PAPER 1 JMASM123 Essay

A New Two Parametric Useful (Weighted) Generalized Entropy for Lifetime Distributions

Bilal Ahmad Bhat* and M.A.K BaigP.G. Department of Statistics, University of Kashmir, Srinagar-190006, India

*[email protected], [email protected]

*Corresponding author: [email protected]

Abstract: In the literature of information theory, the concept of generalized entropy has gained much attention among researchers. Recently, Di Crescenzo and Longobardi (2006) have studied “length-biased” shift-dependent information measure and its dynamic version.

In this paper, we have proposed the concept of weighted generalized entropy of order and type and its dynamic version. We derive the expressions of these two measures corresponding to some well-known lifetime distributions. It is shown that the weighted generalized residual entropy determines the survival function uniquely. Some important properties and inequalities of the proposed residual information measure have also been discussed.

Keywords: Lifetime distributions, Shannon’s entropy, Residual entropy, residual lifetime, length- biased measure.

AMS subject Classification: 94A17, 94A24

Introduction

Shannon (1948) has introduced a very important measure of entropy which plays a vital role in the context of information theory.

For an absolutely continuous non-negative random variable having probability density function (pdf) , Shannon’s entropy (1948) is defined as

(1)

and for a discrete random variable taking values with respective probabilities , and , it is defined as

(2)

There are various generalizations of Shannon’s entropy (1948) which are available in the literature of information theory. Consequently, in this paper we define a new two parametric generalization of this measure. Let be an absolutely continuous non-negative random variable having probability density function (pdf) , then the generalized entropy is defined as

. (3)

where,

, which is the Shannon’s entropy given in (1).

If represents the lifetime of a system that has survived up to time , then the measure (1) is not appropriate in order to ascertain the uncertainty about the residual life of such a system. For such type of cases, Ebrahimi (1996) introduced the concept of residual entropy and is defined as

, (4)

where, represents the survival function of . In the same way, the generalized entropy of the residual lifetime is given by

. (5)

For , , (5) reduces to (4).

Shannon’s entropy has the drawback of considering the outcomes of a random variable equally important with respect to the goal set by the experimenter. However, in real life, it is not always possible that the elementary events of a probabilistic experiment will be of equal importance. An alternative measure which considers both objective probabilities and some qualitative characteristics of the elementary events of a probabilistic experiment was introduced by Belis and Guiasu (1968) and is defined as

. (6)

where the coefficient in the integrand on the right-hand-side of (6) represents the importance of the occurrence of the event and is usually known as weight. This is a “length biased” shift-dependent information measure which assigns larger importance to the larger values of the observed random. Di Crescenzo and Longobardi (2006) have introduced the notion of weighted residual entropy for the lifetime of a system and is defined as

. (7)

F. Mishagha and G. H. Yarib (2011) have introduced the weighted interval information measure and is defined as

(8)

Also, various authors like Kumar et al. (2015), Nourbakhsh, M. and Yari, G. (2016), Abasnejad (2011), Rajesh et al. (2017), Misagh, M. and Yari, G. H. (2011), Mirali et al. (2015) , Yasaei Sekeh (2015), Das, S. (2016), Kayah, S. (2017), Minimol S. (2016), Mirali, M. and Baratpour, S. (2017), Nair, R. S., Sathar, E. I. A. and Rajesh, G. (2017), Sekeh, S. Y., Borzadaran, G. R. M. and Roknabadi, A. H. R. (2012) have proposed different weighted generalized information measures.

In this paper, we propose the concept of two parametric useful (weighted) generalized entropy of order and type and its dynamic version. We derive the expressions of these two measures corresponding to some well-known lifetime distributions. It is also proved that the proposed dynamic information measure determines the survival function uniquely. Some important properties and inequalities of the proposed dynamic measure are also discussed. Finally, some concluding remarks have been mentioned.

Weighted Generalized Entropy

Analogous to the definition (6) of weighted entropy, in this section, we consider the generalized information measure (3) and define its weighted version

Let be an absolutely continuous non-negative random variable with probability density function (pdf) , then the weighted generalized entropy (WGE) is defined as

(9)

where the coefficient on the right-hand-side represents the weight which assigns greater importance to the larger values of the observed random variable In Table 1, we derive the expressions of WGE of some well-known lifetime distributions Here, is an upper incomplete gamma function.

Weighted Generalized Residual Entropy

In this section we discuss the dynamic (residual) version of WGE (8) and also focus on a characterization result which shows that this dynamic measure determines the survival function uniquely. The concept of weighted residual entropy was introduced by Di Crescenzo and Longobardi (2006) and is given by

. (10)

The dynamic (residual) entropy function corresponding to (9) is defined as

. (11)

Obviously, when , (11) reduces to (9).

Table 1. The expressions of WGE for some lifetime distributions

Distribution

Uniform

Exponential

Gamma

Weibull

Pareto

Rayleigh

Lomax

An alternative way of expressing (11) is obtained in the following theorem.

Theorem 1. For all , we have

. (12)

Proof.

(13)

Since,

.

Therefore, due to (11) and (13), (12) is obtained.

The following theorem proves that characterizes the survival function uniquely.

Theorem 2. Let be a non-negative random variable having continuous density function and survival function . Assume that and increasing in , then determines the survival function uniquely.

Proof. From (11), we have

. (14)

Differentiating (14) w.r.t. t, we obtain

, (15)

where, is the failure rate of . Using (14), we can rewrite (15) as

. (16) Rearranging (12), we have

(17)

Differentiating (17) w.r.t. t, we get

(18)

From (16) and (18), we have

Hence, for fixed , is a solution of , where

.

Differentiating both sides w.t.t. , we have

.

For extreme value of , put , which gives

Also

.

Now, for , . Thus attains maximum at . Also, and . Further it can be easily seen that increases for and decreases for So, the unique solution to is given by . Thus, uniquely determines , which in turns determines .

In table 2, we derive the expressions of weighted generalized residual entropy corresponding to some well-known lifetime distributions. Note, and are the upper and lower incomplete gamma functions respectively.

In the following table, we study the behavior of weighted generalized residual information measure given in (10) under the consideration of exponential distribution. In table 2, assuming , and , corresponding to exponential distribution , we get the values of for different values of as follows.

Table 3. Different values of with respect to for fixed , and

6 7 8 9 10 11 12 13 14 15

0.5186 0.5264 0.5333 0.5395 0.5451 0.5502 0.5550 0.5593 0.5634 0.5672

The graph of this table is drawn in Fig.1 and it is obvious that is monotonic increasing in.

Properties of Weighted Generalized Residual Entropy

In this section, we study some important properties and inequalities of weighted generalized residual entropy.

Definition 1. Let and be two non-negative random variables representing the lifetime of two systems, then is said to be smaller than in weighted residual entropy of order and type (denoted by ) if , for all .

Definition 2. A survival function is said to have increasing (decreasing) weighted generalized entropy for residual life of order and type IWGERL (DWGERL) if is increasing (decreasing) in t, .

Table 2. Weighted generalized residual entropy of some lifetime distributions

Distribution

Uniform

Exponential

Gamma

Pareto

Weibull

Theorem 3. Let be a IWGERL (DWGERL) and , then

Proof. From (11), we have

Since is IWGERL (DWGERL) and , therefore, we have

which leads to

.

Example 1. Let be an exponentially distributed random variable wit then from table 2, we have

.

Therefore, if , then is IWGERL.

Theorem 4. Let be the lifetime of a system with p.d.f and survival function, then for , the following inequality holds.

.

Proof. We know that from log-sum inequality

. (19)

where (19) is obtained from (11).

The L.H.S of (19) leads to

. (20)

Substituting (20) in (19) and after some simplifications, we obtain the desired result.

The following lemma will be very useful in proving the next theorems of this section.

Lemma 1. For an absolutely continuous random variable , define , where is a constant. Then

.

Proof. .

Setting , we obtain

By using (11), the required result is obtained.

Theorem 5. Let and be two absolutely continuous non-negative random variables, define and , . Let and . Then , if or is decreasing in .

Poof. Suppose is decreasing in .

Now, implies

. (21)

Further, since , we have,

. (22)

From (21) and (22), we obtain

. (23)

Using (23) and applying lemma 4.1, we have .

Theorem 6. Let be a non-negative random variable with support and having probability density function , survival function , , the following inequality holds.

.

Proof. From log-sum inequality and (11), we have

.

After simplification, the proof is obvious.

Proposition 1. Let be a non-negative random variable having WGRE , then for , the following inequality holds.

.

Proof. Since, , we have

Theorem 7. Let be an absolutely continuous non-negative random variable and IWGERL (DWGERL). Define , where is a constant. Then IWGERL (DWGERL).Proof. Since IWGERL (DWGERL),

Therefore,

.

By applying lemma 4.1, it follows that IWGERL (DWGERL) and hence the theorem is proved.

Conclusion

In this paper, we have introduced and studied the concept of weighted generalized entropy of order and type and its dynamic (residual) version. We establish the expressions of these measures for some well-known lifetime distributions. It is shown that the proposed dynamic measure characterizes the distribution function uniquely. Finally various properties and inequalities of the dynamic measure have also been studied.

References

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