Bushehr University of Medical Sciences Repository - bpums

(1)

Published online in International Academic Press (www.IAPress.org)

Weighted Cumulative Residual (Past) Inaccuracy For Minimum (Maximum) of Order Statistics

Safieh Daneshi^1,2, Ahmad Nezakati¹, Saeid Tahmasebi^3,^∗

1Department of Statistics, Shahrood University of Technology, Shahrood, Iran

2Bushehr University of Medical Sciences, Bushehr, Iran

3Department of Statistics, Persian Gulf University, Bushehr, Iran

Abstract In this paper, we propose a measure of weighted cumulative residual inaccuracy between survival function of the first-order statistic and parent survival functionF^¯. We also consider weighted cumulative inaccuracy measure between distribution of the last- order statistic and parent distributionF. For these concepts, we obtain some reliability properties and characterization results such as relationships with other functions, bounds, stochastic ordering and effect of linear transformation. Dynamic versions of these weighted measures are considered.

Keywords Cumulative inaccuracy, Order statistics, Empirical approach.

AMS 2010 subject classifications62N05, 62B10, 94A17 DOI:10.19139/soic-2310-5070-695

1. Introduction

Let X denote the lifetime of a device or a system with probability density function (pdf) f and cumulative distribution function (cdf)F, respectively. Then, the differential entropy known as Shannon entropy, is defined by Shannon [18] as follows:

H(X) =−

∫ +∞ 0

f(x) logf(x)dx, (1)

where, by convention,0 log 0 = 0. Di Crescenzo and Longobardi [6] introduced the concept of weighted differential entropy which is given by

H^w(X) =−

∫ +∞ 0

xf(x) logf(x)dx, (2)

Recently, new measures of information are proposed in literatures: replacing the pdf by the survival function F¯ = 1−F in Shannon entropy, the cumulative residual entropy (CRE) is defined by Rao et al. [16] as

E(X) =

∫ +∞ 0

F¯(x)Λ(x)dx,

whereΛ(x) =−log ¯F(x). Properties of the CRE can be found in [13], [15], and [22]. A new information measure similar to CRE has been proposed by Di Crescenzo and Longobardi [7] as follows:

CE(X) =

∫ +∞ 0

F(x) ˜Λ(x)dx, (3)

∗Correspondence to: Saeid Tahmasebi (Email: [email protected]). Department of Statistics, Persian Gulf University, Bushehr, Iran.

ISSN 2310-5070 (online) ISSN 2311-004X (print)

(2)

whereΛ(x) =^˜ −logF(x). Analogous to (2), Misagh et al. [12] proposed weighted cumulative residual entropy (WCRE) as

E^w(X) =

∫ +∞ 0

xF(x)Λ(x)dx.¯ (4)

Similarly, Misagh et al. [12] proposed weighted cumulative entropy (WCE) as CE^w(X) =

∫ +∞ 0

xF(x) ˜Λ(x)dx. (5)

Now, suppose that X and Y are two non-negative random variables with reliability functions F^¯(x), G(x)^¯ , respectively. If F(x)^¯ is the actual survival function corresponding to the observations andG(x)^¯ is the survival function assigned by the experimenter, then Kumar and Taneja [11] defined the cumulative residual inaccuracy (CRI) based onF(x)^¯ andG(x)^¯ as follows:

I( ¯F ,G) =¯ −

∫ +∞ 0

F¯(x) log ¯G(x)dx. (6)

In analogy with (6), a measure of cumulative past inaccuracy (CPI) associated withF andGis given by I(F, G) =˜ −

∫ +∞ 0

F(x) logG(x)dx. (7)

Order statistics play an important role in problems such as industrial stress testing, meteorological analysis, hydrology, economics and other related fields. Different order statistics can be used in different applications; for example, the maximum is of interest in the study of floods and other meteorological phenomena while the minimum is often used in reliability and survival analysis, etc. For more details about order statistics and their applications, one may refer to [1]. LetX₁, X₂, ..., X_nbe a random sample of sizenfrom an absolutely continuous cumulative distribution functionF(x). IfX₍₁₎≤X₍₂₎≤...≤X_(n)represent the order statistics of the sampleX₁, X₂, ..., X_n. Then the empirical measure ofF(x)is defined as

Fˆ_n(x) =





0, x < X₍₁₎,

k

n, X(k)≤x≤X(k+1), k= 1,2, ..., n−1 1, x > X(k+1).

Recently Thapliyal and Taneja [21] have introduced the measure of residual inaccuracy of order statistics and proved a characterization result for it. Tahmasebi and Daneshi [19] and Tahmasebi et al. [20] have obtained some results of inaccuracy measures in record values. Eskandarzadeh et al. [5] have discussed the cumulative measure of inaccuracy in k-lower record values and studied characterization results of dynamic cumulative inaccuracy.

Daneshi et al. [4] have proposed a weighted cumulative past (residual) inaccuracy of record values and studied its characterization results. The paper is organized as follows: In Section 2, we consider a measure of weighted cumulative residual inaccuracy (WCRI) between F^¯_X_(1:n) and F^¯ and study its properties. In Section 3, we also propose the weighted cumulative past inaccuracy (WPCI) between F_X_(n:n) and F and obtain an estimator of cumulative inaccuracy using empirical approach .

2. WCRI For Minimum of Order Statistics

In this section, we propose the WCRI betweenF^¯_X_(1:n) andF^¯as follows:

I^w( ¯FX_(1:n),F¯) =−

∫ +∞ 0

xF¯X_(1:n)(x)Λ(x)dx= 1

nE^w(X_(1:n)), (8)

(3)

where

E^w(X_(1:n)) =n

∫ +∞ 0

x[ ¯F(x)]ⁿΛ(x)dx.

Hereafter we present some properties ofI^w( ¯FX_(1:n),F)¯ . Proposition 1

LetX be an absolutely continuous non-negative random variable withI( ¯F_X_(1:n),F¯)<∞, for n≥1. Then, we have

I^w( ¯FX_(1:n),F¯) =

∫ +∞ 0

x[ ¯F(x)]ⁿ (∫ x

0

f(z) F¯(z)dz

) dx

=

∫ +∞ 0

∫ +∞ z

λ(z)x[ ¯F(x)]ⁿdxdz, (9) whereλ(.)is the failure rate function.

Proposition 2

LetX be an absolutely continuous non-negative random variable withI^w( ¯FX_(1:n),F)¯ <∞, forn≥1. Then, we have

I^w( ¯FX_(1:n),F¯) =E (

M_(1:n)^w (Z)( ¯F(Z))ⁿ⁻¹ )

, (10)

where

M_(1:n)^w (z) = 1 ( ¯F(z))ⁿ

∫ +∞ z

x( ¯F(x))ⁿdx is the weighted mean residual lifetime (wmrl) ofX_(1:n).

Proof

From (9), we have

I^w( ¯F_X_(1:n),F¯) =

∫ +∞ 0

∫ +∞ z

λ(z)x[ ¯F(x)]ⁿdxdz

=

∫ +∞ 0

f(z) F(z)¯ dz

[∫ +∞ z

x[ ¯F(x)]ⁿdx ]

=

∫ +∞ 0

f(z)

F(z)¯ [ ¯F(z)]ⁿM_(1:n)^w (z)dz=

∫ +∞ 0

f(z)[ ¯F(z)]ⁿ⁻¹M_(1:n)^w (z)dz. (11) So, the proof is complete.

Proposition 3

Leta, b >0. Forn= 1,2, ...it holds that

I^w( ¯FaX_(1:n)+b,F¯aX+b) =aI^w( ¯FX_(1:n),F).¯ The next propositions give some lower and upper bounds forI^w( ¯FX_(1:n),F¯). Proposition 4

For a non-negative random variableXandn≥1, it holds that

I^w( ¯FX_(1:n),F¯)≥M_(1:n)^w (t)|log ¯F(t)|[ ¯F(t)]ⁿ, (12)

whereM_(1:n)^w (t)is the wmrl ofX_(1:n).

(4)

Proof

The proof follows from Baratpour [2].

Proposition 5

LetX be an absolutely continuous non-negative random variable withI^w( ¯F_X_(1:n),F)¯ <∞, forn≥1. Then, we have

I^w( ¯F_X_(1:n),F)¯ ≥

∫ +∞ 0

x( ¯F(x))ⁿF(x)dx. (13)

Proof

Recalling that₋log ¯F(x)≥F(x), the proof then finally follows.

Proposition 6

I^w( ¯FX_(1:n),F)¯ ≤ E^w(X). (14)

Proof

SinceF^¯(x)≥[ ¯F(x)]ⁿ,x≥0, whenn≥1, the proof then finally follows.

Proposition 7

IfXis IFRA (DFRA), then

I^w( ¯F_X_(1:n),F¯)≤(≥)E(

X²( ¯F(X))ⁿ⁻¹)

. (15)

Proof

SinceXis IFRA (DFRA), ^Λ(x)_x is increasing (decreasing) with respect tox >0, which implies that

F(x)Λ(x)¯ ≤(≥)xf(x), x >0. (16) By multiplyingx[ ¯F(x)]ⁿ⁻¹≥0in (16) and then integrating, the result follows.

Proposition 8

I^w( ¯FX_(1:n),F¯) =E^[h^w^(X)]^, (17)

where

h^w(x) =

∫ x 0

z[

−log ¯F(z)]

[ ¯F(z)]ⁿ⁻¹dz, x≥0.

Proof

From (8) and using Fubini’s theorem, we obtain I^w( ¯FX_(1:n),F¯) =

∫ _∞

0

z[−log ¯F(z)][ ¯F(z)]ⁿ⁻¹F(z)dz¯

=

∫ _∞

0

[∫ _∞

z

f(x)dx ]

z[ ¯F(z)]ⁿ⁻¹[−log ¯F(z)]dz

=

∫ _∞

0

f(x) [∫ x

0

z[ ¯F(z)]ⁿ⁻¹[−log ¯F(z)]dz ]

dx=E^[h^w^(X^)]^.

(5)

Proposition 9

LetXandY be two non-negative random variables with reliability functionsF(x)^¯ ,G(x)^¯ , respectively. IfX ≤^icxY , then

I^w( ¯FX_(1:n),F)¯ ≤I^w( ¯GY_(1:n),G).¯ Proof

Sinceh^w(.)is an increasing convex function forn≥1, it follows by Shaked and Shanthikumar [17] thatX ≤^icxY implies h^w(X)≤^icx h^w(Y). By recalling the definition of increasing convex order and Proposition 8 proof is complete.

Proposition 10

Let X and Y be two non-negative random variables with survival function F^¯(x) and G(x)^¯ , respectively. If X ≤^hrY, then forn= 1,2, ..., it holds that

I^w( ¯FX_(1:n),F)¯

E(X) ≤ I^w( ¯GY_(1:n),G)¯ E(Y) . Proof

By noting that the function h^w(x) =∫x

0 z[ ¯F(z)]ⁿ⁻¹[−log ¯F(z)]dz is an increasing convex function, under the assumptionX≤^hrY, it follows by Shaked and Shanthikumar [17],

E[h^w(X)]

E(X) ≤E[h^w(Y)]

E(Y) . Hence, the proof is completed by recalling (17).

Proposition 11

(i) LetXbe a continuous random variable with survival functionF(.)^¯ that takes values in[0, b], with finiteb. Then, I^w( ¯F_X_(1:n),F¯)≤bI( ¯F_X_(1:n),F).¯

(ii) LetX be a non-negative continuous random variable with survival functionF(.)^¯ that takes values in[a,∞), with finitea >0. Then,

I^w( ¯FX_(1:n),F¯)≥aI( ¯FX_(1:n),F¯).

Assume that X_θ^∗ denotes a non-negative absolutely continuous random variable with the survival function H¯θ(x) = [ ¯F(x)]^θ, x≥0. This model is known as a proportional hazards rate model. We now obtain the weighted cumulative residual measure of inaccuracy betweenH^¯X_(1:n) andH^¯ as follows:

I^w( ¯H_X_(1:n),H¯) = −

∫ +∞ 0

xH¯_X_(1:n)(x) log(_¯ H(x))

dx

= −θ

∫ +∞ 0

x( ¯F(x))^nθlog ¯F(x)dx. (18) Proposition 12

Ifθ≥(≤)1, then for anyn≥1, we have

I^w( ¯HX_(1:n),H¯)≤(≥)θI^w( ¯FX_(1:n),F)¯ ≤(≥)θE^w(X).

Proof

Suppose thatθ≥(≤)1, then it is clear[ ¯F(x)]^θ≤(≥) ¯F(x), and hence (18) yields I^w( ¯HX_(1:n),H)¯ ≤(≥)θE^w(X).

(6)

Theorem 13

I^w( ¯FX_(1:n),F¯) = 0, if and only if,Xis degenerate.

Proof

SupposeXis degenerate at pointa, obviously by definition of degenerate function and definition ofI^w( ¯FX_(1:n),F¯), we haveI^w( ¯F_X_(1:n),F) = 0¯ .

Now, suppose thatI^w( ¯F_X_(1:n),F¯) = 0, i.e.

−

∫ _∞

0

x[ ¯F(x)]ⁿlog ¯F(x)dx= 0. (19)

Then, by noting that integrand function of (19) is non-negative, we conclude that −x[ ¯F(x)]ⁿlog ¯F(x) = 0, for almost allx∈R⁺. Thus,F^¯(x) = 0or1, for almost allx∈R⁺.

Recently, Cali et al. [3] introduced the generalized CPI of ordermdefined as Im(F, G) = 1

m!

∫ +∞ 0

F(x)[−logG(x)]^mdx. (20)

In analogy with the measure defined in (20), we now introduce the weighted generalized CRI (WGCRI) of order mdefined as

I_m^w( ¯F ,G) =¯ 1 m!

∫ +∞ 0

xF¯(x)[−log ¯G(x)]^mdx. (21)

Remark 1

LetXbe a non-negative absolutely continuous random variable with cdfF. Then, the WGCRI of ordermbetween F¯_X_(1:n)andF is

I_m^w( ¯FX_(1:n),F¯) = 1 m!

∫ _∞

0

x[ ¯F(x)]ⁿ[−log ¯F(x)]^mdx

= 1

n^mEm^w(X_(1:n)), (22)

where

Em^w(X) =

∫ _∞

0

x

[−log ¯F(x)]m

m!

F¯(x)dx,

is a weighted generalized cumulative residual entropy (WGCRE) which introduced by Kayal [9].

Remark 2

In analogy with (8), a measure of WCRI associated withF^¯andF^¯_X_(1:n) is given by I^w( ¯F ,F¯X_(1:n)) =−

∫ +∞ 0

xF¯(x) log(_¯

FX_(1:n)(x))

dx=nE^w(X). (23) In the remainder of this section, we study dynamic version ofI^w( ¯FX_(1:n),F¯). LetXbe the lifetime of a system under condition that the system has survived up to aget. Analogously, we can also consider the dynamic version ofI^w( ¯FX_(1:n),F¯)as

I^w( ¯F_X_(1:n),F¯;t) = −

∫ +∞ t

x

F¯_X_(1:n)(x) F¯X_(1:n)(t)log

( _¯ F(x)

F(t)¯ )

dx

= log ¯F(t)M_(1:n)^w (t)−

∫ +∞ t

x

F¯_X_(1:n)(x) F¯X_(1:n)(t)log(_¯

F(x)) dx

(7)

= log ¯F(t)M_(1:n)^w (t)− 1 ( ¯F(t))ⁿ

∫ +∞ t

x( ¯F(x))ⁿlog ¯F(x)dx. (24) Note thatlimt→0I^w( ¯FX_(1:n),F¯;t) =I^w( ¯FX_(1:n),F¯). Sincelog ¯F(t)≤0fort≥0, we have

I^w( ¯FX_(1:n),F¯;t) ≤ − 1 ( ¯F(t))ⁿ

∫ +∞ t

x( ¯F(x))ⁿlog ¯F(x)dx

≤ − 1 ( ¯F(t))ⁿ

∫ +∞ 0

x( ¯F(x))ⁿlog ¯F(x)dx= I^w( ¯FX_(1:n),F¯) ( ¯F(t))ⁿ . Theorem 14

LetX be a non-negative continuous random variable with distribution functionF(.). Let the weighted dynamic cumulative inaccuracy of the 1th order statistics denoted byI^w( ¯FX_(1:n),F;¯ t)<∞,t≥0. ThenI^w( ¯FX_(1:n),F¯;t) characterizes the distribution function.

Proof

From (24) we have

I^w( ¯F_X_(1:n),F¯;t) = log ¯F(t)M_(1:n)^w (t)− 1 ( ¯F(t))ⁿ

∫ +∞ t

x( ¯F(x))ⁿlog ¯F(x)dx. (25) Differentiating both side of (25) with respect totwe obtain:

∂

∂t[I^w( ¯FX_(1:n),F¯;t)] = −λF(t)M_(1:n)^w (t) +nλF(t)I^w( ¯FX_(1:n),F¯;t)

= λ_F(t) [

nI^w( ¯F_X_(1:n),F¯;t)−M_(1:n)^w (t) ]

.

Taking derivative with respect totagain we get

´λF(t) =

(λF(t))² (

nλF(t)M_(1:n)^w (t) +n_∂t^∂I^w( ¯FX_(1:n),F;¯ t)−t )

∂

∂tI^w( ¯FX_(1:n),F¯;t) .

(26) Suppose that there are two functionsFandF^∗such that

I^w( ¯F_X_(1:n),F;¯ t) =I^w( ¯F_X^∗

(1:n),F¯^∗;t) =z(t).

Then for allt, from (26) we get

λ´F(t) =φ(t, λF(t)), λ´F∗(t) =φ(t, λF∗(t)), where

φ(t, y) =y²[nys(t) +n´z(t)−t]

´

z(t) ,

ands(t) =M_(1:n)^w (t). By using Theorem 3.2 and Lemma 3.3 of Gupta and Kirmani [8], we have,λF(t) =λF^∗(t), for allt. Since the hazard rate function characterizes the distribution function uniquely, we complete the proof.

Proposition 15

IfX₍₁₎ ≤X₍₂₎≤...≤X_(n)represent the order statistics of the sampleX₁, X₂, ..., X_n. Then, the empirical measure ofI^w( ¯F_X_(1:n),F¯)is obtained as

Iˆ^w( ¯FX_(1:n),F¯) =−

∫ +∞ 0

x[ ˆF¯n(x)]ⁿlog ˆF¯n(x)dx

(8)

=−

n−1

∑

k=1

∫ X_(k+1) X_(k)

x (

1−k n

)n

log (

1−k n

) dx

=−

n−1

∑

k=1

U_k (

1−k n

)n

log (

1− k n

)

, (27)

whereU_k =^X

2

(k+1)−X_(k)²

2 , k= 1,2, ..., n−1. Theorem 16

Let X be a absolutely continue non-negative random variable whitI^w( ¯FX_(1:n),F)¯ <∞, for alln≥1. Then we have

Iˆ^w( ¯FX_(1:n),F¯) −→I^w( ¯FX_(1:n),F¯) a.s.

Proof

From (27) we have

Iˆ^w( ¯FX_(1:n),F¯) =

∫ _∞

0

x(−log ˆF¯n(x))( ˆF¯n(x))ⁿdx

=

∫ 1 0

x(−log ˆF¯n(x))( ˆF¯n(x))ⁿdx+

∫ _∞

1

x(−log ˆF¯n(x))( ˆF¯n(x))ⁿdx

=: W1+W2, (28)

where

W₁=

∫ 1 0

x(−log ˆF¯_n(x))( ˆF¯_n(x))ⁿdx,

W₂=

∫ _∞

1

x(−log ˆF¯_n(x))( ˆF¯_n(x))ⁿdx.

Using dominated convergence theorem (DCT) and Glivenko-Cantelli, we have

∫ 1 0

x(−log ˆF¯_n(x))( ˆF¯_n(x))ⁿdx−→

∫ 1 0

x(−log ¯F(x))( ¯F(x))ⁿdx as m→ ∞. (29) It follows that

x^pFˆ¯_n(x)≤ 1 n

∑n i=1

X_i^p.

Morever, by using SLLN, _n¹∑n

i=1X_i^p−→E^(X^p⁾ and sup_n(_n¹∑n

i=1X_i^p)<∞, then F^ˆ^¯_n(x)≤ x⁻^p(

sup_n(_n¹∑n i=1X_i^p))

=Cx⁻^p. Now applying the DCT we have lim

n→∞W₂=

∫ _∞

1

x(−log ¯F(x))( ¯F(x))ⁿdx. (30)

Finally by using (28) the result follows.

3. WCPI For Maximum of Order Statistics

We consider the WCPI betweenFX_(n:n)andF as follows:

I˜^w(FX_(n:n), F) =−

∫ +∞ 0

xFX_(n:n)(x) log (F(x))dx= 1

nCE^w(X_(n:n)), (31)

(9)

where

CE^w(X_(n:n)) =n

∫ +∞ 0

x[F(x)]ⁿΛ(x)dx.˜ Hereafter we consider some properties ofI^˜^w(F_X_(n:n), F).

Proposition 17

LetX be an absolutely continuous non-negative random variable withI^˜^w(FX_(n:n), F)<∞, forn≥1. Then, we have

I˜^w(FX_(n:n), F) =

∫ +∞ 0

x[F(x)]ⁿ (∫ _∞

x

f(z) F(z)dz

) xdx

=

∫ +∞ 0

∫ z 0

λ(z)x[F(x)]˜ ⁿdxdz, (32) whereλ(.)^˜ is the reversed failure rate function.

Proposition 18

LetX be an absolutely continuous non-negative random variable withI^˜^w(F_X_(n:n), F)<∞, forn≥1. Then, we have

I˜^w(F_X_(n:n), F) =E

(M˜_(n:n)^w (Z)(F(Z))ⁿ⁻¹ )

, (33)

where

M˜_(n:n)^w (z) = 1 (F(z))ⁿ

∫ z 0

x(F(x))ⁿdx is the weighted mean inactivity time (WMIT) ofX_(n:n).

Proof

By (32), we obtain

I˜^w(FX_(n:n), F) =

∫ +∞ 0

∫ z 0

λ(z)x[F(x)]˜ ⁿdxdz

=

∫ +∞ 0

f(z) F(z)dz

[∫ z 0

x[F(x)]ⁿdx ]

=

∫ +∞ 0

f(z)

F(z)[F(z)]ⁿM˜_(n:n)^w (z)dz=

∫ +∞ 0

f(z)[F(z)]ⁿ⁻¹M˜_(n:n)^w (z)dz. (34) Thus, the proof is complete.

Proposition 19

Leta, b >0. Forn= 1,2, ...it holds that

I˜^w(FaX_(n:n)+b, FaX+b) =aI˜^w(FX_(n:n), F).

The next propositions give some lower and upper bounds forI^˜^w(FX_(n:n), F). Proposition 20

For a non-negative random variableXandn≥1, it holds that

I˜^w(F_X_(n:n), F)≥M˜_(n:n)^w (t)|logF(t)|[F(t)]ⁿ, (35)

whereM^˜_(n:n)(t)is the WMIT ofX_(n:n).

(10)

Proof

The proof follows from Baratpour [2].

Proposition 21

I˜^w(FX_(n:n), F)≥

∫ +∞ 0

x(F(x))ⁿF(x)dx.¯ (36)

Proof

Recalling that−logF(x)≥F¯(x), the proof then finally follows.

Proposition 22

I˜^w(FX_(n:n), F)≤ CE^w(X). (37)

Proof

SinceF(x)≥[F(x)]ⁿ,x≥0, whenn≥1, the proof then finally follows.

Proposition 23 IfXis DRFRA, then

I˜^w(FX_(n:n), F)≤E(

X²(F(X))ⁿ⁻¹)

. (38)

Proof

SinceXis DRFRA, ^Λ(x)^˜_x is decreasing with respect tox >0, which implies that

F(x) ˜Λ(x)≤xf(x), x >0. (39)

By multiplyingx[F(x)]ⁿ⁻¹≥0in (39) and then integrating, the result follows.

Proposition 24

I˜^w(FX_(n:n), F) =E[_˜ h^w(X)]

, (40)

where

˜h^w(x) =

∫ _∞

x

z[−logF(z)] [F(z)]ⁿ⁻¹dz, x≥0.

Proof

From (31) and using Fubini’s theorem, we obtain I˜^w(F_X_(n:n), F) =

∫ _∞

0

z[−logF(z)][F(z)]ⁿ⁻¹F(z)dz

=

∫ _∞

0

[∫ z 0

f(x)dx ]

z[F(z)]ⁿ⁻¹[−logF(z)]dz

=

∫ _∞

0

f(x) [∫ _∞

x

z[F(z)]ⁿ⁻¹[−logF(z)]dz ]

dx=E[_˜ h^w(X)]

.

(11)

Proposition 25

LetXandY be two non-negative random variables with reliability functionsF(x)^¯ ,G(x)^¯ , respectively. IfX ≤^icxY , then

I˜^w(FX_(n:n), F)≤I˜^w(GY_(n:n), G).

Proof

Sinceh^˜^w(.)is an increasing convex function forn≥1, it follows by Shaked and Shanthikumar [17] thatX ≤^icxY implies ^˜h^w(X)≤^icx˜h^w(Y). By recalling the definition of increasing convex order and Proposition 24 proof is complete.

Proposition 26

Let X and Y be two non-negative random variables with survival function F^¯(x) and G(x)^¯ , respectively. If X ≤^hrY, then forn= 1,2, ..., it holds that

I˜^w(FX_(n:n), F)

E(X) ≤ I˜^w(GY_(n:n), G)

E(Y) . Proof

By noting that the function^˜h^w(x) =∫_∞

x z[F(z)]ⁿ⁻¹[−logF(z)]dz is an increasing convex function, under the assumptionX≤^hrY, it follows by Shaked and Shanthikumar [17],

E[_˜ h^w(X)]

E(X) ≤E[_˜ h^w(Y)] E(Y) . Hence, the proof is completed by recalling (40).

Proposition 27

(i) LetXbe a continuous random variable with survival functionF(.)^¯ that takes values in[0, b], with finiteb. Then, I˜^w(F_X_(n:n), F)≤bI(F˜ _X_(n:n), F).

(ii) LetX be a non-negative continuous random variable with survival functionF(.)^¯ that takes values in[a,∞), with finitea >0. Then,

I˜^w(F_X_(n:n), F)≥aI(F˜ _X_(n:n), F).

Assume thatX_θ^∗ denotes a non-negative absolutely continuous random variable with the distribution function H_θ(x) = [F(x)]^θ, x≥0. This model is known as a proportional hazards rate model. We now obtain the weighted cumulative past measure of inaccuracy betweenH_X_(n:n)andH as follows:

I˜^w(HX_(n:n), H) = −

∫ +∞ 0

xHX_(n:n)(x) log (H(x))dx

= −θ

∫ +∞ 0

x(F(x))^nθlogF(x)dx. (41)

Proposition 28

Ifθ≥(≤)1, then for anyn≥1, we have

I˜^w(H_X_(n:n), H)≤(≥)θI˜^w(F_X_(n:n), F)≤(≥)θCE^w(X).

Proof

Suppose thatθ≥(≤)1, then it is clear[F(x)]^θ≤(≥)F(x), and hence (41) yields I˜^w(H_X_(n:n), H)≤(≥)θCE^w(X).

(12)

Theorem 29

I˜^w(FX_(n:n), F) = 0, if and only if, X is degenerate.

Proof

SupposeXis degenerate at pointa, obviously by definition of degenerate function and definition ofI^˜^w(F_X_(n:n), F), we haveI^˜^w(FX_(n:n), F) = 0.

Now, suppose thatI^˜^w(FX_(n:n), F) = 0, i.e.

−

∫ _∞

0

x[F(x)]ⁿlogF(x)dx= 0. (42)

Then, by noting that integrand function of (42) is non-negative, we conclude that −x[F(x)]ⁿlogF(x) = 0, for almost allx∈R⁺. Thus,F(x) = 0or1, for almost allx∈R⁺.

Proposition 30

Let X be an absolutely continuous random variable withI^˜^w(F_X_(n:n), F)<∞, forn≥1. Then, we have I˜^w(F_X_(n:n), F) =

∫ 1 0

F_X⁻¹(u) ψ_n(u) fX

(F_X⁻¹(u))^du,

whereψn(u) =−uⁿlogu, for0< u <1. Note thatψn(0) =ψn(1) = 0. Recently, Cali et al. [3] introduced the generalized CPI of order m defined as

I_m(F, G) = 1 m!

∫ +∞ 0

F(x)[−logG(x)]^mdx. (43)

In analogy with the measure defined in (43), we now introduce the weighted generalized CPI (WGCPI) of orderm defined as

I_m^w(F, G) = 1 m!

∫ +∞ 0

xF(x)[−logG(x)]^mdx. (44)

Remark 3

LetXbe a non-negative absolutely continuous random variable with cdfF. Then, the WGCPI of ordermbetween FX_(n:n) andFis

I˜_m^w(F_X_(n:n), F) = 1 m!

∫ _∞

0

x[F(x)]ⁿ[−logF(x)]^mdx

= 1

n^mCE^wm(X_(n:n)), (45)

where

CE^wm(X) =

∫ _∞

0

x[−logF(x)]^m

m! F(x)dx,

is a weighted generalized cumulative entropy (WGCE) which introduced by Kayal and Moharana [10].

Remark 4

In analogy with (31), a measure of WCPI associated withFandFX_(n:n)is given by I˜^w(F, F_X_(n:n)) =−

∫ +∞ 0

xF(x) log(

F_X_(n:n)(x))

dx=nCE^w(X). (46)

(13)

In the remainder of this section, we study dynamic version ofI^˜^w(F_X_(n:n), F). If a system that begins to work at time 0 is observed only at deterministic inspection times, and is found to be down at time t, then we consider a dynamic version ofI^˜^w(FX_(n:n), F)as

I˜^w(FX_(n:n), F;t) = −

∫ t 0

xFX_(n:n)(x) F_X_(n:n)(t) log

(F(x) F(t) )

dx

= logF(t) ˜M_(n:n)^w (t)−

∫ t 0

xFX_(n:n)(x)

F_X_(n:n)(t) log (F(x))dx

= logF(t) ˜M_(n:n)^w (t)− 1 (F(t))ⁿ

∫ t 0

x(F(x))ⁿlogF(x)dx. (47) Note thatlimt→0I˜^w(FX_(n:n), F;t) = ˜I^w(FX_(n:n), F). SincelogF(t)≤0fort≥0, we have

I˜^w(FX_(n:n), F;t) ≤ − 1 (F(t))ⁿ

∫ t 0

x(F(x))ⁿlogF(x)dx

≤ − 1 (F(t))ⁿ

∫ +∞ 0

x(F(x))ⁿlogF(x)dx=

I˜^w(FX_(n:n), F) (F(t))ⁿ . Theorem 31

LetX be a non-negative continuous random variable with distribution functionF(.). Let the weighted dynamic cumulative inaccuracy of the nth order statistics denoted byI^˜^w(F_X_(n:n), F;t)<∞,t≥0. ThenI^˜^w(F_X_(n:n), F;t) characterizes the distribution function.

Proof

From (47) we have

I˜^w(FX_(n:n), F;t) = logF(t) ˜M_(n:n)^w (t)− 1 (F(t))ⁿ

∫ t 0

x(F(x))ⁿlogF(x). (48) Differentiating both side of (48) with respect totwe obtain:

∂

∂t[ ˜I^w(FX_(n:n), F;t)] = −λ˜F(t) ˜M_(n:n)^w (t)−nλ˜F(t) ˜I^w(FX_(n:n), F;t)

= −λ˜F(t)

[M˜_(n:n)^w (t)−nI˜^w(FX_(n:n), F;t) ]

Taking derivative with respect totagain we get

´˜ λF(t) =

(_˜

λF(t))2(

n˜λF(t) ˜M_(n:n)^w (t) +n_∂t^∂I˜^w(FX_(n:n), F;t)−t )

∂

∂tI˜^w(FX_(n:n), F;t) .

(49) Suppose that there are two functionsFandF^∗such that

I˜^w(F_X_(n:n), F;t) = ˜I^w(F_X^∗

(n:n), F^∗;t) =z(t).

Then for allt, from (49) we get

´˜

λ_F(t) =φ(t, λ_F(t)), λ´˜_F∗(t) =φ(t, λ_F∗(t)),

(14)

where

φ(t, y) =y²[nys(t) +n´z(t)−t]

´

z(t) ,

and˜s(t) = ˜M_(n:n)^w (t). By using Theorem 3.2 and Lemma 3.3 of Gupta and Kirmani [8], we have,λ_F(t) =λ_F∗(t), for allt. Since the hazard rate function characterizes the distribution function uniquely, we complete the proof.

Proposition 32

IfX₍₁₎≤X₍₂₎≤...≤X_(n)denote the order statistics of the sampleX₁, X₂, ..., X_n. Then, the empirical measure ofI^˜^w(FX_(n:n), F)is given by

ˆ˜

I^w(F_X_(n:n), F) =−

∫ +∞ 0

x[ ˆF_n(x)]ⁿlog ˆF_n(x)dx

=−

n−1

∑

k=1

∫ X_(k+1) X_(k)

x (k

n )n

log (k

n )

dx

= −1 nⁿ

n∑−1 k=1

kⁿU_klog (k

n )

, (50)

whereUk = ^X

2

(k+1)−X_(k)²

2 , k= 1,2, ..., n−1. Example 1

Consider the random sampleX₁, X₂, ..., X_nfrom a Weibull distribution with density function f(x) = 2λexp(−λx²).

ThenYk =X_k²has an exponential distribution with mean_λ¹. In this case, the sample spacings2Uk=X_(k+1)² −X_(k)² are independent and exponentially distributed with mean _λ(n¹₋_k) (for more details see Pyke [14]). Now from (50) we obtain

E^[^I^ˆ^˜^w^(F^X(n:n), F)] = −1 nⁿ

n∑−1 k=1

kⁿ

2λ(n−k)logk

n, (51)

and

V ar[Iˆ˜^w(FX_(n:n), F)] = 1 n²ⁿ

n−1

∑

k=1

k²ⁿ 4λ²(n−k)²

( logk

n )2

. (52)

We have computed the values ofE^[^I^ˆ^˜^w^(F^X(n:n), F)]andV ar[Iˆ˜^w(FX_(n:n), F)]for sample sizesn= 10,15,20and λ= 0.5,1,2 in Table 1. We can easily see that E^[^I^ˆ^˜^w^(FX_(n:n), F)] is decreasing in n. Also, we consider that limn→∞V ar[Iˆ˜^w(FX_(n:n), F)] = 0.

Example 2

Let X1, X2, ..., Xn be a random sample from a population with pdf f(x) = 2x, 0< x <1. Then the sample spacings2U_kare independent of beta distribution with parameters 1 andn(for more details see Pyke [14]). Now from (50) we obtain

E^[^I^ˆ^˜^w^(FX_(n:n), F)] = −1 nⁿ

n−1

∑

k=1

kⁿ

2(n+ 1)logk

n, (53)

and

V ar[Iˆ˜^w(FX_(n:n), F)] = 1 n²ⁿ⁻¹

n−1

∑

k=1

kⁿ 4(n+ 1)²(n+ 2)

( logk

n )2

. (54)