Extended cumulative entropy based on kth lower record values for the coherent systems lifetime

Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016) first proposed generalized cumulative entropy based on lower record values. Motivated by Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016), recently, Tahmasebi and Eskandarzadeh (Stat. Probab. Lett. 126:164–172, 2017) proposed an extended cumulative entropy (ECE) based on kth lower record values. In this paper, we obtain some properties of ECE. We study this measure of information for the coherent systems lifetime with identically distributed components. We define the conditional ECE for the system lifetime and discuss some properties of it. We also use this idea to propose a measure of extended cumulative past inaccuracy. Finally, we propose the estimators of these measures using empirical approach. In addition, we study large sample properties of these estimators.


Introduction
Information measures have a fundamental role in various areas of science such as probability and statistics, financial analysis, engineering, and information theory; see, e.g., Cover and Thomas [7]. One of the most important measures of uncertainty in probability and statistics is the entropy of a random phenomenon. Let X denote the random lifetime of a system or a component with probability density function (pdf ) f and a survival function F = 1 -F, respectively. Shannon [29] introduced a measure of uncertainty associated with X as follows: residual entropy (CRE) given by where Λ(x) = -logF(x). An information measure similar to E(X) is the cumulative entropy (CE) defined as follows (see Di Crescenzo and Longobardi [8]): whereΛ(x) = -log F(x). Note that CE(X) ≥ 0 and that CE(X) = 0 if and only if X = c. The CE can be seen as a dispersion measure (see Toomaj et al. [37]). More properties on CE in past lifetime are available in Di Crescenzo and Longobardi [8] and Navarro et al. [20].
Recently Di Crescenzo and Toomaj [10] discussed some properties of a new weighted distribution based on stochastic orders and introduced the reversed relevation transform in connection with CE function. Some new connections of the CRE and the residual lifetime are given by Kapodistria and Psarrakos [14] using the relevation transform. Psarrakos and Navarro [23] generalized the concept of CRE relating this concept with the mean time between record values and with the concept of relevation transform, and they also considered a dynamic version of this new measure. Sordo and Psarrakos [30] provided comparison results for the cumulative residual entropy of systems and their dynamic versions. Toomaj et al. [37] used the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Kayal [15] proposed generalized cumulative entropy based on lower record values and obtained various results of it. Cali et al. [6] studied the generalized cumulative past information in coherent systems.
Let {X n , n ≥ 1} be a sequence of independent and identically distributed random variables with cumulative distribution function (cdf ) F and pdf f . An observation X j will be called a lower record value if its value is less than the values of all previous observations. Thus, X j is a lower record value if X j < X i for every i < j. For a fixed positive integer k, Dziubdziela and Kopocinski [12] defined the sequence {L n(k) , n ≥ 1} of kth lower record times for the sequence {X n , n ≥ 1} as follows: L n+1(k) = min{j > L n(k) : X k:L n(k) +k-1 > X k:k+j-1 }, where X j:m denotes the jth order statistic in a sample of size m. Then {X n(k) := X k:L n(k) +k-1 } is called a sequence of kth lower record values of {X n , n ≥ 1}. The pdf of X n(k) is given by Dziubdziela and Kopocinski [12] as follows: 3) The cdf of Equation (1.3) can be obtained as Now, let X be a nonnegative absolutely continuous random variable with cdf F. Then, Tahmasebi and Eskandarzadeh [31] defined further extension of CE of X as follows: du, for n = 1, 2, . . . , k ≥ 1, (1.5) where ϕ n,k (u) = u k (-log u) n ≥ 0, 0 < u < 1. Note that ϕ n,k (0) = ϕ n,k (1) = 0. For k = 1, Equation (1.5) reduces to the generalized cumulative entropy due to Kayal [15]. Equation (1.5) is a new CE which is presented on the idea of GCRE introduced by Psarrakos and Navarro [23] and is obtained relating the concept of CE with the mean time between lower k-record values and with the concept of relevation transform (see Krakowski [18] and Baxter [3]). They called it extended cumulative entropy (ECE). In reliability theory, the performance characteristics of the coherent systems are of great importance. Accordingly, this paper is organized as follows. In Sect. 2, we present general properties of ECE including stochastic ordering, linear transformations, and bounds. In Sect. 3, we study CE n,k (T) when T is the lifetime of a coherent system with identically distributed components. In Sect. 4, we also obtain some results on the conditional ECE of a system lifetime. In Sect. 5, we propose an extended cumulative past inaccuracy (ECPI) measure and study a measure of distance symmetric in coherent and mixed systems. Finally, in Sect. 6, the estimators of ECE and ECPI using empirical approach are presented. Throughout this paper, the terms 'increasing' and 'decreasing' are used in a nonstrict sense.

General properties on the ECE
In this section, we study some general results of ECE. For that we first present the following example.
Example 2.1 Let X denote the lifetime of a system or a unit.
i. If X has the Fréchet distribution with F(x) = e -θ x , x > 0, θ > 0, then for n > 1 we have CE n,k (X) = k 2 θ n(n-1) = k 2 CE n,1 (X). ii. If X has a uniform distribution in (0, b), then we have iii. If X has an inverse Weibull distribution with the cdf where Γ (·) is the complete gamma function.
In the following, we state various results of ECE. It includes basic properties such as stochastic orderings, bounds, the effect of linear transformations, and a two-dimensional Proposition 2.2 Let X and Y be two nonnegative random variables with pdfs f and g, , then X ≤ disp Y due to Ahmed et al. [1]. Therefore, from Theorem (2.1) the desired result follows.

Proposition 2.3 Let X be a nonnegative random variable with decreasing pdf f such that
Proof The nonnegative random variable X has a decreasing pdf if and only if U ≤ c X, where U ∼ Uniform(0, 1) (see Shaked and Shanthikumar [28]). Hence, from Proposition 2.2 the desired result follows.

Proposition 2.4
Suppose that X and Y are two independent nonnegative random variables. If X and Y have log-concave densities, then Proof The proof is similar to that of Theorem 3.2 of Di Crescenzo and Toomaj [11].

Proposition 2.5 Let X be a random variable with cdf F. Further, let Y
Proposition 2.6 Let X be a nonnegative random variable with cdf F and the reversed hazard rate r(z), z > 0. Then, for any k ≥ 1 and n = 1, 2, . . . , we have Remark 2.1 Let X be a nonnegative random variable with cdf F, then we have where CE n,1 (X) is the generalized cumulative entropy (see Kayal [15]).
Remark 2.2 Let X be a nonnegative absolutely continuous random variable. Then Let X and Y denote the lifetimes of two components of a system with joint distribution function F(x, y), respectively. Then the bivariate ECE is defined as follows: whereΛ(x, y) = -log F(x, y). Using the binomial expansion in (2.1), we obtain the following proposition.

Proposition 2.7
Suppose that the nonnegative random variables X and Y are independent with joint distribution function F(x, y), then Proposition 2.8 Let X be a symmetric random variable with respect to the finite mean ] n dx is the generalized cumulative residual entropy (see Tahmasebi et al. [32]).
The concept of elasticity in life expectancy is an important feature in life tables. It should be noted that V 1 (X) = E 1,1 (X) E(X) is the elasticity in life expectancy with respect to proportional hazards models with survival functionF k (x) = [F(x)] k (see Leser [19] and Rao [25]). Recently, by using E n-1,k (X), Psarrakos and Toomaj [24] obtained the following approximation: where V n (X) = n E n,1 (X) E n-1,1 (X)n + 1 is the elasticity of expected interepoch intervals in a nonhomogeneous Poisson process (NHPP) with respect to a proportional hazards models.
Let us now investigate the ECE within the proportional reversed hazards model (PRHM). We recall that two random variables X and X * θ satisfy the PRHM if their distribution functions are related by the following identity, for θ > 0: For instance, for some properties of such a model associated with aging notions and the reversed relevation transform, see Gupta and Gupta [13] and Di Crescenzo and Toomaj [10], respectively. In this case, we assume that X and X * θ are nonnegative absolutely continuous random variables. Due to Equation (2.1) and noting that Λ * θ (x) = θ Λ(x), we obtain the ECE measure for X * θ as follows, for θ > 0: Proposition 2.9 Let X and X * θ be nonnegative absolutely continuous random variables satisfying the PRHM as specified in ] k for all x ≥ 0, and then the thesis immediately follows from (1.5).

ECE of coherent systems
A system is said to be coherent if it does not have any irrelevant components and its structure function is monotone. Now, let the component lifetimes have the common distribution F X . Suppose that T is the lifetime of a coherent system with m identically distributed (id) components, then its distribution function F T can be written as where q : [0, 1] → [0, 1] is a distortion function that depends on the structure of a system and the copula of the component lifetime. Note that the function q is a continuous increasing function such that q(0) = 0 and q(1) = 1 (for more details on coherent systems, see Burkschat and Navarro [5] and Navarro et al. [21]). A special case of coherent systems is the k-out-of-n system, where the system fails when the kth component failure occurs.
For example, for a 2-out-of-3 system with i.i.d. components, we have q(u) = 3u 2 -2u 3 . Also, for a parallel system with lifetime T = max(X 1 , X 2 , X 3 , . . . , X m ), we have q(u) = u m . Hence, the ECE of the random lifetime T can be obtained as follows: For example, for a parallel system with i.i.d components of U(0, 1), we have As an application of Equation (3.1), we have the following example.
Proposition 3.1 Let T be the lifetime of a coherent system with i.d components and with a distortion function q. If ϕ n,k (q(u)) ≥ (≤)ϕ n,k (u), then we have

Proposition 3.2 Assume that the components have cdf F X and pdf f X and support S. Let T be the lifetime of a coherent system with i.d components and with a distortion function q.
( Example 3.2 (i) Let T be the lifetime of a coherent system with i.d components having an exponential distribution with mean θ , then L = 1 θ and (ii) Let T be the lifetime of a coherent system with i.d components having a Pareto type
In Table 1, we give the distortion functions for all the coherent systems with 1-4 i.i.d. components. Also, in Table 2, we give CE n,2 (T) for these systems when the components have a uniform distribution in (0, 1).
It is clear that if D 2,n ≤ 1, then CE n,k (T 2 ) ≤ CE n,k (T 1 ). Now, let us have two coherent systems with i.i.d components. Suppose that T 1 = X 2:2 = max(X 1 , X 2 ) is the lifetime of a 2 Table 2 CE n,2 (T) and bounds for CE n,2 (T) obtained from the coherent systems given in Table 1   component parallel system with q 1 (u) = u 2 and T 2 is the lifetime of a 2-out-of-3 system with q 2 (u) = 3u 2 -2u 3 , then from the previous proposition we obtain CE 2,2 (T 2 ) ≤ 8CE 2,2 (T 1 ).
In the following example, we consider a parallel system with dependent and identically distributed (d.i.d) components and obtain the bounds of CE n,k (T).
Example 3.3 Let T = max(X 1 , X 2 ) be the lifetime of a parallel system with d.i.d components. If the component lifetimes are dependent with the FGM copula as 2 ]. So, from Proposition 3.3, we obtain An application of (3.1) is the comparison of the ECE of two coherent systems when two systems have the same structure with different i.d. component lifetimes. Thus we have the following theorem.

Proposition 3.5 Let T 1 and T 2 be the lifetimes of two coherent systems with the same structure and with i.d. components having common distributions F and G, respectively. If
X ≤ disp Y , then for any k ≥ 1 and n = 1, 2, . . . , we have CE n,k (T 1 ) ≤ CE n,k (T 2 ).
Proof Since both systems have a common distortion function q and the same structure, the proof follows from Equation (3.1) and the assumption on the dispersive order.

Corollary 3.1
Under the assumptions of Proposition 3.5, if X ≤ hr Y and X or Y is DFR, then CE n,k (T 1 ) ≤ CE n,k (T 2 ).

Corollary 3.2 Under the assumptions of Proposition
then CE n,k (T 1 ) ≤ CE n,k (T 2 ).
Proof Since CE n,k (X) ≤ CE n,k (Y ), we have from (1.5) that So, the proof is completed.
Remark 3.1 Under the assumptions of Theorem 3.1, if q is strictly increasing in (0, 1), then Proof The proof follows from Theorem 2.9 of Navarro et al. [21].

Conditional ECE
Suppose that X is the lifetime of a system on a probability space (Ω, F, P) such that E|X| < ∞. We denote by E(X|G) the conditional expectation of X given sub σ -field G, where G ⊂ F . Here, we define the conditional ECE of X and discuss some properties of it. Definition 4.1 Let X be the lifetime of a system with cdf F X . Then, for a given σ -field F , the conditional ECE is defined as follows: The following proposition says that the conditional ECE has the "super-martingale property". Proposition 4.1 Let X ∈ L p for some p > 2, then for σ -fields G ⊂ F E CE n,k (X|F )|G ≤ CE n,k (X|G). (4.1) Proof The proof follows by applying Jensen"s inequality for the convex function t k (-log t) n , 0 < t < 1 as follows: From the Markov property for the lifetime random variables T 1 , T 2 , and T 3 , we have the following lemma.

Lemma 4.2 If T 1 → T 2 → T 3 is a Markov chain, then
Proof (i) By using the Markov property and the definition of CE n,k (T 3 |T 2 , T 1 ), the result follows.
Theorem 4.1 Let X ∈ L p for some p > 2 be the lifetime of a system and F be a σ -field. Then E(CE n,k (X|F )) = 0 iff X is F -measurable.
Proof Suppose that E(CE n,k (X|F )) = 0, then CE n,k (X|F ) = 0. Now using the definition of CE n,k (X|F ), we conclude that E(I (X≤x) |F ) = 0 or 1. Hence, using relation (24) of Rao et al. [26], T is F -measurable. Supposing that T is F -measurable, again using relation (24) of Rao et al. [26], we have P(X ≤ x|F ) = 0 or 1 for almost all x ∈ R + , so the result follows.

Extended cumulative past inaccuracy measure
Let X and Y be two nonnegative random variables with distribution functions F(x), G(x), respectively. If F(x) is the actual cdf corresponding to the observations and G(x) is the cdf assigned by the experimenter, then the cumulative past inaccuracy measure between F(x) and G(x) is defined by Thapliyal and Taneja [33] as follows: When G(x) = F(x), then (5.1) becomes cumulative entropy which is studied by Di Crescenzo and Longobardi [8].
In analogy with the measure defined in Equation (5.1), we now introduce the extended cumulative past inaccuracy (ECPI) defined as As applications of Equation (5.2), we have the following properties.
Therefore, the stated results follow.
Proof The proof is similar to that of Proposition 7 of Cali et al. [6].

Proposition 5.2 Let X and Y be nonnegative random variables with distribution functions
F and G, respectively. If X ≤ dcx Y , then for n, k ≥ 1 we have Proof Since CE n,k (X) and I n,k (G, F) can be expressed as mean value ofh n,k (·), the proof follows by noting thath n,k (·) is a decreasing convex function. Park et al. [22] have recently suggested an extension of KL information in terms of the distribution function, which can be called cumulative Kullback-Leibler information (CKL), as follows: This measure of information is also studied in Di Crescenzo and Longobardi [9]. In the following, we define a symmetric version of the CKL(X, Y ).

Definition 5.1
Let X and Y be nonnegative random variables with distribution functions F and G, respectively. Then the symmetric CKL is defined as follows: Note that SCKL(X, Y ) ≥ 0 and symmetric. It is noted that when we are comparing systems pairwise, then we can find a system in which its distribution is closer to the distribution of the parallel system or the series system. Thus, in the following we will propose a measure of distance symmetric to the mixed system. Proof Proof is similar to the proof of Lemma 1 in Toomaj et al. [37] and hence it is omitted.
A mixed system is a stochastic mixture of coherent systems. Hence, any coherent system is a mixed system (Toomaj and Doostparast [36]). Toomaj and Doostparast [35] obtained an expression for the Shannon entropy of mixed r-out-of-n systems when the lifetimes of the components are independent and identically distributed. Toomaj [34] discussed the Renyi entropy of mixed system's lifetime. Kayal [16] studied a generalized entropy of mixed systems whose component lifetimes are independent identically distributed.
If T is the lifetime of any arbitrary mixed system, then it is well known that X 1:n ≤ st T ≤ st X n:n (see Samaniego [27] and Barlow and Proschan [2]). Therefore, we can find a system in which its structure (or distribution) is similar (or closer) to the distribution of the parallel system or the series system.

Proposition 5.3
Let T be the lifetime of a mixed(or coherent) system based on i.i.d. component lifetimes X 1 , X 2 , . . . , X n , then SCKL(T, X i:n ) ≤ SCKL(X 1:n , X n:n ) for i = 1, n.
Note that | DS(T)| ≤ 1. If DS(T) is closer to 1(-1), then the distribution of T is closer to the distribution of the parallel(series) system. In Table 3

Empirical measures of ECE and ECPI
Let X 1 , X 2 , . . . , X m be a random sample of size m from an absolutely continuous cumulative distribution function F(x). If X 1:m ≤ X 2:m ≤ · · · ≤ X m:m represent the order statistics of the sample X 1 , X 2 , . . . , X m , then the empirical measure of F(x) for i = 1, 2, . . . , m -1 is defined as follows: Thus the empirical measure of ECE is obtained as where U i = X i+1:m -X i:m .The following example provides an application of the empirical measure of ECE to real data. Then, from the data set, we compute CE 5,1 (F m ) = 0.1564, CE 5,2 (F m ) = 0.5280, CE 5,3 (F m ) = 0.6701, CE 5,4 (F m ) = 0.8043, and CE 5,5 (F m ) = 0.9892. Figure 1 shows the function CE n,2 (F m ) for n > 1. It decreases in empirical measure of ECE for different values of n ≥ 2. Let us remember that the well-known theorem of Glivenko-Cantelli states that Using this result, the following theorem asserts that CE n,k (F m ) converges almost surely to CE n,k (X). The proof of which follows the same lines as given in Theorem 9 of Rao et al. [26]. Theorem 6.1 Let X be a nonnegative and absolutely continuous random variable with cdf F. Then, for any random X in L p for some p > 2, we have Using (6.2)-(6.6) the result follows. According to Equation (5.2), we define the empirical ECPI as follows: Proof The proof is similar to that of Theorem 5 of Calì et al. [6].

Conclusions
In this paper, we have obtained various properties of ECE. This concept of cumulative entropy can be applied in measuring the uncertainty contained in the associated past lifetime. We studied this measure of uncertainty for the coherent systems lifetime with identically distributed components. We also discussed the conditional ECE of a system lifetime. Moreover, we proposed a measure of ECPI and its empirical version. We studied a measure of distance symmetric in coherent and mixed systems. Finally, we proposed estimators of these measures by using empirical approach and studied numerical results of ECE in lifetime data. Further, it has been shown that the empirical measure of ECE and ECPI converges to normal distribution, when a random sample is taken from continuous distribution.