Y2 Y W if H^UUW) tol) or H^W (g; 0 is decreasing in t > 0. Poof. Suppose H(U'U^) (g; t) is decreasing in t. Now, uWG<REV implies
HtZ^i'iH^g^). (15)
Further, since — > — , we have
a- a,2
H
(U'W) (g;-r) < HiU:W) (g;i2).
axJ (uw) v a2)
Combining (15) and (16), we obtain
H
^ in- M C (n. t\
(16)
(17)
Using Lemma 4.1 in (17), we have Y1 wcfE Y2 .
Theorem 4.2. For a r.v U having support (0,k],k > 0 , p.d.f g(u) and s.f G(t),t > 0, then for ^ < ] < i, i>1, the following upper bound of H^^Jg; t) holds
H%WW)(g;t)-
2u(jî-u)
rk( g(u)\
Jt (um)
2R-
(k( g(u)\ n S VlG(t))
Proof. From log-sum inequality and (8), we have
+ l°g(k - t)
du
V' 1 V' 1 o'i 1
Jtiu&^u&^duïJÏiufàP-1dul°g
Jt (ug(u)) ß du ftk(c(t))2ß du
2—— 1
= Jtk(uGu)ß du[2i(ß-!)H^W(g;t)-]°g(k-0].
After simplification, we get the desired result. Theorem 4.3. Let G be a IWGRE (DWGRE) and i>], then
¿c(t) < (>) Proof. From (14), we have
[2ß-1)exp[2u^-u)H$%(g;t)}
ß
4.-1 tß
2(V-ß)
2i(ß- i)±H^uW\(g; t) = (2U- i)Äc(t) - exp {2i(ß - V)H^uW)(g;f)} (tÄc(t))2~ß
Since G is IWGERL (DWGERL), therefore, we have
*c(t) which leads to
V 21 ——1 )
(2U -l)- t2-ß-%(r (t)exp {21(1 - tiHUW)(g; 0}
> (<)0.
¿c(t) < (>)
2ßr1)exV{2u(v-u)H$$)(g;t)}
ß
t ß
2(V-ß)
Theorem 4.4. If U is IWGERL (DWGERL), then
H<ùUW)(g;t)<(>)
2u(^-u)
\og
t2ßr1(1+^G(t)\2(-ß-1)
—gV )
where mc (t) is the mean residual life function of U. Proof. From (14), we have
bH(UUW)(g;t) =
2u(h-u)
(2U -l)Äc(t) - (tXc(t))2ß-1exp {21(1 - ß)H(UW(g; t)}
Using Äc (t) = dJ G ), we have
—G(t)
bH(U*M;t) =
2u(ß-U)
H-4
\ —G(t) I
( —G(t) )
2—-1 ß
exp
{21(1 - ß)H(U'W!)(g;t)}
Since, G is IWGERL (DWGERL), therefore, after simplification, we have
H^U,W)(g;t)>(<)\°g
t2l-1f1+l—G(.t)\2(-ß-
^ßr1\ —G(t) )
V
1
V
1
1
1
Theorem 4.5. Let U be the lifetime of a system with p.d.f g(u) and s.f G(t), t > 0, then H(U'w) (d> ^
attains a lower bound as follows
« (*; t)>lUU-U-№— 1) ^logudu + 2(1—U) Hu(g; t)]. (18)
Proof. From log-sum inequality, we have
Cg(u)\og—3(u\ du > rg(u)du\og—St
-ylL—A - Jt c/ v ' ° 9—_1
(umrm)"- *u
= G(t) [logG(t) — \og{2V(si — V)H^UU'W)-(3; 0}]. (19)
where (19) is obtained from (8). The L.H.S of (19) leads to
2(1— U) jt°°g(u) \ogg(u)du — (2^1) J" g(u) logudu + (2 U—1) G(t) log G(t). (20)
Using (20) in (19), we obtain (18).
5. Application
To illustrate the effectiveness and importance of our proposed UM's, we consider a real life data set. The data set represents the remission times (in months) of a random sample of 128 bladder cancer patients given in Lee and Wang [15] and is given as follows:
0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28,
9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19,
2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.
Afaq et al. [16] have shown that the length biased Lomax distribution (LD) provides a better fit for this data. Now, in order to compute the entropy of this data set, it is necessary to apply the weighted entropy technique rather than the simple entropy. For the weighted entropy, we need to consider the basic model (i.e LD) of the length biased LD. The MLEs of the parameters of LD from this data set are obtained as: 8 = 8.431393(shape parameter) and A = 70.289624(scale parameter) respectively. Now, for r = 1.5,n = 2.5 and t = 10, we have H(U'U^)(g) = 1.638 and H(U'Uv)(9;t) = 1.694. Similarly, r = 2.5,^ = 3 and t = 20, we obtain H^te) = 1-164 and H^U^t) = 1.481
respectively.
6. Conclusion
In this article, we have introduced the concepts of weighted generalized entropy and also its dynamic (residual) version which is known as weighted generalized residual entropy. It has been shown that the proposed residual entropy uniquely determines the survival function. The various important properties and the relationship with other well-known reliability measures of the proposed residual entropy are also obtained. Finally, a real data set has been used to investigate the usefulness of the proposed entropy functions.
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