# A Generalized Measure of Cumulative Residual Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Cumulative Entropy

#### 2.1. Generalized Cumulative Residual Entropy

**Definition**

**1.**

#### 2.2. Generalized Cumulative Entropy

**Definition**

**2.**

#### 2.3. Generating Function

**Definition**

**3.**

**Definition**

**4.**

## 3. Sharma–Taneja–Mittal Entropy

#### 3.1. Sharma–Taneja–Mittal Cumulative Residual Entropy

**Definition**

**5.**

**Definition**

**6.**

#### 3.2. Sharma–Taneja–Mittal Cumulative Entropy

**Definition**

**7.**

**Definition**

**8.**

## 4. Connection between Entropy and Extropy

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CE | Cumulative entropy |

CRE | Cumulative residual entropy |

STM | Sharma–Taneja–Mittal |

WCRTE | Weighted cumulative residual Tsallis entropy |

WCTE | weighted cumulative Tsallis entropy |

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Entropy Measure | $\mathit{w}\left(\mathit{u}\right)$ | $\mathit{\varphi}\left(\mathit{x}\right)$ |
---|---|---|

Cumulative residual entropy | 1 | x |

Weighted cumulative residual entropy | $\frac{1}{2}$ | ${x}^{2}$ |

Cumulative residual Tsallis entropy | ${\overline{F}}^{\alpha -1}\left(u\right)$ | x |

Weighted cumulative Tsallis residual entropy | ${\overline{F}}^{\alpha -1}\left(u\right)$ | $\frac{{x}^{2}}{2}$ |

Entropy Measure | $\mathit{w}\left(\mathit{u}\right)$ | $\mathit{\varphi}\left(\mathit{x}\right)$ |
---|---|---|

Cumulative entropy | 1 | x |

Weighted cumulative entropy | $\frac{1}{2}$ | ${x}^{2}$ |

Cumulative Tsallis entropy | ${F}^{\alpha -1}\left(u\right)$ | x |

Weighted cumulative Tsallis entropy | ${F}^{\alpha -1}\left(u\right)$ | $\frac{{x}^{2}}{2}$ |

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**MDPI and ACS Style**

Kattumannil, S.K.; Sreedevi, E.P.; Balakrishnan, N.
A Generalized Measure of Cumulative Residual Entropy. *Entropy* **2022**, *24*, 444.
https://doi.org/10.3390/e24040444

**AMA Style**

Kattumannil SK, Sreedevi EP, Balakrishnan N.
A Generalized Measure of Cumulative Residual Entropy. *Entropy*. 2022; 24(4):444.
https://doi.org/10.3390/e24040444

**Chicago/Turabian Style**

Kattumannil, Sudheesh Kumar, E. P. Sreedevi, and Narayanaswamy Balakrishnan.
2022. "A Generalized Measure of Cumulative Residual Entropy" *Entropy* 24, no. 4: 444.
https://doi.org/10.3390/e24040444