A Fantastic Brace Parametric Conducive (Weighted) Generalized Entropy ce Epoch Dispensations

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

*[email protected], [email protected]

*Selfsimilar author: [email protected]

Abstract: In the scholarship of counsel hypothesis, the concept of generalized entropy has gained ample heed unordered researchers. Recently, Di Crescenzo and Longobardi (2006) enjoy elaborate length-biased shift-dependent counsel mete and its dynamic rendering.

In this dissertation, we enjoy contemplated the concept of momented generalized entropy of classify and emblem and its dynamic rendering. We trace the expressions of these brace metes selfsimilar to some well-unconcealed epoch dispensations. It is shown that the momented generalized residual entropy determines the birth exercise choicely. Some leading properties and inequalities of the contemplated residual counsel mete enjoy too been argueed.

Keywords: Epoch dispensations, Shannons entropy, Residual entropy, residual epoch, length- biased mete.

AMS material Classification: 94A17, 94A24

## Introduction

Shannon (1948) has introduced a very leading mete of entropy which plays a animate role in the tenor of counsel hypothesis.

Ce an categorically uniform non-negative casual wavering having presumption inobservance exercise (pdf) , Shannons entropy (1948) is mark-outd as

(1)

and ce a discrete casual wavering preamble prizes ingenuityh appertaining probabilities , and , it is mark-outd as

(2)

There are sundry generalizations of Shannons entropy (1948) which are available in the scholarship of counsel hypothesis. Consequently, in this dissertation we mark-out a fantastic brace parametric generalization of this mete. Let be an categorically uniform non-negative casual wavering having presumption inobservance exercise (pdf) , then the generalized entropy is mark-outd as

. (3)

## where,

, which is the Shannons entropy dedicated in (1).

If represents the epoch of a rule that has survived up to duration , then the mete (1) is referable misapply in classify to detect the misgiving about the residual duration of such a rule. Ce such emblem of cases, Ebrahimi (1996) introduced the concept of residual entropy and is mark-outd as

, (4)

where, represents the birth exercise of . In the similar practice, the generalized entropy of the residual epoch is dedicated by

. (5)

### Ce , , (5) reduces to (4).

Shannons entropy has the nauseousness of because the outcomes of a casual wavering similar leading ingenuityh deference to the view regular by the trialer. However, in existent duration, it is referable regularly practicable that the rudimental circumstances of a probabilistic trial conquer be of correspondent signification. An opinion mete which cogitates twain external probabilities and some innate characteristics of the rudimental circumstances of a probabilistic trial was introduced by Belis and Guiasu (1968) and is mark-outd as

. (6)

where the coefficient in the integrand on the right-hand-side of (6) represents the signification of the circumstance of the circumstance and is usually unconcealed as moment. This is a length biased shift-dependent counsel mete which assigns larger signification to the larger prizes of the observed casual. Di Crescenzo and Longobardi (2006) enjoy introduced the referableion of momented residual entropy ce the epoch of a rule and is mark-outd as

. (7)

F. Mishagha and G. H. Yarib (2011) enjoy introduced the momented gap counsel mete and is mark-outd as

(8)

Also, sundry authors love 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) enjoy contemplated incongruous momented generalized counsel metes.

In this dissertation, we move the concept of brace parametric conducive (weighted) generalized entropy of classify and emblem and its dynamic rendering. We trace the expressions of these brace metes selfsimilar to some well-unconcealed epoch dispensations. It is too proved that the contemplated dynamic counsel mete determines the birth exercise choicely. Some leading properties and inequalities of the contemplated dynamic mete are too argueed. Finally, some spent remarks enjoy been mentioned.

### Weighted Generalized Entropy

Analogous to the determination (6) of momented entropy, in this exception, we cogitate the generalized counsel mete (3) and mark-out its momented rendering

Let be an categorically uniform non-negative casual wavering ingenuityh presumption inobservance exercise (pdf) , then the momented generalized entropy (WGE) is mark-outd as

(9)

where the coefficient on the right-hand-side represents the moment which assigns elder signification to the larger prizes of the observed casual wavering In Board 1, we trace the expressions of WGE of some well-unconcealed epoch dispensations Here, is an loftier spoilt gamma exercise.

### Weighted Generalized Residual Entropy

In this exception we argue the dynamic (residual) rendering of WGE (8) and too convergence on a characterization end which shows that this dynamic mete determines the birth exercise choicely. The concept of momented residual entropy was introduced by Di Crescenzo and Longobardi (2006) and is dedicated by

. (10)

The dynamic (residual) entropy exercise selfsimilar to (9) is mark-outd as

. (11)

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

Board 1. The expressions of WGE ce some epoch dispensations

## Distribution

## Uniform

## Exponential

## Gamma

## Weibull

## Pareto

## Rayleigh

## Lomax

An opinion practice of expressing (11) is achieveed in the succeedingcited theorem.

### Theorem 1. Ce full , we enjoy

. (12)

## Proof.

(13)

## Since,

.

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

The succeedingcited theorem proves that characterizes the birth exercise choicely.

Theorem 2. Let be a non-negative casual wavering having uniform inobservance exercise and birth exercise . Assume that and increasing in , then determines the birth exercise choicely.

### Proof. From (11), we enjoy

. (14)

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

, (15)

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

. (16) Rearranging (12), we enjoy

(17)

### Differentiating (17) w.r.t. t, we achieve

(18)

### From (16) and (18), we enjoy

Hence, ce unroving , is a elucidation of , where

.

Differentiating twain sides w.t.t. , we enjoy

.

Ce remote prize of , dispose , which gives

Also

.

Now, ce , . Thus attains apex at . Too, and . Further it can be abundantly seen that increases ce and decreases ce So, the choice elucidation to is dedicated by . Thus, choicely determines , which in turns determines .

In board 2, we trace the expressions of momented generalized residual entropy selfsimilar to some well-unconcealed epoch dispensations. Referablee, and are the loftier and inferior spoilt gamma exercises appertainingly.

In the succeedingcited board, we con-over the comportment of momented generalized residual counsel mete dedicated in (10) below the cogitateation of exponential dispensation. In board 2, magnificent , and , selfsimilar to exponential dispensation , we achieve the prizes of ce incongruous prizes of as follows.

Board 3. Incongruous prizes of ingenuityh deference to ce unroving , 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 board is drawn in Fig.1 and it is explicit that is monotonic increasing in.

Properties of Momented Generalized Residual Entropy

In this exception, we con-over some leading properties and inequalities of momented generalized residual entropy.

Determination 1. Let and be brace non-negative casual waverings representing the epoch of brace rules, then is said to be smaller than in momented residual entropy of classify and emblem (denoted by ) if , ce full .

Determination 2. A birth exercise is said to enjoy increasing (decreasing) momented generalized entropy ce residual duration of classify and emblem IWGERL (DWGERL) if is increasing (decreasing) in t, .

Board 2. Momented generalized residual entropy of some epoch dispensations

## Distribution

## Uniform

## Exponential

## Gamma

## Pareto

## Weibull

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

### Proof. From (11), we enjoy

Since is IWGERL (DWGERL) and , accordingly, we enjoy

## which leads to

.

Example 1. Let be an exponentially reserved casual wavering ingenuity then from board 2, we enjoy

.

### Therefore, if , then is IWGERL.

Theorem 4. Let be the epoch of a rule ingenuityh p.d.f and birth exercise, then ce , the succeedingcited disparity holds.

.

Proof. We understand that from log-sum disparity

. (19)

### where (19) is achieveed from (11).

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

. (20)

Substituting (20) in (19) and succeeding some simplifications, we achieve the desired end.

The succeedingcited lemma conquer be very conducive in proving the proximate theorems of this exception.

Lemma 1. Ce an categorically uniform casual wavering , mark-out , where is a fixed. Then

.

## Proof. .

## Setting , we achieve

By using (11), the required end is achieveed.

Theorem 5. Let and be brace categorically uniform non-negative casual waverings, mark-out and , . Let and . Then , if or is decreasing in .

### Poof. Suppose is decreasing in .

## Now, implies

. (21)

### Further, since , we enjoy,

. (22)

### From (21) and (22), we achieve

. (23)

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

Theorem 6. Let be a non-negative casual wavering ingenuityh buttress and having presumption inobservance exercise , birth exercise , , the succeedingcited disparity holds.

.

Proof. From log-sum disparity and (11), we enjoy

.

Succeeding simplification, the essay is explicit.

Proposition 1. Let be a non-negative casual wavering having WGRE , then ce , the succeedingcited disparity holds.

.

### Proof. Since, , we enjoy

Theorem 7. Let be an categorically uniform non-negative casual wavering and IWGERL (DWGERL). Mark-out , where is a fixed. Then IWGERL (DWGERL).Proof. Since IWGERL (DWGERL),

## Therefore,

.

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

## Conclusion

In this dissertation, we enjoy introduced and elaborate the concept of momented generalized entropy of classify and emblem and its dynamic (residual) rendering. We prove the expressions of these metes ce some well-unconcealed epoch dispensations. It is shown that the contemplated dynamic mete characterizes the dispensation exercise choicely. Finally sundry properties and inequalities of the dynamic mete enjoy too been elaborate.

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