Fractional generalized cumulative entropy and its dynamic version
FFractional generalized cumulative entropy and its dynamic version
Antonio Di Crescenzo ∗ Suchandan Kayal † Alessandra Meoli ‡ Abstract
Following the theory of information measures based on the cumulative distribution func-tion, we propose the fractional generalized cumulative entropy, and its dynamic version.These entropies are particularly suitable to deal with distributions satisfying the propor-tional reversed hazard model. We study the connection with fractional integrals, and somebounds and comparisons based on stochastic orderings, that allow to show that the proposedmeasure is actually a variability measure. The investigation also involves various notions ofreliability theory, since the considered dynamic measure is a suitable extension of the meaninactivity time. We also introduce the empirical generalized fractional cumulative entropyas a non-parametric estimator of the new measure. It is shown that the empirical measureconverges to the proposed notion almost surely. A central limit theorem is also establishedunder the exponential distribution. The stability of the empirical measure is addressed, too.An example of application to real data is finally provided.
Keywords:
Cumulative entropy, fractional calculus, stochastic orderings, estimation.
Let X be a discrete random variable taking values in { x i ; i = 1 , . . . , n } and having probabilitymass function p i = P ( X = x i ), where 0 < p i < i = 1 , . . . , n. The entropy of X is given by(see Shannon [1]) S ( X ) = − n (cid:88) i =1 p i ln p i , (1)where ‘ln’ denotes natural logarithm. It is well known that the entropy (1) quantifies the uncer-tainty contained in the probability distribution associated to X , and is particularly importantin coding theory (see Cover and Thomas [2] for specific details). In 2009, Ubriaco [3] extendedthe notion of entropy to the following version based on fractional calculus: S α ( X ) = n (cid:88) i =1 p i ( − ln p i ) α , ≤ α ≤ . ∗ Dipartimento di Matematica, Universit`a degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano(SA), Italy, Email: [email protected] † Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India, Email:[email protected], [email protected] ‡ Dipartimento di Matematica, Universit`a degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano(SA), Italy, Email: [email protected] a r X i v : . [ m a t h . P R ] F e b learly, for α = 1 the fractional entropy S α ( X ) reduces to the classical entropy (1). The authorestablished that the fractional entropy is stable in the sense of Lesche and thermodynamicstability criteria. Moreover, the fractional entropy is nonadditive, positive and concave in nature.Machado [4] showed that for the description of a complex system, the fractional entropy is quiteappealing since it allows high sensitivity to the signal evolution. Recently, motivated by thefractional entropy, Machado and Lopes [5] proposed the similar measure named fractional Renyientropy and discussed various properties.There have been various developments of uncertainty measures in continuous domain too.The continuous analogue of (1) is known as the differential entropy. For a nonnegative absolutelycontinuous random variable X with probability density function (PDF) f ( x ), the differentialentropy is H ( X ) = − (cid:90) ∞ f ( x ) ln f ( x ) dx. (2)However, differently from the classical entropy that takes nonnegative values in the case ofdiscrete random variables, the differential entropy may assume negative values. For example, forthe random variable uniformly distributed in the interval [0 , a ], the differential entropy is equalto ln a , and then is negative for 0 < a <
1. In order to avoid this fact, various different measureshave been proposed in the recent past. Indeed, Rao et al. [6] introduced a measure of uncertaintysimilar to H ( X ), for which the PDF is replaced by the survival function ¯ F ( x ) = P ( X > x ) inthe right-hand-side of (2). This is known as the cumulative residual entropy; for a nonnegativerandom variable it is defined as
CRE ( X ) = − (cid:90) ∞ ¯ F ( x ) ln ¯ F ( x ) dx (3)and assumes nonnegative values. Along this line Di Crescenzo and Longobardi [7] proposed andstudied a similar measure, named cumulative entropy. This is defined in terms of the cumulativedistribution function (CDF) F ( x ) = P ( X ≤ x ), i.e. (see also Navarro et al. [8]) CE ( X ) = − (cid:90) l F ( x ) ln F ( x ) dx, (4)where (0 , l ) is the support of X . The corresponding dynamic measure for the past lifetime isbased on the conditional distribution function P ( X ≤ x | X ≤ t ) = F ( x ) F ( t ) , 0 ≤ x ≤ t , and isnamed cumulative past entropy: CE ( X ; t ) = − (cid:90) t F ( x ) F ( t ) ln F ( x ) F ( t ) dx, (5)for t ∈ (0 , l ). Recently, stimulated by the purpose of constructing a fractional version of CRE ( X ), in analogy with S α ( X ) the following measure has been introduced in Xiong et al. [9]: E α ( X ) = (cid:90) ∞ ¯ F ( x )[ − ln ¯ F ( x )] α dx, ≤ α ≤ . This is called fractional cumulative residual entropy of X . Among the results on this measurepresented in [9] we mention the asymptotics of its empirical version and suitable applications tofinancial data. 2e remark that a better correspondence with other useful measures can be obtained byincluding a further term. Namely, for any nonnegative random variable X one can consider the fractional generalized cumulative residual entropy , defined as CRE α ( X ) := 1Γ( α + 1) E α ( X ) = 1Γ( α + 1) (cid:90) ∞ ¯ F ( x )[ − ln ¯ F ( x )] α dx, α ≥ . (6)If α is a positive integer, say α = n ∈ N , then CRE n ( X ) identifies with the generalized cumula-tive residual entropy, that has been introduced by Psarrakos and Navarro [10]. We recall that,if α = n ∈ N , then CRE n ( X ) is a dispersion measure that is strictly related to the (upper)record values of a sequence of independent and identically distributed random variables. It isalso related to the relevation transform and to the interepoch intervals of a non-homogeneousPoisson process (see, for instance, Toomaj and Di Crescenzo [11] and references therein for somerecent results on this measure).Along the lines of the above mentioned researches, in this paper we propose a fractionalversion of the generalized cumulative entropy. The new measure is defined similarly as in (6), byreplacing the survival function with the CDF of X . Fractional versions of various informationmeasures have been proposed in the recent years. Indeed, more advanced mathematical toolsare suitable to handle complex systems and anomalous dynamics. Various characteristics offractional calculus allow the related measures to better capture long-range phenomena andnonlocal dependence in certain random systems. For instance, we recall the recent papers byZhang and Shang [12] and Wang and Shang [13], finalized to study new fractional modificationson the discrete version of the cumulative residual entropy (3), which are useful to analyze timeseries and have been applied in the context of data from the stock market. Further applicationsof multiscale fractional measures to the analysis of time series has been successfully exploitedin Dong and Zhang [14]. Different information measures based on fractional calculus havebeen investigated in Yu et al. [15], where the fractional entropy and other related notions havebeen obtained by replacing the Riemann integral with the Riemann-Liouville integral operator,leading to new tools of interest in image analysis.It is worth mentioning that the fractional measure proposed in this paper is particularlysuitable to be adopted in the context of the proportional reversed hazard model, as alreadyseen for various information measures derived from the cumulative entropy (4). Moreover, itexhibits a nice connection with the Riemann-Liouville fractional integral with respect to anotherfunction.The rest of the paper is organized as follows. In Section 2, we discuss some properties of thefractional generalized cumulative entropy and provide some examples from typical distributionsof interest. In particular, we analyze the proposed measure under the proportional reversedhazard model. We also point out the above mentioned connection with fractional integrals ofthe Riemann-Liouville type. Section 3 is devoted to various bounds for the proposed measures.Some comparisons are also studied by means of suitable stochastic orderings. This allows to showthat the fractional generalized cumulative entropy is actually a variability measure. In Section4, we propose the dynamic version of the considered measure. We provide some examplessatisfying the proportional reversed hazard model and arising from the analysis of first-hittingtime distributions in customary stochastic processes. In Section 5, we propose a non-parametricestimator of the new measure. We discuss its statistical characteristics, with special care on theasymptotic properties. Such properties allow the empirical measure to be successfully adoptedto describe the information content in experimental data, such as for time-series and in signalanalysis. Accordingly, the section concludes with an example of application to a real dataset.3inally, some final remarks complete the paper in Section 6.Throughout the paper, N denotes the set of positive integers, and N = N ∪ { } . Let X be a nonnegative random variable with support (0 , l ) and CDF F . Then, in analogy withthe measure given in (6), the fractional generalized cumulative entropy of X is defined by CE α ( X ) = 1Γ( α + 1) (cid:90) l F ( x )[ − ln F ( x )] α dx, α > , (7)provided that the integral in the right-hand-side is finite. Clearly, for α = 1, the measure CE α ( X ) reduces to the cumulative entropy given in (4). From (7) it is not hard to see thatlim α → + CE α ( X ) = (cid:26) l − E ( X ) , < l < + ∞ + ∞ , l = + ∞ . The fractional generalized cumulative entropy is nonnegative and nonadditive. If 0 < α < CE α ( X ) ≥
0; moreover one has CE α ( X ) = 0 if and only if X is degenerate. We recall that if X is absolutely continuous, then the function in the square brackets in the right-hand side of Eq.(7) corresponds to the cumulative reversed hazard rate function of X , i.e. T ( x ) = − ln F ( x ) = (cid:90) lx τ ( x ) dx, < x < l, where τ ( x ) = f ( x ) F ( x ) is the reversed hazard rate of X and f ( x ) is the PDF of X .The fractional generalized cumulative entropy is provided in Table 1 for some distributions,where Γ denotes the complete gamma function, and where E α ( β ) = (cid:90) ∞ e − βt t − α dt (8)is the exponential integral function. For these cases, CE α ( X ) is decreasing in α and tends to 0as α → ∞ .When α is a positive integer, say n ∈ N , then CE n ( X ) identifies with the generalizedcumulative entropy, defined and studied by Kayal [16]. In this case, CE n ( X ) is strictly relatedto the lower records of a sequence of i.i.d. random variables, and to the recursive reversedrelevation transform (see also Di Crescenzo and Toomaj [17]). Remark 2.1
It is not hard to see that if X has bounded support (0 , l ) and possesses a symmetricdistribution, such that F ( x ) = ¯ F ( l − x ) for all ≤ x ≤ l , then from Eqs. (6) and (7) one has CE α ( X ) = CRE α ( X ) for all α > .For instance, this is true for the uniform distribution (cf. Case (i) of Table 1 and Example 1 of[9]). F ( x ) CE α ( X ) , α > xl , 0 ≤ x ≤ l l α +1 (ii) power (cid:16) xl (cid:17) b , 0 ≤ x ≤ l , b > l b α ( b + 1) α +1 (iii) Fr´echet e − b x − η , x > b, η > b /η η Γ( α + 1) Γ (cid:16) α − η (cid:17) , < η < α (iv) boundedFr´echet exp (cid:8) b (1 − lx ) (cid:9) , 0 < x ≤ l , b > l bα (cid:16) e b ( α + b ) E α ( b ) − (cid:17) Table 1: Fractional generalized cumulative entropy for some distributions.We point out that, even though the cases of interest usually deal with absolutely continuousrandom variables, the fractional generalized cumulative entropy can also refer to discrete randomvariables. For instance, if U is uniformly distributed on { , , . . . , n } then from (7) we have CE α ( U ) = 1Γ( α + 1) n − (cid:88) k =1 (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α , α > . (9)Various information measures already known are expressed as the expectation of a givenfunction of the random variable of interest. For instance, we recall that for the cumulativeresidual entropy (3) one has (cf. Theorem 2.1 of Asadi and Zohrevand [18]) CRE ( X ) = E [mrl( X )] , where, for all t ≥ F ( t ) > t ) := E [ X − t | X > t ] = 1¯ F ( t ) (cid:90) ∞ t ¯ F ( x ) dx (10)is the mean residual life of a nonnegative lifetime X . Similarly, for the cumulative entropy (4)we have (see Theorem 3.1 of [7]) CE ( X ) = E [˜ µ ( X )] , where, for all t ≥ F ( t ) > µ ( t ) := E [ t − X | X ≤ t ] = 1 F ( t ) (cid:90) t F ( x ) dx (11)is the mean inactivity time of X , and deserves interest in reliability theory. Hereafter westate s similar result for the fractional generalized cumulative entropy, by expressing it as theexpectation of a decreasing function of X , for α fixed. Proposition 2.1
Let X be a nonnegative random variable with support (0 , l ) and CDF F . If ξ α ( x ) := 1Γ( α + 1) (cid:90) lx [ − ln F ( t )] α dt < ∞ , α > , (12) then CE α ( X ) = E [ ξ α ( X )] , α > . (13)5 roof. From (7), by Fubini’s theorem and (12) we obtain CE α ( X ) = 1Γ( α + 1) (cid:90) l [ − ln F ( t )] α dt (cid:90) t dF ( x )= 1Γ( α + 1) (cid:90) l (cid:18)(cid:90) lx [ − ln F ( t )] α dt (cid:19) dF ( x ) = (cid:90) l ξ α ( x ) dF ( x ) , so that the relation (13) holds.We remark that for α = 1, the result given in Proposition 2.1 corresponds to Proposition 3.1of [7].We recall that the measure defined in (6) is shift-independent. That is, for the affine trans-formation Y = cX + b , c > b ≥
0, one has
CRE α ( Y ) = CRE α ( cX + b ) = c CRE α ( X ) for all α > F Y ( x ) = F X ( x − bc ), x ∈ R , andthus is omitted. Proposition 2.2
Let Y = cX + b , where c > and b ≥ . Then, CE α ( Y ) = CE α ( cX + b ) = c CE α ( X ) for all α > . Note that the result of Proposition 2.2 can be extended to the case of random variablestaking values on R , after suitable extension of the definition of the measure CE α ( X ).In analogy with the normalized cumulative entropy proposed in [7], it is possible to define anormalized version of CE α ( X ). Let us assume that the cumulative entropy (4) is finite and non-zero. Then, the normalized fractional generalized cumulative entropy of a nonnegative randomvariable X with support (0 , l ) is defined as N CE α ( X ) = CE α ( X )( CE ( X )) α = 1Γ( α + 1) (cid:82) l F ( x )[ − ln F ( x )] α dx (cid:16)(cid:82) l F ( x )[ − ln F ( x )] dx (cid:17) α , α > . (14)Clearly, one has lim α → + N CE α ( X ) = (cid:90) l F ( x ) dx, lim α → N CE α ( X ) = 1 . For instance, Table 2 provides the normalized fractional generalized cumulative entropy for somedistributions. The corresponding plots are given in Figure 1.With reference to the relation (7) we also point out that, if X is absolutely continuous withsupport (0 , l ), CDF F and PDF f , then the fractional generalized cumulative entropy can beexpressed as CE α ( X ) = 1Γ( α + 1) (cid:90) u ( − ln u ) α f ( F − ( u )) du, α > . (15)6istribution N CE α ( X ) , α > (cid:16) l (cid:17) − α (ii) power (cid:16) lb + 1 (cid:17) − α (iii) Fr´echet b (1 − α ) /η Γ (cid:16) α − η (cid:17) η − α Γ( α + 1) (cid:16) Γ (cid:16) − η (cid:17)(cid:17) α , < η < α ≤ l b ) − α α e b ( α + b ) E α ( b ) − e b (1 + b ) E ( b ) − α Table 2: Normalized fractional generalized cumulative entropy for the same distributions ofTable 1. α α ( X ) α α ( X ) Figure 1: The normalized fractional cumulative entropy for the distributions given in case (ii)of Table 2, with l = 1, on the left, and in case (iii) of Table 2, with η = 1, on the right, as afunction of α , for b = 0 .
5, 1, 2, 4, 8, from top to bottom near the origin in the case on the left,and viceversa on the right.
Example 2.1
Let X have half-logistic distribution distribution on (0 , ∞ ) , with PDF and CDFgiven respectively by f ( x ) = 4 e − x (1 + e − x ) , F ( x ) = 1 − e − x e − x , x ≥ . Hence, noting that f ( F − ( u )) = 1 − u , making use of (15) and of the integral representationsof the Riemann zeta function (cf. Section 23.2 of Abramowitz and Stegun [19]) ζ ( s ) = ∞ (cid:88) k =1 k − s , s > , in this case one has CE α ( X ) = ζ ( α + 1)2 α +1 , α > . .1 Proportional reversed hazard model The proportional reversed hazard model is expressed by a nonnegative absolutely continuousrandom variable X θ whose CDF is a power of a baseline function F ( x ), which in turn is the CDFof a nonnegative absolutely continuous random variable X , i.e. (see, for instance, Di Crescenzo[20], Gupta and Gupta [21], and Li et al. [22]) F X θ ( x ) = [ F ( x )] θ , x ∈ R , θ > . (16)This model is often encountered in first-hitting-time problems of Markov processes and in theanalysis of the reliability of parallel systems. The PDF and the reversed hazard rate function of X θ are given respectively by f X θ ( x ) = θ [ F ( x )] θ − f ( x ) (17)and τ X θ ( x ) = f X θ ( x ) F X θ ( x ) = θ f ( x ) F ( x ) , (18)where f ( x ) is the PDF of the baseline distribution. We now evaluate the fractional generalizedcumulative entropy of X θ . Proposition 2.3
Under the proportional reversed hazard model (16), the fractional generalizedcumulative entropy of X θ , θ > , can be expressed as CE α ( X θ ) = E θ ( α ) − E θ ( α + 1) , α > , (19) where E θ ( α ) := 1Γ ( α ) E (cid:104) X θ [ − ln F X θ ( X θ )] α − (cid:105) , α > . (20) Proof.
Recalling (7) and (16), we have CE α ( X θ ) = 1Γ ( α + 1) (cid:90) l [ F ( x )] θ (cid:104) − ln [ F ( x )] θ (cid:105) α dx. Integrating by parts, we get CE α ( X θ ) = 1Γ ( α + 1) (cid:26) − θ (cid:90) l x [ F ( x )] θ − f ( x ) (cid:104) − ln [ F ( x )] θ (cid:105) α dx + α θ (cid:90) l x [ F ( x )] θ − f ( x ) (cid:104) − ln [ F ( x )] θ (cid:105) α − dx (cid:27) (21)= 1Γ ( α + 1) (cid:26) − (cid:90) l xf X θ ( x ) [ − ln F X θ ( x )] α dx + α (cid:90) l xf X θ ( x ) [ − ln F X θ ( x )] α − dx (cid:27) , where the last equality is due to (17). The final result then follows from (20).8or instance, it is not hard to see that if F ( x ) = e − x − , x >
0, then (20) yields E θ ( α ) = θα − ,for α >
1, and thus (19) gives CE α ( X θ ) = θα ( α − , α >
1, this being in agreement with case (iii)of Table 1.We observe that CE α ( X θ ) can be alternatively rewritten as the sum of weighted fractionalgeneralized cumulative entropies. In fact, from (16) and (18), Eq. (21) becomes CE α ( X θ ) = 1Γ ( α + 1) (cid:26) − (cid:90) l xF X θ ( x ) τ X θ ( x ) [ − ln F X θ ( x )] α dx + α (cid:90) l xF X θ ( x ) τ X θ ( x ) [ − ln F X θ ( x )] α − dx (cid:27) . Moreover, the fractional generalized cumulative entropy of X θ satisfies a recurrence relation.Indeed, from (19) we have CE α +1 ( X θ ) = E θ ( α + 1) − E θ ( α + 2)= E θ ( α ) − E θ ( α + 2) − CE α ( X θ ) . (22)More generally, in the next proposition we express the fractional generalized cumulative entropyof X θ of order α + n in terms of the same measure of order α . Proposition 2.4
Under the proportional reversed hazard model (16), for α > and θ > , andfor any integer n ≥ , CE α + n ( X θ ) = ( − n CE α ( X θ ) + E θ ( α + n ) − E θ ( α + n + 1) + ( − n − [ E θ ( α ) − E θ ( α + 1)] . (23) Proof.
The proof is by induction on n . Let us consider the case n = 2. Applying the recurrencerelation (22) twice yields CE α +2 ( X θ ) = E θ ( α + 1) − CE α +1 ( X θ ) − E θ ( α + 3)= CE α ( X θ ) + E θ ( α + 2) − E θ ( α + 3) − E θ ( α ) + E θ ( α + 1) , which coincides with (23). Now let us assume that Eq. (23) holds for some n >
2. Then, due to(22), CE α + n +1 ( X θ ) = E θ ( α + n ) − CE α + n ( X θ ) − E θ ( α + n + 2)= E θ ( α + n ) − ( − n − [ E θ ( α ) − E θ ( α + 1)] + ( − n − CE α ( X θ ) − E θ ( α + n ) + E θ ( α + n + 1) − E θ ( α + n + 2)= ( − n +1 CE α ( X θ ) + E θ ( α + n + 1) − E θ ( α + n + 2) + ( − n [ E θ ( α ) − E θ ( α + 1)] . The validity of Eq. (23) for n implies its validity for n + 1. Therefore, it is true for all n ≥ Let us now pinpoint the connection between the generalized measures defined above and somenotions of fractional calculus.The growing interest on the theory and applications of Fractional Calculus has led severalauthors to introduce new notions of fractional integrals. In this area, for instance we refer9he reader to the book by Samko et al. [23]. We recall that for any sufficiently well-behavedfunction φ locally integrable in the interval I = ( a, b ], the (Riemann-Liouville) left- and right-sided fractional integrals of order α of φ , for a < x < b and α >
0, are defined respectivelyas I αa + φ ( x ) = 1Γ( α ) (cid:90) xa φ ( y )( x − y ) − α dy, I αb − φ ( x ) = 1Γ( α ) (cid:90) bx φ ( y )( x − y ) − α dy. These notions have been extended to the case of integral with respect to another function.Indeed, if g is an increasing and positive monotone function on I , having a continuous derivative g (cid:48) on ( a, b ), then the left- and right-sided fractional integrals of order α of φ with respect to g ,for a < x < b and α >
0, are given respectively by (see Section 2.5 of Kilbas [24]) I αa +; g φ ( x ) = 1Γ( α ) (cid:90) xa g (cid:48) ( y ) φ ( y )[ g ( x ) − g ( y )] − α dy, I αb − ; g φ ( x ) = 1Γ( α ) (cid:90) bx g (cid:48) ( y ) φ ( y )[ g ( x ) − g ( y )] − α dy. (24)It is worth mentioning that both the fractional generalized cumulative entropy and the fractionalgeneralized cumulative residual entropy can be expressed in terms of the integrals given in (24).Indeed, from (7) it is not hard to see that CE α ( X ) = lim a → lim x → l I α +1 a +; g φ ( x ) , α > , for g ( x ) = ln F ( x ) , φ ( x ) = [ F ( x )] [ f ( x )] − , provided that the PDF f is positive, continuous and integrable in (0 , l ). Similarly, under thesame assumptions for f on (0 , ∞ ), from (6) one has CRE α ( X ) = lim x → lim b →∞ I q +1 b − ; g φ ( x ) , α > , for g ( x ) = − ln ¯ F ( x ) , φ ( x ) = [ ¯ F ( x )] [ f ( x )] − . These remarks justify the fractional nature of the measures introduced so far.
The aim of this section is two-fold: obtaining some bounds of the proposed fractional measure,and providing results based on stochastic comparisons.The cumulative entropy of the sum of two nonnegative independent random variables islarger than the maximum of their individual cumulative entropies (cf. [7]). Below, we show thata similar inequality holds for the fractional generalized cumulative entropy. The proof followsfrom Theorem 2 of [6], therefore it is omitted.
Proposition 3.1
For any pair of nonnegative absolutely continuous independent random vari-ables X and Y , we have CE α ( X + Y ) ≥ max { CE α ( X ) , CE α ( Y ) } . In the following proposition, we obtain a bound of the fractional entropy (7) in terms of thecumulative entropy (4). 10 roposition 3.2 If X is a nonnegative random variable with support (0 , l ) , < l < ∞ , andwith finite cumulative entropy, then CE α ( X ) ≤ l − α Γ( α + 1) ( CE ( X )) α , if < α ≤ ≥ l − α Γ( α + 1) ( CE ( X )) α , if α ≥ . Proof.
For 0 < α ≤
1, the CDF of X satisfies F ( x ) ≤ ( F ( x )) α for all x ∈ (0 , l ), so that Eq.(7) yields CE α ( X ) ≤ α + 1) (cid:90) l [ − F ( x ) ln F ( x )] α dx = (cid:90) l ϕ α ( η ( x )) dx, where η ( x ) = − F ( x ) ln F ( x ) ≥ , ϕ α ( t ) = t α Γ( α + 1) , with ϕ α ( t ) concave in t ≥
0, for 0 < α ≤
1. Hence, from the integral Jensen inequality we obtain CE α ( X ) ≤ l ϕ α (cid:18) l (cid:90) l η ( x ) dx (cid:19) = l − α Γ( α + 1) (cid:18) − (cid:90) l F ( x ) ln F ( x ) dx (cid:19) α , this giving the proof due to (4). The case α ≥ N CE α ( X ) ≤ l − α Γ( α + 1) , if 0 < α ≤ ≥ l − α Γ( α + 1) , if α ≥ . The next proposition provides some bounds for the fractional generalized cumulative entropyof bounded distributions.
Proposition 3.3
For any random variable X with support (0 , l ) and with finite CE α ( X ) , for α > , we have(a) CE α ( X ) ≥ D α e H ( X ) , with D α = exp { (cid:82) ln( x ( − ln x ) α ) dx } and H ( X ) given by (2);(b) CE α ( X ) ≥ α + 1) (cid:90) l F ( x )[1 − F ( x )] α dx ;(c) CE α ( X ) ≤ l Γ( α + 1) (cid:16) αe (cid:17) α , for < α ≤ . Proof.
The first inequality can be reached applying the log-sum inequality (see, for instance,[2]). Recalling (7), the second inequality can be obtained by using ln u ≤ u − < u ≤ u ( − log u ) α is nonnegative and concave in u ∈ (0 ,
1) for all α ∈ (0 , u ( − log u ) α ≤ ( − log θ ) α − [ αθ − u ( α + log θ )] for all u ∈ (0 , α ∈ (0 ,
1] and θ ∈ (0 , CE α ( X ) ≤ α + 1) ( − log θ ) α − { αθl − [ l − E ( X )]( α + log θ ) } . By taking θ = e − α we finally obtain the third inequality.For α = 1, the results (a) and (b) of Proposition 3.3 become the relations given in Propo-sitions 4.2 and 4.3 of [7], respectively. Clearly, by resorting to Fubini’s theorem the inequalitygiven in (b) can be expressed as CE α ( X ) ≥ E [ ψ α ( X )] , where ψ α ( x ) = α +1) (cid:82) lx [1 − F ( y )] α dy, α > Let us now present some ordering properties of the fractional generalized cumulative entropy.We refer the reader to the book of Shaked and Shanthikumar [25] for the notions on stochasticorders recalled in this section.We first study whether there is any connection with the usual stochastic order. We recallthat a random variable X with CDF F is said to be smaller than another random variable Y with CDF G in the usual stochastic order if F ( x ) ≥ G ( x ) for all x , and write X ≤ st Y . Inthe following example we show that in general the usual stochastic ordering does not imply theordering of fractional generalized cumulative entropies. Example 3.1
Consider two random variables having power distribution, with CDF F ( x ) =( x/l ) b and G ( x ) = ( x/l ) d , where ≤ x ≤ l and b, d > . Further, for b ≤ d we have X ≤ st Y. Moreover, recalling (ii) of Table 1, CE α ( X ) = l b α ( b + 1) α +1 , CE α ( Y ) = l d α ( d + 1) α +1 , α > . However, for some values of the parameters and some choices of α the condition CE α ( X ) ≤ CE α ( Y ) is not fulfilled (see Figure 2). Now, we obtain some stochastic ordering properties of the considered measure. We show thatmore dispersed random variables produce larger fractional generalized cumulative entropies. Let X and Y be nonnegative random variables with CDF’s F and G , respectively. Denote the right-continuous inverses of F and G as F − and G − , respectively. Then, X is said to be smallerthan Y in the dispersive order, denoted by X ≤ d Y , if F − ( v ) − F − ( u ) ≤ G − ( v ) − G − ( u ) forall 0 < u ≤ v < Theorem 3.1
Let X and Y be nonnegative absolutely continuous random variables with PDF’s f and g , and CDF’s F and G , respectively. Then, X ≤ d Y implies that CE α ( X ) ≤ CE α ( Y ) ,for all α > . Proof.
From the representation given in (15), for α > CE α ( X ) − CE α ( Y ) = 1Γ( α + 1) (cid:90) u ( − ln u ) α (cid:20) f ( F − ( u )) − g ( G − ( u )) (cid:21) du. CE α ( X ) − CE α ( Y ) as considered in Example 3.1 for b, d ∈ (0 , l = 3,and (a) α = 0 .
25, (b) α = 0 .
5, (c) α = 1, (d) α = 2.The thesis then immediately follows recalling that X ≤ d Y if and only if f ( F − ( u )) ≥ g ( G − ( u ))for all u ∈ (0 ,
1) (see Section 3.B of [25]).
Remark 3.1
It is worth mentioning that the fractional generalized cumulative entropy is ac-tually a variability measure (following Bickel and Lehmann [26]), thanks to previously givenresults. Indeed, under suitable assumptions the following properties hold:(P1) CE α ( X + b ) = CE α ( X ) for all constants b ,(P2) CE α ( cX ) = c CE α ( X ) for all c > ,(P3) CE α ( a ) = 0 for any degenerate random variable at a ,(P4) CE α ( X ) ≥ for all X ,(P5) X ≤ d Y implies CE α ( X ) ≤ CE α ( Y ) . In order to provide a further comparison result, we recall that a random variable X is said tobe smaller than Y in the hazard rate order, denoted as X ≤ hr Y , if ¯ G ( x ) / ¯ F ( x ) is nondecreasingwith respect to x , where ¯ F = 1 − F and ¯ G = 1 − G are respectively the survival functions of X and Y . Moreover, X is said to be decreasing failure rate (DFR) if ¯ F is logconvex. Theorem 3.2
Let the random variables X and Y satisfy the same assumptions of Theorem 3.1.Further, assume that X ≤ hr Y and let X or Y be DFR. Then, we have CE α ( X ) ≤ CE α ( Y ) forall α > . Proof.
The proof follows from Theorem 2.1(b) of Bagai and Kochar [27] and the result givenin Theorem 3.1. 13n various applied contexts it is appropriate to compare random measures arising from possi-bly ordered systems. Let us then face the following problem: to express the fractional generalizedcumulative entropy of X in terms of suitable quantities depending on X and Y , where the latterrandom variables are ordered in the usual stochastic order sense. Proposition 3.4
Let X and Y be nonnegative random variables with finite but unequal means,with CDF’s F and G respectively, and such that X ≤ st Y or Y ≤ st X . If condition (12) holdsand if E [ ξ α ( Y )] < ∞ , then for α > CE α ( X ) = E [ ξ α ( Y )] + E [ ξ (cid:48) α ( Z )] [ E ( X ) − E ( Y )] , (25) where Z is an absolutely continuous nonnegative random variable with PDF f Z ( x ) = G ( x ) − F ( x ) E ( X ) − E ( Y ) , x > . Proof.
Recalling Eq. (13), i.e. CE α ( X ) = E [ ξ α ( X )], the proof follows from the probabilisticanalogue of the mean value theorem given in Theorem 4 . ξ (cid:48) α ( x ) = − α +1) [ − ln F ( x )] α ≤ x >
0, and α > In this section we develop a dynamic version of the fractional generalized cumulative entropy.To this aim we take as reference a notion from reliability theory. Suppose that a system, startedat time 0, is seen to be failed at a pre-specified inspection time, say t ∈ (0 , l ). In this case theuncertainty relies on the past, in the sense that the unknown system failure instant has occurredin (0 , t ), previous than the inspection time. Let X be the random variable that denotes thefailure instant. We can consider the fractional generalized cumulative entropy of the past time X ( t ) := [ X | X ≤ t ] , t ∈ (0 , l ) . Various measures have been proposed in the literature for X ( t ) , such as the cumulative pastentropy given in (5). Indeed, one has CE ( X ; t ) = CE ( X ( t ) ), for t ∈ (0 , l ). Here we can definethe dynamic fractional generalized cumulative entropy , for α >
0, as CE α ( X ; t ) := CE α ( X ( t ) ) = 1Γ( α + 1) (cid:90) t F ( x ) F ( t ) (cid:20) − ln (cid:18) F ( x ) F ( t ) (cid:19)(cid:21) α dx, t ∈ (0 , l ) . (26)Clearly, if α → CE α ( X ; t ) tends to cumulative past entropy CE ( X ; t ) given in (5).Moreover, from (26) we have that CE α ( X ; t ) reduces to the fractional generalized cumulativeentropy (7) when t → l . Example 4.1 (i) Let X have power distribution in the interval (0 , l ) with parameter b , as inthe case (ii) of Table 1. Then, the dynamic fractional generalized cumulative entropy is given by CE α ( X ; t ) = t b α ( b + 1) α +1 , t ∈ (0 , l ) . .0 0.5 1.0 1.5 2.0 2.5 3.0 t CE α ( X;t ) t CE α ( X;t ) Figure 3: The dynamic fractional generalized cumulative entropy (27) of the Fr´echet distributionwith parameters b and 1, for α = 0 .
25, 0 .
5, 1, 2, 4 (from top to bottom), with b = 1 (left) and b = 3 (right). (ii) Let X have Fr´echet distribution with parameters b and 1, i.e. F ( x ) = e − b x − , x > , with b > . Then, from (26) we have CE α ( X ; t ) = bα (cid:18)(cid:16) α + bt (cid:17) E α (cid:16) bt (cid:17) e bt − (cid:19) , t > , (27) where E α is defined in (8). In this case, some plots of CE α ( X ; t ) are given in Figure 3. Similarly to Proposition 2.2, we get the following result concerning the effect of an affinetransformation.
Proposition 4.1
Let Y = cX + b , where c > and b ≥ . Then, CE α ( cX + b ; t ) = c CE α (cid:0) X ; t − bc (cid:1) for all α > and all t ∈ ( b, b + c l ) . Remark 4.1
Under the same conditions specified in Remark 2.1, from Eq. (26) one can verifythat for a symmetric distribution the following relation holds: CE α ( X ; t ) = CRE α ( X ; l − t ) for all α > and all t ∈ (0 , l ) ,where, in analogy with (6), CRE α ( X ; t ) := 1Γ( α + 1) (cid:90) lt ¯ F ( x )¯ F ( t ) (cid:20) − ln (cid:18) ¯ F ( x )¯ F ( t ) (cid:19)(cid:21) α dx, t ∈ (0 , l ) (28) is the dynamic fractional generalized cumulative residual entropy of X . It is worth mentioning that the dynamic measures CE α ( X ; t ) and CRE α ( X ; t ) not only pro-vide respectively a generalization of the cumulative past entropy and of the cumulative residualentropy, attained in the limit α →
1. They also constitute a further extension of well-knownmeasures of interest in reliability theory. Indeed, from Eqs. (26) and (28) we have respectivelylim α → + CE α ( X ; t ) = ˜ µ ( t ) , t ∈ (0 , l ) , and lim α → + CRE α ( X ; t ) = mrl( t ) , t ∈ (0 , l ) , µ ( t ) is the mean inactivity time (11), and where mrl( t ) is the mean residual life of X ,defined in (10).Similarly to Proposition 3.2, we obtain the following bounds for the dynamic measure definedin (26), for t ∈ (0 , l ): CE α ( X ; t ) ≤ t − α Γ( α + 1) ( CE ( X ; t )) α , if 0 < α ≤ ≥ t − α Γ( α + 1) ( CE ( X ; t )) α , if α ≥ , with CE ( X ; t ) given in (5). Moreover, following the same arguments of the proof of Proposition3.3 we obtain the following bounds for the dynamic fractional generalized cumulative entropy.The proof is omitted being similar. Proposition 4.2
For any random variable X with support (0 , l ) and with finite CE α ( X ; t ) , for α > and t ∈ (0 , l ) we have the following bounds:(a) CE α ( X ; t ) ≥ D α e H ( X ; t ) , where D α = exp { (cid:82) ln( x ( − ln x ) α ) dx } and H ( X ; t ) is the cumu-lative past entropy (5);(b) CE α ( X ; t ) ≥ α + 1) (cid:90) t F ( x ) F ( t ) (cid:18) − F ( x ) F ( t ) (cid:19) α dx .(c) CE α ( X ; t ) ≤ t Γ( α + 1) (cid:16) αe (cid:17) α , for < α ≤ . Next, we introduce the class of distributions based on the monotonicity property of thedynamic fractional generalized cumulative entropy. It was proved in [7] that the class of distri-butions having decreasing dynamic cumulative entropy is empty. A similar property holds for CE α ( X ; t ), whereas this measure can be increasing. First, we present the following definition. Definition 4.1
A nonnegative random variable X is said to have increasing dynamic fractionalgeneralized cumulative entropy (IDFCE) if CE α ( X ; t ) is increasing in t . For instance, from Case (i) of Example 4.1 we have that the power distribution is IDFCE.The following result shows that the above defined class is preserved under affine increasingtransformations. The proof is immediate due to Proposition 4.1.
Proposition 4.3
Let Y = cX + b , where c > and b ≥ . If X is IDFCE, then Y is IDFCE. Hereafter we consider the dynamic fractional generalized cumulative entropy under the pro-portional reversed hazard model. Specifically, let X θ , θ >
0, be a random variable defined in(0 , l ) that satisfies the proportional reversed hazard model with baseline CDF F ( x ), i.e. havingCDF F θ ( x ) = [ F ( x )] θ . Hence, from (26) one has CE α ( X θ ; t ) = θ α Γ( α + 1) (cid:90) t (cid:18) F ( x ) F ( t ) (cid:19) θ (cid:20) − ln (cid:18) F ( x ) F ( t ) (cid:19)(cid:21) α dx, t ∈ (0 , l ) . (29)Then, it is not hard to see that in this case, for any t ∈ (0 , l ) the following bounds hold: CE α ( X θ ; t ) (cid:26) ≥ θ α CE α ( X ; t ) if 0 < θ ≤ ≤ θ α CE α ( X ; t ) if θ ≥ . Let us now consider two examples dealing with the dynamic fractional generalized cumulativeentropy. 16 t CE α ( X θ ;t ) t CE α ( X θ ;t ) Figure 4: The function CE α ( X θ ; t ) for Example 4.2 with λ = 1, µ = 2, and θ = 1 (left) and θ = 4 (right), for α = 0, 0 .
2, 0 .
5, 1, 2 (from top to bottom).
Example 4.2
Let X θ be a random variable with support (0 , ∞ ) , having CDF F θ ( x ) = (cid:32) µ (1 − e − ( λ − µ ) x ) λ − µ e − ( λ − µ ) x (cid:33) θ , x ∈ (0 , ∞ ) , for < λ < µ and θ > , and satisfying the proportional reversed hazard model. We remarkthat for θ ∈ N , X θ may be viewed as the first-entrance time into the absorbing state 0 for alinear birth-death process over N , with birth rates λ n = λ n and death rates µ n = µ n , n ∈ N ,having initial state θ ∈ N at time 0 (cf. Example 5.2 of Di Crescenzo and Ricciardi [29]). Thecorresponding dynamic fractional generalized cumulative entropy, evaluated by means of (29), isprovided in Figure 4. It is shown that it is increasing in t and decreasing in α . Example 4.3
Consider the random variable X θ having CDF F θ ( x ) = (cid:18) λx λx (cid:19) θ , x ∈ (0 , ∞ ) , with λ > and θ > . Clearly, it satisfies the proportional reversed hazard model. If θ ∈ N , then X θ has the same distribution of the first-crossing time of the Geometric Counting Process withparameter λ > through the constant boundary θ (cf. Eq. (23) of Di Crescenzo and Pellerey[30]). Making use of (29) we can evaluate its dynamic fractional generalized cumulative entropy(see Figure 5). Also in this example, CE α ( X θ ; t ) is increasing in t and decreasing in α . We note that similar results can be obtained for
CRE α ( X ; t ) under a dual model. Assumethat X ∗ θ , θ >
0, is a random variable defined in (0 , l ) which satisfies the proportional hazardmodel with baseline survival function ¯ F ( x ), i.e. having survival function ¯ F ∗ θ ( x ) = [ ¯ F ( x )] θ . Then,due to (28) the dynamic fractional generalized cumulative residual entropy of X is given by CRE α ( X ∗ θ ; t ) = θ α Γ( α + 1) (cid:90) lt (cid:18) ¯ F ( x )¯ F ( t ) (cid:19) θ (cid:20) − ln (cid:18) ¯ F ( x )¯ F ( t ) (cid:19)(cid:21) α dx, t ∈ (0 , l ) . (30)Also in this case we obtain suitable bounds for (30), i.e. CRE α ( X ∗ θ ; t ) (cid:26) ≥ θ α CRE α ( X ; t ) if 0 < θ ≤ ≤ θ α CRE α ( X ; t ) if θ ≥ . t CE α ( X θ ;t ) t CE α ( X θ ;t ) Figure 5: The function CE α ( X θ ; t ) of Example 4.3 with λ = 1, and θ = 1 (left) and θ = 2(right), for α = 0, 0 .
2, 0 .
5, 1, 2 (from top to bottom).Finally, in analogy with (14), we note that the normalized dynamic fractional generalizedcumulative entropy can defined as
N CE α ( X ; t ) = CE α ( X ; t )( CE ( X ; t )) α , t ∈ (0 , l ) ( α > . This section is devoted to the nonparametric estimate of the fractional generalized cumulativeentropy.Consider a random sample X , . . . , X n of size n from a distribution with CDF F. Then, theordered sample values denoted by X n ≤ . . . ≤ X n : n represent the order statistics of the randomsample. As well known, the empirical distribution function based on the random sample is givenby ˆ F n ( x ) = 1 n n (cid:88) i =1 I { X i ≤ x } = , x < X n ,kn , X k : n ≤ x < X k +1: n , ( k = 1 , . . . , n − , x ≥ X n : n , where I A is the indicator function of A , i.e. I A = 1 if A is true and I A = 0 otherwise. Thus, theempirical measure of the fractional generalized cumulative entropy given by (7) can be expressedas CE α ( ˆ F n ) = 1Γ( α + 1) (cid:90) l ˆ F n ( x )[ − ln ˆ F n ( x )] α dx = 1Γ( α + 1) n − (cid:88) k =1 V k +1 (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α , (31)where V = X n , V k +1 = X k +1: n − X k : n , k = 1 , . . . , n − , are the sample spacings. When α = 1 , then CE α ( ˆ F n ) reduces to the empirical cumulative entropyproposed in [7] and in Di Crescenzo and Longobardi [31]. Moreover, when α is a positive integer18hen CE α ( ˆ F n ) identifies with the empirical generalized cumulative entropy treated in [16] and[17].Next, we discuss the asymptotic property of the empirical fractional generalized cumulativeentropy given by (31). We first shown that CE α ( ˆ F n ) converges to the actual value of the measureintroduced in (7). Proposition 5.1
Let X ∈ L p , p > . Then, the empirical fractional generalized cumulativeentropy converges to the fractional generalized cumulative entropy almost surely. That is, for α > CE α ( ˆ F n ) a.s. −−→ CE α ( X ) , as n → ∞ . Proof.
From Glivenko-Cantelli theorem, it can be established thatsup x | ˆ F n ( x ) − F ( x ) | a.s. −−→ , as n → ∞ .Thus, the rest of the proof proceeds as in Theorem 9 of [6].In the following, we consider two distributions and study the empirical fractional generalizedcumulative entropy. Example 5.1
Consider a random sample X , . . . , X n from the exponential distribution withparameter λ. Since the sample spacings are independent, thus, V k +1 follows the exponentialdistribution with parameter λ ( n − k ) . So, from (31) the expectation and variance of the empiricalfractional generalized cumulative entropy are respectively obtained as E (cid:104) CE α ( ˆ F n ) (cid:105) = 1Γ( α + 1) 1 λ n − (cid:88) k =1 n − k (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α , and V ar (cid:104) CE α ( ˆ F n ) (cid:105) = 1[Γ( α + 1)] λ n − (cid:88) k =1 n − k ) (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α . Figure 6 shows the above quantities as a function of α , for some choices of n . In particular,both mean and variance are decreasing in α . Moreover, it is shown that the mean E (cid:104) CE α ( ˆ F n ) (cid:105) approaches the fractional generalized cumulative entropy as n grows, more rapidly for larger α . Example 5.2
For the uniformly distributed identical and independent random observations inthe interval (0 , , the sample spacings V k +1 follow beta distribution with parameters and n with E ( V k +1 ) = 1 / ( n + 1) . Thus, E (cid:104) CE α ( ˆ F n ) (cid:105) = 1Γ( α + 1) 1 n + 1 n − (cid:88) k =1 (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α ≡ E ( V k +1 ) · CE α ( U ) , with CE α ( U ) given in Eq. (9), and V ar (cid:104) CE α ( ˆ F n ) (cid:105) = 1[Γ( α + 1)] n ( n + 1) ( n + 2) n − (cid:88) k =1 (cid:18) kn (cid:19) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α . .0 0.5 1.0 1.5 2.0 α α Figure 6: With reference to Example 5.1, for λ = 1, on the left: the upper curve is the fractionalgeneralized cumulative entropy, the other curves give E (cid:104) CE α ( ˆ F n ) (cid:105) for n = 10 h , h = 1, 2, 3, 4,6 (from bottom to top); on the right: V ar (cid:104) CE α ( ˆ F n ) (cid:105) for n = 10 h , h = 1, 2, 3, 4 (from top tobottom in proximity of α = 0 . α α Figure 7: For Example 5.2, on the left: the upper curve is the fractional generalized cumulativeentropy, the other curves show E (cid:104) CE α ( ˆ F n ) (cid:105) for n = 5, 10, 25, 100 (from bottom to top); on theright: V ar (cid:104) CE α ( ˆ F n ) (cid:105) for the same choices of n (from top to bottom). Such mean and variance are shown in Figure 7 for α ∈ (0 , , with various choices of n . Similarlyas Example 5.1, both quantities are decreasing in α . Note that in this case we have lim n →∞ E [ CE α ( ˆ F n )] = 12 α +1 = CE α ( X ) , lim n →∞ V ar [ CE α ( ˆ F n )] = 0 . Indeed, the considered nonparametric estimator is consistent to the fractional generalized cumu-lative entropy when the random sample is taken from the U (0 , distribution. Next, we present a central limit theorem for the empirical fractional generalized cumulativeentropy, when the i.i.d. random observations are available from the exponential distribution.
Theorem 5.1
Let X , . . . , X n be a random sample from the exponential distribution with pa-rameter λ . Then, for any α > Z n := CE α ( ˆ F n ) − E [ CE α ( ˆ F n )] (cid:16) V ar [ CE α ( ˆ F n )] (cid:17) / → N (0 , n distribution as n → ∞ . Proof.
From [7], we note that the empirical fractional generalized cumulative entropy can beexpressed as the sum of independent exponential random variables U k with mean E [ U k ] = 1Γ( α + 1) kλn ( n − k ) (cid:20) − ln (cid:18) kn (cid:19)(cid:21) α . The rest of the proof follows using similar arguments in [17]. Thus, it is omitted.Let us now discuss the stability of the empirical fractional generalized cumulative entropy,by taking as reference the Section 3.3 of [9].
Definition 5.1
Let X (cid:48) , . . . , X (cid:48) n be any small deformation of the random sample X , . . . , X n taken from a population with CDF F ( x ) . Then, the empirical fractional generalized cumulativeentropy is stable if for all (cid:15) > , there exists δ > such that, for all n ∈ N , n (cid:88) k =1 (cid:12)(cid:12) X k − X (cid:48) k (cid:12)(cid:12) < δ ⇒ (cid:12)(cid:12)(cid:12) CE α ( ˆ F n ) − CE α ( ˆ F (cid:48) n ) (cid:12)(cid:12)(cid:12) < (cid:15). Based on the above definition, below we present sufficient condition for the stability of CE α ( ˆ F n ). Theorem 5.2
The empirical fractional generalized cumulative entropy of an absolutely contin-uous random variable X is stable provided that X has a distribution on a finite interval. Proof.
Assume that X has distribution in a nonnegative finite interval. The empirical frac-tional generalized cumulative entropy is written as CE α ( ˆ F n ) = 1Γ( α + 1) n − (cid:88) k =1 ( X k +1: n − X k : n ) ˆ F n ( X k : n ) (cid:104) − ln (cid:16) ˆ F n ( X k : n ) (cid:17)(cid:105) α , α > . Then, the proof proceeds as for Theorem 5 of [9] and thus it is omitted.As example, we now analyze a real data set and compute the empirical fraction generalizedcumulative entropy for different values of α . Example 5.3
We consider the following data set from Chowdhury et al. [32], concerning ob-servations taken from [33] on the number of casualties in n = 44 different plane crashes: { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } . Based on the given dataset, we compute the values of the fractional generalized cumulative en-tropy (31), shown in Figure 8. Hence, we deal with a linear combination of terms of the type ϕ ( α ; x ) := x α Γ( α + 1) , α > , x = − ln (cid:18) kn (cid:19) > . .0 0.5 1.0 1.5 2.0 α α ( F n ) n α ( F n ) Figure 8: The values of the fractional generalized cumulative entropy for the data set of Example5.3. On the left: for 0 ≤ α ≤ n = 44. On the right: for n varying, with reference to thefirst n data of the sample, with α = 0 , . , . , . , . , , . Note that if < x ≤ e − γ = 0 . ... (where γ is the Euler-Mascheroni constant) then thefunction ϕ ( α ; x ) is decreasing in α ; moreover if < x ≤ exp (cid:8) − (cid:0) γ + (cid:112) (2 / π (cid:1)(cid:9) = 0 . ... then ϕ ( α ; x ) is convex in α . If the sample spacings are slowly varying, since the larger coefficientsin the sum on the right-hand side of (31) are given by large k , and thus for x close to 0, thenin the linear combination for the fractional generalized cumulative entropy the prevailing termsare decreasing convex in α . This remark justifies its form in the left plot of Figure 8, where itis shown as a function of α . Furthermore, when it is treated as a function of n , i.e. referring tothe first n data of the sample, the right plot of Figure 8 shows a jagged trend, that is smootherfor larger values of α. In this paper, we have defined the fractional generalized cumulative entropy and its dynamicversion. Various properties including bounds and ordering results have been studied. It is shownthat the usual stochastic ordering does not imply the ordering between the considered entropies.However, we have shown that the dispersive order implies the ordering of the considered measure.Thus, the fractional generalized cumulative entropy actually constitutes a variability measure.This fact discloses the possibility of applications in risk theory involving the proportional hazardsmodel, for instance along the line addressed by Psarrakos and Sordo [34].A nonparametric estimator of the fractional measure has been proposed based on the empir-ical distribution function. Various statistical properties of the empirical fractional generalizedcumulative entropy have been studied, including asymptotic results for large samples. In par-ticular, we focus on the convergence of the estimator and on the central limit theorem when arandom sample is taken from the exponential distribution. A stability criteria of the proposedmeasure has been studied, too.
Acknowledgements
Antonio Di Crescenzo and Alessandra Meoli are members of the research group GNCS of IN-dAM (Istituto Nazionale di Alta Matematica). This research is partially supported by MIUR22 PRIN 2017, project ‘Stochastic Models for Complex Systems’, No. 2017JFFHSH. SuchandanKayal gratefully acknowledges the partial financial support for this research work under a grantMTR/2018/000350, SERB, India.
References [1] Shannon CE. A note on the concept of entropy. Bell System Tech J 1948;27(3):379–423.[2] Cover TM, Thomas JA. Elements of information theory. New York: John Wiley & Sons; 1991.[3] Ubriaco MR. Entropies based on fractional calculus. Phys Lett A 2009;373(30):2516–2519.https://doi.org/10.1016/j.physleta.2009.05.026[4] Machado JT. Fractional order generalized information. Entropy 2014;16(4):2350–2361.https://doi.org/10.3390/e16042350[5] Machado JT, Lopes AM. Fractional R´enyi entropy, Eur Phys J Plus 2019;134(5):217.https://doi.org/10.1140/epjp/i2019-12554-9[6] Rao M, Chen Y, Vemuri BC, Wang F. Cumulative residual entropy: a new measure of information.IEEE Trans Inf Theory 2004;50(6):1220–1228. https://doi.org/10.1109/TIT.2004.828057[7] Di Crescenzo A, Longobardi M. On cumulative entropies. J Statist Plann Inference2009;139(12):4072–4087. https://doi.org/10.1016/j.jspi.2009.05.038[8] Navarro J, del Aguila Y, Asadi M. Some new results on the cumulative residual entropy. J StatistPlann Inference 2010;140:310–322. https://doi.org/10.1016/j.jspi.2009.07.015[9] Xiong H, Shang P, Zhang Y. Fractional cumulative residual entropy. Comm Nonlinear Sci NumSimul 2019;78:104879. https://doi.org/10.1016/j.cnsns.2019.104879[10] Psarrakos G, Navarro J. Generalized cumulative residual entropy and record values. Metrika2013;27:623–640. https://doi.org/10.1007/s00184-012-0408-6[11] Toomaj A, Di Crescenzo A. Generalized entropies, variance and applications. Entropy 2020;22:709.https://doi.org/10.3390/e22060709[12] Zhang B, Shang P. Uncertainty of financial time series based on discrete fractional cumulativeresidual entropy, Chaos 2019:29(10):103104. https://doi.org/10.1063/1.5091545[13] Wang Y, Shang P. Complexity analysis of time series based on generalized fractional or-der cumulative residual distribution entropy. Phys A: Stat Mech Appl 2020;537:122582.https://doi.org/10.1016/j.physa.2019.122582[14] Dong, K, Zhang X. Multiscale fractional cumulative residual entropy of higher-order moments for estimating uncertainty. Fluct Noise Lett 2020;2050038 (16 pages).https://doi.org/10.1142/S0219477520500388[15] Yu S, Huang TZ, Liu X, Chen W. Information measures based on fractional calculus. Inf Proc Lett2012;112:916–921. https://doi.org/10.1016/j.ipl.2012.08.019[16] Kayal S. On generalized cumulative entropies, Prob Engin Inform Sciences 2016;30(4), 640–662.https://doi.org/10.1017/S0269964816000218[17] Di Crescenzo A, Toomaj A. Further results on the generalized cumulative entropy. Kybernetika2017;53(5):959–982. https://doi.org/10.14736/kyb-2017-5-0959