On weighted cumulative residual Tsallis entropy and its dynamic version

Accepted Manuscript On Weighted Cumulative Residual Tsallis Entropy and its dynamic version A.H. Khammar, S.M.A. Jahanshahi

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S0378-4371(17)30971-8 https://doi.org/10.1016/j.physa.2017.09.079 PHYSA 18684

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Physica A

Received date : 29 November 2016 Revised date : 16 April 2017 Please cite this article as: A.H. Khammar, S.M.A. Jahanshahi, On Weighted Cumulative Residual Tsallis Entropy and its dynamic version, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.079 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

OBJECT: Highlights of manuscript PHYSA-162074

• Deriving the weighted form of the cumulative residual entropy measure and call it Weighted Cumulative Residual Tsallis Entropy (WCRTE). • Proposing ageing classes based on the dynamic version of the WCRTE and showing that it can uniquely determine the survival function and Rayleigh distribution. • Obtaining Several properties, including linear transformations, bounds and related results to stochastic orders for considered measures. • Identifying classes of distributions in which some well-known distributions are maximum dynamic version of WCRTE. • Proposing the empirical WCRTE to estimate the WCRTE.

1

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On Weighted Cumulative Residual Tsallis Entropy and its Dynamic Version A.H. Khammar1 , S. M. A. Jahanshahi

∗b,

a Department b Department

of Statistics, University of Birjand, Birjand, Iran of Statistics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract Recently, Sati and Gupta [1] have introduced a generalized cumulative residual entropy based on the non-additive Tsallis entropy. The cumulative residual entropy, introduced by Rao et al. [2] is a generalized measure of uncertainty which is applied in reliability and image alignment and non-additive measures of entropy. This entropy finds justifications in many physical, biological and chemical phenomena. In this paper, we derive the weighted form of this measure and call it Weighted Cumulative Residual Tsallis Entropy (WCRTE). Being a ”lengthbiased” shift-dependent information measure, WCRTE is related to the differential information in which higher weight is assigned to large values of observed random variables. Based on the dynamic version of this new information measure, we propose ageing classes and it is shown that it can uniquely determine the survival function and Rayleigh distribution. Several properties, including linear transformations, bounds and related results to stochastic orders are obtained for these measures. Also, we identify classes of distributions in which some well-known distributions are maximum dynamic version of WCRTE. The empirical WCRTE is finally proposed to estimate the new information measure. Keywords: Cumulative residual Tsallis entropy, Characterization, Empirical entropy, Maximum Tsallis entropy, Stochastic orders, Weighted generalized entropy ∗ Corresponding

author Email addresses: [email protected] (A.H. Khammar), [email protected] (S. M. A. Jahanshahi ∗ )

Preprint submitted to Journal of Physica A: Statistical Mechanics ...

April 8, 2017

2010 MSC: 94A17, 62N05, 60E15, 62E10

1. Introduction In 1948, Claude Shannon introduced a criterion for measuring of uncertainty and called it entropy. Nowadays, this criterion has recieved a special place in such sciences as economics, physics, computer, telecommunications, communi5

cation theory and reliability. A well known generalization of Shannon’s entropy is Tsallis entropy that was first introduced by Havrda and Charvat [3] in the context of cybernetics theory. Then, Tsallis [4] exploited its non-extensive features and placed it in a physical setting. This measure is defined as [ ] ∫ +∞ 1 α Sα (X) = 1− f (x)dx , α ̸= 1, α > 0, α−1 0

(1.1)

where X is a random variable (rv) having an absolutely continuous cumula10

tive distribution function (cdf) F (t) = P (X ≤ t), t ≥ 0, with the probability density function (pdf) f (t). Entropy (1.1) approaches the Shannon entropy as α → 1. Moreover, the Tsallis entropy is a non-additive entropy as for any two independent rvs X and Y Sα (X, Y ) = Sα (X) + Sα (Y ) + (1 − α)Sα (X)Sα (Y ). From 2000 on, an increasingly wide spectrum of natural, artificial and social

15

complex systems have been identified which confirm the predictions and consequences derived from this non-additive entropy, such as non-extensive statistical mechanics (Tsallis [5]), which generalizes the Boltzmann-Gibbs’s theory. In a recent conference (Recent Innovations in Info-Metrics: An Interdisciplinary Perspective 2014, American University, Washington DC), Tsallis presented a

20

classification of physical systems according to their complexities and identified the systems where additive entropy (Shannon entropy) is and is not applicable, hence a non-additive measure. Ebrahimi [6] proposed a measure which measures the uncertainty about the remaining lifetime of a system if it is working at time t that is known as the 2

25

residual entropy function. Nanda and Paul [7] have introduced the generalized residual entropy and have redefined (1.1) for a unit surviving up to age t as [ ] ∫∞ α f (x)dx 1 t Sα (X; t) = 1− , (1.2) α−1 F¯ α (t) where Sα (X; t) is the residual Tsallis entropy. Note that F¯ (t) = 1 − F (t) is the survival function (sf) of X. Moreover, (1.2) is a non-additive residual entropy and when t = 0, (1.2) reduces to (1.1). For more details, see Nanda Paul [7]

30

and Kumar Taneja [8]. Rao et al. [2] defined a new measure of uncertainty based on the survival function of X, and called it cumulative residual entropy (CRE). Shannon entropy is defined only for distributions with densities, but CRE is based on cdf rather than pdf, and is thus, generally more stable since the distribution function is

35

more regular being defined in an integral form unlike the density function, which is defined as the derivative of the distribution. Note that CRE measure is always non-negative and can be applied in reliability and image alignment. For more properties of CRE, one can refer to Wang and Vemuri [9] and Rao [10]. The dynamic form of CRE is proposed by Asadi and Zohrevand [11] entitled

40

the dynamic cumulative residual entropy (DCRE). Recently, Sati and Gupta [1] have proposed the cumulative residual entropy based on Tsallis entropy as ηα (X) =

[ ] ∫ ∞ 1 1− F¯ α (x)dx , α ̸= 1, α > 0. α−1 0

(1.3)

and called it the cumulative residual Tsallis entropy (CRTE). Measure (1.3) approaches the CRE as α → 1. Also, they have considered the dynamic version 45

of CRTE which is defined as

[ ] ∫∞ α F¯ (x)dx 1 t ηα (X; t) = 1− , α ̸= 1, α > 0. α−1 F¯ α (t)

(1.4)

and have called it the dynamic cumulative residual Tsallis entropy (DCRTE). By considering a relationship between DCRTE and mean residual life of a component, Sati and Gupta [1] have characterized some specific lifetime distributions. 3

50

Also, Kumar [12] defined a cumulative residual entropy based on Tsallis entropy which is identical to DCRTE and studied some properties and characterization results for it. Sometimes in statistical modelling, standard distributions are not suitable for our data and we need to study weighted distributions. In recent years, this

55

concept has been applied in many areas of statistics, such as analysis of family size, human heredity, world life population study, renewal theory, biomedical and statistical ecology, reliability modelling, etc. For recent works on weighted distributions, we refer to Navarro et al. [13], Nair and Sunoj [14], Pakes et al. [15], Di Crescenzo and Longobardi [16], Oluyede and Terbeche [17], Sunoj and

60

Maya [18] and Jahanshahi et al. [19]. In particular, in the context of theoretical neurobiology, some measures of uncertainity based on the notion of the weighted entropy (see Johnson and Glantz [20] and Belis and Guiasu [21]). Guiasu [22] has shown that weighted entropy has been used to balance the amount of information and the degree of ho-

65

mogeneity associated with a partition of data in classes. Among others, Di Crescenzo and Longobardi [16] have defined and studied the notion of weighted entropy for residual and past lifetimes of a component via the notion of weighted residual and past entropies, extending some properties previously introduced by Belzunce et al. [23] and Nanda and Paul [7] to characterize a distribution func-

70

tion via weighted dynamic measures, Misagh and Yari [24] have studied the weighted differential information measure of two-sided truncated random variables. Based on the CRE Mirali et al. [25] have introduced a new information measure called weighted cumulative residual entropy (WCRE). Being, ”lengthbiased” and shift-dependent, the WCRE assigns larger weights to larger values

75

of random variable X as εw (X) = −

∫

∞

xF¯ (x) log F¯ (x)dx.

(1.5)

0

The designation of εw (X) as WCRE arises from coefficient x. For more details see Mirali et al. [25]. Suchismita Das [26] has introduced the concept of weighted

4

Tsallis entropy and its dynamic version as follows: ] [ ∫ ∞ 1 w α fw (x)dx , α ̸= 1, α > 0, Sα (X) = 1− α−1 0 and Sαw (X; t) 80

where fw (t) =

t E(X) f (t)

[ ] ∫∞ α fw (x)dx 1 t = 1− . α−1 F¯wα (t) E(X|X>t) ¯ F (t), E(X)

and F¯w (t) =

(1.6)

(1.7)

0 < E(X) < ∞ are the

probability density function and survival function of a length-biased weighted rv Xw associated to the rv X. When w(t) = t, Xw is called the length (or size) biased random variable and X is denoted by Xl . This means that, when the weight function depends on the lengths of the component, the resulting distri85

bution is called ”length-biased weighted function”. In this paper, we propose generalizations of CRTE and DCRTE called weighted cumulative residual Tsallis entropy (WCRTE) and dynamic weighted cumulative residual Tsallis entropy (DWCRTE), respectively. 2. Weighted cumulative residual Tsallis entropy

90

In analogy with (1.5), we define the Weighted cumulative residual Tsallis entropy (WCRTE) of a non-negative rv X as [ ] ∫ ∞ 1 ηαw (X) = 1− xF¯ α (x)dx , α ̸= 1, α > 0. α−1 0

(2.1)

The factor x in the integral on right-hand side yields a ”length-biased” shiftdependent information measure assigning greater importance to larger values of the rv X. An alternative expression for the WCRTE of X is the following [ ] ∫ ∫ ′ 1 1 ∞ ¯α α − 1 ∞ ¯α ηαw (X) = 1− F (x)m(x)dx − xF (x)m (x)dx , α−1 α 0 α 0 95

where m(t) = E(X − t | X > t) = ′

∫∞ t

F¯ (x)dx F¯ (t)

is the mean residual life function

(MRL) and m (t) is the first derivative of m(t) with respect to t. Example 2.1. Let X and Y be two rvs with pdfs     1, 1, 0
b
From (1.3), we obtain ηα (X) = ηα (Y ) =

1 α−1

can see that

[ ] a 1 − α+1 . On the other hand, we

[ ] 1 a2 1− , α−1 (α + 1)(α + 2) [ ] 1 a(b − a)(α + 2) + a2 (α + 1) ηαw (Y ) = 1− . α−1 (α + 1)(α + 2) ηαw (X) =

Hence, even though ηα (X) = ηα (Y ), the WCRTE about the predictability of X by the pdf f (t) is greater (or smaller) than the predictability of Y by the pdf 100

g(t) for α > 1 (0 ≤ α < 1). Table 1 provides some well-known families of distributions and their CRTE and WCRTE.

Table 1:

CRTE and WCRTE for some well-known families of distributions Distribution function

ηα (X)

w (X) ηα

1 [1 − b−a ] α−1 α+1

2 1 [1 − b(b−a)(α+2)−(b−a) (α+1) ] α−1 (α+1)(α+2)

1 [1 − 1 ] α−1 θα

] 1 [1 − 1 α−1 2(θα)2

] 1 [1 − b α−1 aα−1

1 [1 + b2 β ((a + 1)(α − 1) − 1, 2)] α−1

( 1) 1 [1 − β (α+1), θ ] α−1 θ

( 2) 1 [1 − β (α+1), θ ] α−1 θ

Uniform distribution ¯ (t) = b−t , F b−a

a < t < b

Exponential distribution ¯ (t) = e−θt , F θ > 0

Pareto distribution ( ) ¯ (t) = 1 + t −a F b x ≥ β > 0, λ > 0 Power distribution ¯ (t) = 1 − tθ F a > 1,

0 ≤ x ≤ b

Next we will prove some properties of WCRTE. 105

Theorem 2.1. ηαw (X) < ∞ if for some p > 2, E(X p ) < ∞. Proof. It is enough to show that

∫∞ 0

xF¯ α (x)dx < ∞. Using Markov inequality

6

in the second inequality, we have ∫ ∞ ∫ α ¯ xF (x)dx =

0

The last integral is finite if p > p>

xF¯ α (x)dx +

∫

∞

xF¯ α (x)dx )α ∫ 1 ∫ ∞ ( E(X p ) dx ≤ x dx + x xp 0 1 ∫ ∞ )α 1 ( = + E(X p ) x1−pα dx, 2 1

0

2 α.

1

2 α.

1

For p > 2, we can choose α < 1 to satisfy

Thus, the result follows.

Let us now discuss the effect of linear transformations on the WCRTE. Lemma 2.1. If Y = aX + b, with a > 0 and b ≥ 0, then ηαw (Y ) = 110

1−b−a + aηαw (X) + bηα (X). α−1

Proof. The result follows by noting that F¯aX+b (t) = F¯X ( t−b a ), t ≥ 0, and using (2.1). Let Xθ∗ be an absolutely continuous non-negative rv with sf F¯θ∗ (t) = [F¯ (t)]θ , t ≥ 0.

(2.2)

The model (2.2) was first proposed by Lehman [27] in contrast to the proportional hazards rate model. It is more flexible to accommodate both monotonic 115

as well as non-monotonic failure rates even though the baseline failure rate is monotonic. For more details on the applications and properties of the proportional hazards rate model see Mudholkar et al. [28], Gupta et al. [29] and Di Crescenzo [30], among others. Now, look at the following lemma that contains an upper and a lower bound for the WCRTE of Xθ∗ depending on ηαw (X).

120

Lemma 2.2. There holds ηαw (Xθ∗ ) ≥ ηαw (X) ≥ ηαw (θX), 0 < θ ≤ 1, 0 < α < 1 and θ ≥ 1, α > 1, the inequality being reversed for 0 < θ ≤ 1, α > 1 and θ ≥ 1, 0 < α < 1. 7

Proof. The result follows by noting that ηαw (Xθ∗ ) =

αθ−1 w α−1 ηαθ (X)

and using fact

¯θ

that F (t) ≤ (≥)F¯ (t), x ≥ 0, when θ ≥ 1(0 < θ ≤ 1). 125

The following result immediately follows from Lemma 2.2 and by recalling that the right-hand side of (2.2) is the distribution function of the maximum of iid random variables (when the power is integer). Corollary 2.1. Let X1 , X2 , ..., Xn be iid random variables, with n a positive integer. Then, ηαw (nX1 ) ≤ ηαw (max{X1 , X2 , ..., Xn }).

130

In order to provide some bounds for the WCRTE of a non-negative rv X, let us define the weighted mean residual lifetime (WMRL) of X: ∫∞ xF¯ (x)dx m∗F (t) = t ¯ . F (t) ∫∞ In particular, m∗F (0) = 0 xF¯ (x)dx.

(2.3)

Theorem 2.2. Let X be a non-negative continuous rv with WMRL function m∗F (t), WCRTE ηαw (X) and WCRE εw (X), such that ηαw (X) < ∞. Then (i) ηαw (X) ≥

1 [1 − m∗F (0)], α > 0. α−1

(ii) w ∗ ηαw (X) ≤ (≥)εw α (X) ≤ (≥)ε (X) = mF (t), α > 1, (0 ≤ α < 1), 135

where εw α (X) =

1 1−α

log

∫∞ 0

xF¯ α (x)dx is the weighted survival entropy of

order α (see Rajesh et al. [31]). Proof. (i) Since F¯ α (t) ≤ (≥)F¯ (t), t ≥ 0, when α ≥ 1, (0 < α ≤ 1), the result follows. (ii) Since − log u ≥ 1 − u, u > 0 we can conclude that ηαw (X) ≤ (≥)εw α (X), 140

α > 1, (0 ≤ α < 1). On the other hand, Rajesh et al. [31] expressed that w ∗ εw α (X) ≤ (≥)ε (X) = mF (t), α > 1, (0 ≤ α < 1). Thus the result follows.

8

The next lemma gives some bounds for the WCRTE, depending on CRTE. The proof is omitted. Lemma 2.3. Let X be a non-negative continuous rv with sf F¯ (t) that 145

(i) takes values in [0, b], with finite b. Then ( ) 1 − (α − 1)ηαw (X) ≤ (≥)b 1 − (α − 1)ηα (X) , α > 1, (0 ≤ α < 1).

(ii) takes values in [a, ∞], with finite a. Then

( ) 1 − (α − 1)ηαw (X) ≥ (≤)a 1 − (α − 1)ηα (X) , α > 1, (0 ≤ α < 1).

3. Dynamic weighted cumulative residual Tsallis entropy Sati and Gupta [1] have introduced the concept of dynamic weighted Tsallis cumulative residual entropy (DWCRTE) which can be defined as ] [ ∫∞ α F¯w (x)dx 1 t , α > 0, α ̸= 1. ηα (W ; t) = 1− α−1 F¯wα (t) 150

(3.1)

In this section, we shall focus on new weighed version of DCRTE of non-negative random variables. In order to introduce this new shift-dependent dynamic information measure, we now make use of (2.1) to define a weighted information for residual lifetimes. The dynamic weighted cumulative residual Tsallis entropy (DWCRTE) is defined by ηαw (X; t)

155

[ ] ∫∞ xF¯ α (x)dx 1 t 1− , α > 0, α ̸= 1. = α−1 F¯ α (t)

(3.2)

Also, we define a weighted information for past lifetimes that is the weighted version of entropy (1.4) and call it the dynamic weighted cumulative past Tsallis entropy (DWCPTE) of a non-negative rv X as [ ] ∫t xF α (x)dx 1 w 0 η¯α (X; t) = 1− , α > 0, α ̸= 1. α−1 F α (t)

(3.3)

It is easy to see that limt→+0 ηαw (X; t) = limt→∞ η¯αw (X; t) = ηαw (X). There are several equivalent expressions for the DWCRTE. An alternative ex160

pression for the DWCRTE of X is provided in the next lemma, that is in terms of DCRTE. 9

Lemma 3.1. Let X be an absolutely continuous non-negative rv with DWCRTE and DCRTE functions ηαw (X; t) and ηα (X; t), respectively. Then, for all t ≥ 0, we have ηαw (X; t)

[ ( ) =1 − (α − 1) tα 1 − (α − 1)ηα (X; t) ] ( ¯ )α ∫ ∞ ( ) α−1 F (y) +α y 1 − (α − 1)ηα (X; y) dy . F¯ (t) t

Proof. )α )α )( ¯ ∫ ∞( ¯ ∫ ∞(∫ x F (x) F (x) α−1 x ¯ dx = y dy dx F (t) F¯ (t) t t 0 [∫ ]( ¯ )α ∫ ∞ ∫ x t F (x) α−1 α−1 =α y dy + y dy dx F¯ (t) t 0 t )α ) ∫ ∞( ∫ ∞( ¯ ∫ ∞ α F (x) α−1 ¯ α (x)dx dy, dx + = tα y F F¯ (t) F¯ α (t) t t y (3.4) where

and

∫ ∫

∞ t

∞

t

)α (¯ F (x) dx = 1 − (α − 1)ηα (X; t) F¯ (t)

[ ] F¯ α (x)dx = F¯ α (t) 1 − (α − 1)ηα (X; t) .

Using both expressions in (3.4) and from (1.4), we get the stated result. 165

The following lemma describes the alternative expression to the DWCRTE of symmetric random variable X, that is in terms of DWCRTE. Lemma 3.2. Let X be a rv with support in [0, b], and symmetric with respect to 2c , i.e. F¯ (t) = F (c − t) for 0 ≤ t ≤ c. Then, ( ) ( ) 1−(α−1)ηαw (X; t) = c 1−(α−1)¯ ηα (X; c−t) + 1−(α−1)¯ ηαw (X; c−t) , 0 ≤ t ≤ c, where η¯α (X; t) =

170

1 α−1

[

1−

∫t 0

F α (x)dx F α (t)

]

, α ̸= 1, α > 0, is the dynamic cumulative

past Tsallis entropy, that is identical to the relation (4.5) of Kumar [12].

10

Proof. From (3.2) and the symmetry property F¯ (t) = F (c − t), we have [ ] ∫c xF α (c − x)dx 1 w t ηα (X; t) = 1− α−1 F α (c − t) [ ] ∫ c−t α (c − y)F (y)dx 1 = 1− 0 α−1 F α (t) [ ] ∫ c−t α ∫ c−t c 0 F (y)dx yF α (y)dx 1 0 = 1− + dy , α−1 F α (c − t) F α (c − t) which completes the proof. In order to attain the relationship between WCRTE and DWCRTE, we have the following theorem. The proof is omitted. Lemma 3.3. For all t ≥ 0, it holds that ∫ t ) ( w 1 − (α − 1)ηα (X) = xF¯ α (x)dx − F¯ α (t) 1 − (α − 1)ηαw (X; t) . 0

In the following, we extend Lemma 2.1, Lemma 2.2 and Theorem 2.2 for

175

DWCRTE. The proofs are omitted. Lemma 3.4. If Y = aX + b, with a > 0 and b ≥ 0, then ηαw (Y ) =

1−b−a t−b t−b + aηαw (X; ) + bηα (X; ). α−1 a a

keeping in mind that for any x ≥ t, it holds

F¯ (x) F¯ (t)

(3.5)

≤ 1, Lemma 2.2 can be

stated in following. 180

Lemma 3.5. For all t ≥ 0, it holds that ηαw (Xθ∗ ; t) ≥ ηαw (X; t) ≥ ηαw (θX; t), 0 < θ ≤ 1, 0 < α < 1 and θ ≥ 1, α > 1, the inequality being reversed for 0 < θ ≤ 1, α > 1 and θ ≥ 1, 0 < α < 1. Theorem 3.1. Let X be a non-negative continuous rv with WMRL function m∗F (t) and DWCRTE function ηαw (X; t) < ∞. Then (i)

185

ηαw (X; t) ≥

1 [1 − m∗F (t)], α > 0. α−1 11

(ii) w ηαw (X; t) ≤ (≥)εw α (X; t) ≤ (≥)ε (X; t), α > 1, (0 ≤ α < 1),

where εw (X; t) is the dynamic version of WCRE and εw α (X; t) =

1 1−α

log

∫∞ t

xF¯ α (x)dx F¯ α (t)

is the dynamic weighted survival entropy of order α (see Rajesh et al. [31]). 190

Theorem 3.2. The following inequality holds: [ ] ( )1( )1 η2w (X; t) ≥ 1 − 1 − (α − 1)ηαw (X; t) α 1 − (β − 1)ηβw (X; t) β , α, β > 2.

¯ Proof. Let X and Y be two non-negative rvs with sfs F¯ (t) and G(t). Using Holder inequality, we have ∫

∞ t

)β ] β [∫ ∞( 1 1 ¯ ¯ ¯ (x) )α ] α [ ∫ ∞ ( 1 G(x) 1 F x α F¯ (x) x β G(x) β α x ¯ dx ≤ x ¯ dx dx , ¯ F¯ (t) G(t) F (t) G(t) t t

where

1

1 α

+

1 β

1

¯ = 1 and α, β ∈ [1, ∞]. If F¯ (x) = G(x), (3.2) gives the stated

result. 4. Stochastic comparisons and ageing classes. The Cauchy distribution is a well-known distribution in probability and 195

statistics. In physics, it is known as Lorentz or the Breit-Wigner distribution, as it is described as the solution to the differential equation describing forced resonance. Consider the comparison of two Cauchy distributions. In general, the simplest way to compare two distribution functions is by their associated means and variances or standard deviations. However, for Cauchy distributions, means

200

and variances do not exist! Moving apart from Cauchy distributions, means and variances are not very informative in comparing distributions. Hence, we need to take advantage of stochastic orders as an informative method for comparing distributions. In previous years stochastic orders have attracted an increasing number of authors, who used them in several areas of probability and statis-

205

tics, with applications in many fields, such as reliability theory, queueing theory, survival analysis, operations research, mathematical finance, risk theory, management science and biomathematics. Also, stochastic orders are often invoked 12

not only to provide useful bounds and inequalities but also to compare stochastic systems. For a comprehensive discussion on various concepts of stochastic 210

ordering one can see Shaked and Shanthikumar [32]. In this section, we study some ordering properties of WCRTE and DWCRTE for random variables. We need the following definitions in which X and Y denote rvs with cdfs F (t) and ¯ G(t), pdfs f (t) and g(t) and sfs F¯ (t) and G(t). Definition 4.1. The rv X is said to be st

215

¯ • stochastically smaller than Y (denoted ⩽) if F¯ (t) ≤ G(t) for all t ≥ 0. hr

• smaller than Y in hazard rate order (denoted ⩽) if λF (t) ≥ λG (t) for all t ≥ 0, where λF (t) =

f (t) F¯ (t)

is the hazard rate function of X. lr

• smaller than Y in likelihood ratio order (denoted ⩽) if

g(t) f (t)

is increasing in t.

• smaller than Y in the weighted cumulative residual entropy of order α > 0 220

(denoted

W CRE

⩽ ) if ηαw (X) ≤ ηαw (Y ).

• smaller than Y in the dynamic cumulative residual entropy of order α > 0 (denoted

DCRE

⩽ ) if ηα (X; t) ≤ ηα (Y ; t) for all t > 0.

• smaller than Y in the dynamic weighted cumulative residual entropy of order α > 0 (denoted

DW CRE

⩽

) if ηαw (X; t) ≤ ηαw (Y ; t) for all t > 0.

The connection between the earlier mentioned stochastic orders is described

225

in the following diagram (see Shaked and Shanthikumar [32]) fr

lr

disp

st

X ⩽ Y ⇒ X ⩽ Y ⇒ X ⩽ Y ⇐ X ⩽ Y. The following theorem describes the relationship between WCRE and stochastic orderings. The proof is omitted. Theorem 4.1. Let X and Y be two non-negative absolutely continuous rvs with st

230

¯ sfs F¯ (t) and G(t). If X ⩽ Y , then X

W CRE

⩾

W CRE

( ⩽ )Y , α > 1 (0 ≤ α < 1).

Example 4.1. Let X and Y be two rvs with sfs F¯ (t) = 1 − t, 0 < t < 1 and st

¯ G(t) = 1 − t2 , 0 < t < 1, respectively. It can be shown that X ⩽ Y , but X

W CRE

⩾

W CRE

( ⩽ )Y , α > 1 (0 ≤ α < 1).

13

Remark 4.1. For some families of distributions such as exponential or Pareto, 235

ηαw (X) can be easily computed in closed form and thus, the ordering can be directly obtained. But the ordering for other distributions can be obtained by application of Theorem 4.1 (ordering within parametric families) and using dispersion, likelihood ratio and stochastic ordering. For example, it can be easily shown that, if X has a gamma distribution with shape parameter θ, then for lr

240

st

θ0 < θ1 , Xθ0 ⩽ Xθ1 , and thus Xθ0 ⩽ Xθ1 . So we have Xθ0

W CRE

⩾

W CRE

( ⩽ )Xθ1 ,

for α > 1 (0 ≤ α < 1). If X has a Weibull distribution with shape parameter disp

st

β, then for β0 < β1 , Xβ0 ⩾ Xβ1 , and thus Xβ0 ⩾ Xβ1 . Therefore, Xβ0 W CRE

W CRE

⩾

( ⩽ )Xβ1 , for α > 1 (0 ≤ α < 1). The following theorem describes the relationship between DWCRE and haz245

ard rate orderings. The proof is omitted. Theorem 4.2. Let X and Y be two non-negative absolutely continuous rvs ¯ with sfs F¯ (t) and G(t) and the hazard functions λF (t) and λG (t), respectively. hr

If X ⩾ Y , then X

DW CRE

⩽

In particular, X 250

W CRE

⩽

(

DW CRE

⩾

W CRE

)Y , α > 1 (0 ≤ α < 1).

( ⩾ )Y , α > 1 (0 ≤ α < 1).

Theorem 4.2 can be used in order statistics and record values as following corollaries. For comprehensive discussion on various concepts of order statistics and record values see Arnold et al. [33] and David and Nagaraja [34]. Note that the same results can be hold for the WCRE ordering. Corollary 4.1. Let X1 , X2 ..., Xn be iid non-negative rvs having sf F¯ (t). If

255

Xi:n denotes the ith order statistic in this sample of size n, then the lifetime of a series system is determined by X1:n and the lifetime of a parallel system is determined by Xn:n with distribution functions F1:n (t) and Fn:n (t), respectively. So, we have following results: (a) Suppose that ηαw (Xi:n ; t) is a decreasing (increasing) function of i for all

260

α > 1 (0 ≤ α < 1), then Xi:n (b) X1:n

DW CRE

⩾

X1:n−1 (

DW CRE

DW CRE

min{X1 , X2 ..., Xn }. (c) Xn−1:n−1

DW CRE

⩾

⩾

Xn:n (

⩽

⩽

), α > 1 (0 ≤ α < 1).

), α > 1 (0 ≤ α < 1). That is X1:n =

DW CRE

⩽

DW CRE

Xi+1:n (

), α > 1 (0 ≤ α < 1). That is Xn:n = 14

max{X1 , X2 ..., Xn }. 265

(d) It is know that, λF1:n (x) = nλF (x) ≥ λF (x). Thus, X DW CRE

(

⩽

DW CRE

⩾

X1:n

), α > 1 (0 ≤ α < 1).

It is known that, λFn:n (t) = θ(t)λF (t), where n[F (t)/F¯ (t)]n−1 θ(t) = ∑n−1 n! ≤ 1. ¯ i i=0 i!(n−i)! .[F (t)/F (t)] Thus, λFn:n (t) ≤ λF (t). Hence Xn:n

DW CRE

⩾

DW CRE

X (

⩽

), α > 1 (0 ≤ α < 1). hr

¯ (e) Suppose that Y1 , Y2 ..., Yn be iid non-negative rvs having sf G(t). If X1 ⩾ Y1 , 270

then Xi:n

DW CRE

⩾

DW CRE

⩽

Yi:n (

), α > 1 (0 ≤ α < 1).

Corollary 4.2. Let {Xn , n ≥ 1} be a sequence of iid non-negative rvs from the cdf F (t) with hazard function λF (t), then we have the following results: (a) suppose that Un , n ≥ 1 denotes the nth upper record with distribution function Fn (t), then it is known that λFn (t) = θ(t)λF (t), where θ(t) =

275

[− log F¯ (t)]n−1 /(n − 1)! ≤ 1. ∑n−1 1 ¯ k k=0 k! [− log F (t)]

That is, λFn (t) ≤ λF (t). Hence Un hr

Also, Un ⩽ Um for n < m, thus Un

DW CRE

DW CRE

⩾

X (

⩾

Um (

DW CRE

⩽

), α > 1 (0 ≤ α < 1).

DW CRE

⩽

), α > 1 (0 ≤ α < 1).

(b) Suppose that Vn denotes the nth upper record of a sequence of random varihr

hr

ables {Yn , n ≥ 1}. If X1 ⩽ Y1 , then Un ⩽ Vn and we have Un DW CRE

( 280

⩽

DW CRE

⩾

Vn

), α > 1 (0 ≤ α < 1).

The following lemma gives the values of the functions ηαw (X; t) under linear transformations and is applied to obtain several results in the sequel. Lemma 4.1. Let Y = ϕ(X) and ϕ(t) be any strictly increasing function with limt→∞ ϕ(t) = ∞. Then, for any t ≥ 0, we have ηαw (Y ; t) = ηαw (ϕ(X); ϕ−1 (t)), where ηαw (ϕ(X); t) =

1 α−1

[

1−

∫∞ t

ϕ(x)F¯ α (x)dx F¯ α (t)

15

] , α > 0, α ̸= 1.

(4.1)

Proof. From (3.2), we have ηαw (Y

285

[ ] ∫∞ y F¯ϕα (x)dx 1 t ; t) = 1− α−1 F¯ϕα (t) [ ( ∫ ∞ ¯ −1 ) ] d −1 y F ((ϕ (y)) | dy ϕ (y) | α 1 t = 1− , α−1 P r(Y > t)

where Fϕ (t) is cdf of ϕ(X). Let ϕ(t) be strictly increasing. By setting y = ϕ(x), we get the stated result. The following theorem shows that DCRE order implies DWCRE order under an increasing transformation. Theorem 4.3. Let X and Y be two non-negative random variables such that

290

X

DCRE

⩽

DCRE

Y ( ⩾ ) and take ϕ(t) a strictly increasing function with ϕ(0) = 0

′

and ϕ (0) >1. Then X

DW CRE

⩽

DW CRE

Y (

⩾

∞

2

), for 0 ≤ α < 1 (α > 1).

Proof. From (4.1), we have [ ] (α−1) ηαw (X; t)−ηαw (Y ; t) =

∫

ϕ (x)

ϕ−1 (t)

[(

)α ( )α ] ¯ G(x) F¯ (x) − ¯ −1 dx. ¯ −1 (y)) G((ϕ F ((ϕ (y))

Since ϕ(x) > t, we can obtain [ ] [ ] (α − 1) ηαw (X; t) − ηαw (Y ; t) > t2 (α − 1) ηαw (X; ϕ−1 (t)) − ηαw (Y ; ϕ−1 (t)) > 0.

The last inequality is obtained by assumption that 0 ≤ α < 1 and X 295

The proof for α > 1 and X

DCRE

⩾

DCRE

⩽

Y.

Y is simillar.

The following theorem shows that, under certain conditions, DWCRE ordering between two non-negative random variables is closed under increasing linear transformation. The proof is omitted. Corollary 4.3. For two non-negative rvs X1 and X2 , let us define Y1 = a1 X1 + 300

b1 and Y2 = a2 X2 + b2 , where a1 ≥ a2 > 0, b1 ≥ b2 ≥ 0. (i) Let α > 1 and X1

DCRE

⩽

(ii) Let 0 ≤ α < 1 and X1

X2 . Then Y1

DCRE

⩾

DW CRE

⩽

X2 . Then Y1

16

Y2 if X1

DW CRE

⩾

DW CRE

⩽

Y2 if X1

X2 . DW CRE ⩾

X2 .

The applications of classes of life distributions can be observed in reliability, engineering, biological science, maintenance and biometrics. Therefore, statisti305

cians and reliability analysts are interested in modeling survival data and using classifications of life distributions based on some aspects of aging. See for example, Barlow and Proschan [35], Zacks [36] and Lai and Xie [37]. Sati and Gupta [1] have defined two non-parametric classes of distributions based on the monotonicity properties of DCRTE. Here, we define new aging classes based on

310

DWCRTE as follows: Definition 4.2. The r.v. X is said to have increasing (decreasing) DWCRTE, denoted by IDWCRTE (DDWCRTE) property, if ηαw (X; t) is increasing (decreasing) with respect to t. The following example shows that there exist distributions which are not

315

monotone in terms of ηαw (X; t). Example 4.2. Let X 2]. From (3.2), we can see [ be uniformly ( distributed on [0,)] t (α+2)−(α+1)(1− ) 1 t 2 that ηαw (X; t) = α−1 1 − 4(1 − 2 ) , which is not monotone (α+1)(α+2) for α = 0.3 in t ∈ [0, 0.6] as shown in Figure 1. Also, for α = 4.3, we see that

ηαw (X; t) is not monotone in t ∈ [0, 3] as shown in figure 1. So ηαw (X; t) is not 320

monotone. Note that differentiating (3.2) with respect to t, we can conclude that (α − 1)

d w η (X; t) = t − αλF (t)[1 − (α − 1)ηαw (X; t)]. dt α

Hence, the rv X is IDWCRTE (DDWCRTE) for all t ≥ 0, if and only if [ ] 1 t w ηα (X; t) ≥ (≤) 1− , α > 1, α−1 αλF (t) [ ] 1 t 1− , 0 ≤ α < 1. ηαw (X; t) ≤ (≥) α−1 αλF (t) Remark 4.2. The above relations are equivalent to t λF (t) ≤ (≥) [1 − (α − 1)ηαw (X; t)]−1 , α > 1, α t λF (t) ≥ (≤) [1 − (α − 1)ηαw (X; t)]−1 , 0 ≤ α < 1. α 17

(4.2)

(4.3) (4.4)

alpha = 4.3

0.40 0.30

0.35

WCRTE

0.48 0.46

0.25

0.44

WCRTE

0.50

0.45

alpha = 0.3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.5

t

1.0

1.5

2.0

2.5

3.0

t

w (X; t) for 0 ≤ t ≤ 0.6, 0 ≤ α < 1 (left panel) and 0 ≤ t ≤ 3, α > 1 Figure 1: Graph of ηα

(right panel).

Since the distribution function and the hazard rate function are equivalent (in the sense that having known one, the other can be obtained uniquely), from above relation one can give bounds to the distribution function, as well. 325

The next theorem shows that under some conditions the IWDCRTE (DDWCRTE) property is invariant under linear transformations. Theorem 4.4. Let X be a non-negative rv and Y = aX + b, where a > 0, b ≥ 0. Then Y is IWDCRTE (DDWCRTE), if X is IDCRTE (DDCRTE) and IWDCRTE (DDWCRTE).

330

Proof. The proof follows directly from the relation (3.5). Suppose that a random variable has a known pdf and hence, it determines the distribution completely. But in many cases, the explicit density is unknown and must be estimated. The classical procedure consists of fitting an analytical function to the observations. Another more logical approach is to apply

335

the maximum entropy (ME) proposed by Jaynes [38]. ME principle states that when some information is given about a random variable, the least biased probability distribution can be obtained by maximizing the Shannon entropy subject to the given constraints. ME principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, 18

340

statistical mechanics and logical inference in particular. For the non-additive case, we can make use of the maximum Tsallis entropy (MTE). On the other hand, Tsallis distributions (the ones derived from MTE) are of great interest in many branches of science since they are similar to generalized Pareto distributions. For more properties of MTE, one can refer to Dober and Bolle [39],

345

Dukkiputi et al. [40] and Karmeshu and Sharma [41]. Asadi et al. [42] have proposed a general approach for formulating constraints that define a set of distributions ΩF in which a given distribution with a monotone pdf is the maximum dynamic entropy (MDE) model. The constraints are formulated in terms of differential inequalities describing the evolution of the

350

hazard function. Motivated by this approach. we identify classes of distributions that have maximum DWCRTE. The maximum DWCRTE is the distribution with survival function F¯ ∗ (t) in set of distributions ΩF = F (t) such that for all t ≥ 0

355

ηαw (F ; t) ≤ ηαw (F ∗ ; t), F ∈ ΩF .

In general, for any distribution F (t), the relative growth of the hazard function is given by

′

′

λF (t) f (t) = + λF (t). λF (t) f (t)

(4.5)

For any survival function F¯ (t), we can identify a set of distributions ΩF = F (t) so that

′

′

λF (t) λ ∗ (t) ≤ F , λF (t) λF ∗ (t)

(4.6)

where the right-hand side is given by (4.5). Solution of the differential inequality 360

(4.6) with appropriate initial conditions gives λF ∗ (t) which is dominated by λF (t) for all distributions in ΩF . Thus, by Theorem 4.2, ηαw (F ∗ ; t) dominates ηαw (F ; t). This is a general procedure that can be used to identify a set of distributions in which a given F¯ ∗ (t) is the maximum DWCRTE model. Table 2 displays the constraints defining the maximal sets of distributions ΩF in which

365

some well-known distributions are maximum DWCRTE, for α > 1. For 0 <

19

α < 1 the inequalities are reversed. Abbasnejad et al. [43] present a similar table based on the maximum dynamic survival entropy of order α. Table 2:

Maximum dynamic weighted cumulative residual Tsallis entropy distributions Maximum DWCRTE density with support [0, ∞]

Constraints for maximal set

Exponential f ∗ (t) = θe−θt ,

   λF (0) = θ  

θ > 0

    λ′ (t) ≤ 0 F

Weibull β f ∗ (t) = θβtβ−1 e−θt ,

   λF (1) = θβ    

θ > 0, 0 < β < 1

     

Pareto (Type II)

f ∗ (t) = θβ θ (t + β)−θ−1 ,

  λF (0) = θ   β   

θ, β > 0

     

Generalized Pareto 1 β+1 β − β −2 f ∗ (t) = (1 + t) , θ θ

θ, β > 0

     

     

Half Cauchy

     

Half logistic

′ λF (t) ≤ (1 − 2t arctan(t−1 ))λF (t) λF (t)

 β  λF (0) =   1+θ   

θ, β > 0

     

Half normal

f ∗ (t) =

′ λF (t) β ≤ θ λF (t)

   λF (0) = 2  π   

2 π(1+t2 )

√

′ λF (t) βλF (t) ≤ − β+1 λF (t)

   λF (0) = θ    

β β 2 t − t f ∗ (t) = θe θ exp{ θ (1 − e θ )}, 0 < β ≤ θ 2 β

2 β(θ+1)eβt f ∗ (t) = , θ+eβ t

′ λF (t) λ (t) ≤ F θ λF (t)

 β+1   λF (0) =  θ   

Extreme value

f ∗ (t) =

′ λF (t) β−1 ≤ t λF (t)

′ λF (t) ≤ β − λF (t) ≤ 0 λF (t)

 √ 2  λF (0) =   π   

−t2 2e 2 π

     

′ λF (t) ≤ λF ∗ (t) − t λF (t)

Remark 4.3. Recently, Mirali et al. [25] have proposed a general class of distributions that have maximum WCRE under some constraints and have charactrized 20

370

some well-known distributions. 5. Charactrization Based on DWCRTE The Rayleigh distribution is a special case of the Weibull distribution with a scale parameter of 2 and a suitable model for life-testing studies. The square of a Rayleigh rv with a shape parameter σ = 1 is equal to a chi square rv

375

with 2 degrees of freedom. Also, the square root of an exponential rv has the Rayleigh distribution. Hence, by applying this transformation to the data, all Rayleighity tests can be utilized for the purpose of testing the goodnessof-fit to the exponential distribution. Also, the Rayleigh distribution is widely used in the physical sciences to model wind speed, wave heights and sound/light

380

radiation and has been used in medical imaging science, to model noise variance in magnetic resonance imaging. For more details on Rayleigh distribution refer to Johnson et al. [44]. The origin and other aspects of this distribution can be found in Siddiqui [45], Miller and Sackrowttz [46] and Jahanshahi et al. [47]. This model has the sf given by F¯ (t) = exp

385

(

) −t2 . 2σ 2

(5.1)

In this section, we show that DWCRTE can determine the survival function. Moreover, we show that the Rayleigh distribution can be characterized through the relationship between their WDCRTE and the weighted mean residual lifetime (WMRL). Theorem 5.1. Let X and Y be two non-negative absolutely continuous rvs

390

¯ with sfs F¯ (t) and G(t) and the hazard functions λF (t) and λG (t), respectively. ¯ If ηαw (X; t) = ηαw (Y ; t), then F¯ (t) = G(t). d w Proof. Assumption ηαw (X; t) = ηαw (Y ; t) implies that (α − 1) dt ηα (X; t) = (α − d w ηα (Y ; t). From (4.2), the last quality gives 1) dt

t − αλF (t)[1 − (α − 1)ηαw (X; t)] = t − αλG (t)[1 − (α − 1)ηαw (Y ; t)], ¯ which reduces to λF (t) = λG (t) or equivalently F¯ (t) = G(t). This completes 395

the proof. 21

In the following theorem we show that the Rayleigh distribution can be characterized in terms of DWCRTE. Theorem 5.2. Let X be an absolutely continuous rv. Then the relation ηαw (X; t) = k, where k is a constant and holds if and only if X has the Rayleigh distribution. 400

Proof. The if part of the theorem can be easily obtained using (3.2). For the only if part, let the DWCRTE of X be a constant. Then

d w dt ηα (X; t)

= 0 or

equivalently t − αλF (t)[1 − (α − 1)ηαw (X; t)] = 0, where obtained by (4.2) and implies that λF (t) =

t σ2 .

This is the hazard rate

of Rayleigh distribution. Hence the proof is completed. 405

In the following theorem, we characterize Rayleigh distribution using a relationship between DWCRTE and WMRL. Theorem 5.3. Let X be an absolutely continuous rv. Then the relation 1 − (α − 1)ηαw (X; t) =

m∗F (t) , α

(5.2)

holds if and only if X has the survival function given in (5.1). Proof. If X has a Rayleigh distribution with survival function given in (5.1), 410

it can be easily shown that (5.2) holds with m∗F (t) = σ 2 . Conversely, let (5.2) hold. Then using (3.2) and (2.3), we have ∫ ∞ ∫ αF¯ (t) xF¯ α (x)dx = F¯ α (t) t

∞

xF¯ (x)dx.

t

Differentiating both sides with respect to t, we get ∫ ∞ ∫ −αf (t) xF¯ α (x)dx−αtF¯ α (t)F¯ (t) = −αf (t)F¯ α−1 (t) t

∞

xF¯ α (x)dx−tF¯ (t)F¯ α (t).

t

The above equation simplifies to λF (t)m∗F (t) = t. On the other hand, differentiating (2.3) with respect to t, we get d ∗ m (t)F¯ (t) − f (t)m∗F (t) = −tF¯ (t). dt F 22

(5.3)

415

From (5.3), the above equation simplifies to

d ∗ dt mF (t)

= 0 or equivalently m∗F (t) =

t k, where k is a constant.(Using ) again (5.3), this fact implies that λF (t) = k or 2 equivalently F¯ (t) = exp −tk , which is the sf of Rayleigh distribution and the

result follows.

6. Estimator of WCRTE 420

In this section we consider the problem of estimating the WCRTE. Let X1 , X2 , . . . , Xn be non-negative, absolutely continuous iid random variables that constitute a random sample from a population having distribution function F (x). According to (2.1), we define the empirical WCRTE as [ ] ∫ ∞ 1 w ¯ α ¯ ηα (Fn ) = 1− xFn (x)dx , α ̸= 1, α > 0. α−1 0 where

n

Fn (t) = 425

(6.1)

1∑ I (Xi ≤ t), n i=1

t ∈ R,

is the empirical distribution of the sample and X(1) < X(2) < · · · < X(n) are

the order statistics and F¯n (t) = 1 − Fn (t). Equation (6.1) can be expressed as ] [ n−1 ∑ ∫ X(i+1) 1 ηαw (F¯n ) = 1− xF¯nα (x)dx . (6.2) α−1 X (i) i=1 From (6.2), we get ηαw (F¯n )

[ ( )α ] n−1 ∑ 1 i 1− U(i+1) 1 − , = α−1 n i=1

(6.3)

where U(1) = X(1) ,

2 2 U(i) = X(i) − X(i−1) ,

i = 2, . . . , n.

The following theorem deals with a central limit theorem for the empirical 430

WCRTE for random samples from the Weibull distribution with pdf 2

f (t) = 2λte−λt ,

t > 0, λ > 0.

(6.4)

We point out that the following asymptotic result is analogous to Theorem 7.1 of Di Crescenzo and Longobardi [48]. 23

Theorem 6.1. Let X1 , · · · , Xn be a random sample from Weibull distribution with pdf (6.4), then Zn = 435

ηαw (X) − E [ηαw (X)] √ , V ar [ηαw (X)]

converges in distribution to a standard normal variable, as n → ∞. Proof. First, it should be mentioned that Y = X 2 has an exponential distribution with parameter λ. Thus, (6.3) can be written as [ ] n−1 ∑ 1 1− ηαw (F¯n ) = Yi , α−1 i=1

)α ( where Yi = U(i+1) 1 − ni , i = 1, 2, . . . , n − 1 is independent rvs. The U(i)

are independent and exponentially distributed with parameter λ(n − i). Thus, 440

mean and variance of Yi are )α ( )α ( i 1 i EU(i+1) = 1 − , EYi = 1− n n λ(n − i) ( )2α ( )2α i i 1 V ar(Yi ) = 1− V ar(U(i+1) ) = 1 − 2. n n λ2 (n − i) In this regard, from (6.3) we obtain the mean and variance of the WCRTE as follows:

[ ] n−1 α 1 1 ∑ (1 − i/n) E = 1− , α−1 λ i=1 n−i ( )2 n−1 ∑ (1 − i/n)2α 1 V ar [ηαw (X)] = 2 . λ(α − 1) (n − i) i=1 [ηαw (X)]

[ ] 3 3 Since E |Zi − E(Zi )| = 2e−1 (6 − e)[E(Zi )] for any exponentially dis-

tributed random variable Zi , by setting

k

αi,k = E|Zi − E(Zi )| , 445

the following approximations hold for large n, )2 n n ( ∑ c2 1 ∑ n−i ≈ 2 , αi,2 = 2 2 log 4λ n n 4λ n i=1 i=1 n ∑ i=1

αi,3 = −

)3 n ( (6 − e)c3 2(6 − e) ∑ n−i ≈− , log 8eλ3 n3 i=1 n 4eλ3 n2 24

where ck :=

∫

1 0

Since,

  2 k=2 k (log(1 − x)) dx =  −6 k = 3. 1/3

(α1,3 + · · · + αn,3 )

1/2

(α1,2 + · · · + αn,2 )

≈ n−1/6 → 0 as n → ∞,

therefore, Lyapunovs condition of the central limit theorem is satisfied (see Gut 450

[49]), which completes the proof.

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