EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5796 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Comparative Analysis of Entropy Measures for Exponentiated Exponential and Truncated Exponentiated Exponential Distributions Javid Gani Dar1,∗, Irsa Sajjad2, Shahid Tamboli3 1 Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Lavale, Pune, Maharashtra, India 2 Department of Mathematics and Statistics, Central South University Changsha, Hunan, China 3 Department of Mechanical Engineering, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Lavale, Pune, India Abstract. Our research explores several entropy measurements for both EE (Exponentiated Ex- ponential) and TEE (Truncated Exponentiated Exponential) lifetime models that scientists com- monly use in their reliability projects. Our research develops various measures of entropy such as Shannon and Rényi to precisely measure uncertainty and unpredictability within these statistical distributions. The research analyzes entropy behavior across changing distribution parameters and contrasts EE with TEE results. The results show what exactly connects distribution parameters to their entropy measurements and shows why each distribution is special. This study enhances statistical and information theoretical methods for general use by showing all known uncertainty characteristics of EE and TEE distributions. 2020 Mathematics Subject Classifications: 16N60, 16W10, 47B47 Key Words and Phrases: Exponentiated Exponential Distribution, Truncated Exponenetiated, Exponential Distribution, Entropy 1. Introduction In probability theory, information theory plays a vital role, which is the basis of re- liability theory, communication systems, and financial characteristics. Claude Shannon gave the idea of information theory in 1948, which is also known as Shannon Entropy [1]. It is a connection point between several fields such as Mathematics, Statistics, Com- puter Science, Physics, Engineering, etc. In addition, it has diverse applications including natural language processing, linguistics, statistical Inference, fuzzy entropy, cryptography, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5796 Email addresses: javid.dar@sitpune.edu.in (J. Dar), irsasajjad27@gmail.com (I. Sajjad) , shahidt@sitpune.edu.in (S. Tamboli) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 2 of 17 bioinformatics, quantum computer science, plagiarism detection, human vision [2–4] and thermal physics [5]. Moreover, the subfields of information theory include algorithm com- plexity theory, theoretical information security, source coding, information measurements, and gray system. In literature, many authors contributed to measuring entropies and their generaliza- tion. [6] defined the seven entropy measures of Truncated Rayleigh distribution instead of Rayleigh distribution and computed the relative loss of entropy. We initiated this in- vestigation due to the increasing need to understand entropy measures for two special distribution types. Scientists and researchers depend heavily on Exponentiated Exponen- tial and Truncated Exponentiated Exponential distributions because they handle lifetime data, reliability analysis, and survival studies effectively. Deep knowledge of the entropy measures for these distributions helps us better explain their randomness characteristics. This research examines entropy values of these distributions to show their actual perfor- mance characteristics for different applications. New statistical information will benefit areas of practical application while helping researchers understand distribution qualities better. 2. Several Entropy measures The mathematical expression of Shannon entropy ΞS(y) = − ∫ +∞ −∞ g(y) ln g(y)dy (1) The only shortcoming of Shannon entropy is that it may produce negative results for some probability models, which is not possible for uncertainty measures. Many authors contributed to overcoming this problem and proposed different entropy measures. [7] gave the idea of randomness and uncertainty and gave a new generalized entropy. The Renyi entropy [8] is as follows: ΞR(y) = 1 1− ς log( ∫ +∞ −∞ [g(y)]ςdy), ς > 0, ς ̸= 1 (2) It gives a non-negative result due to the constant ς involved in the expression. [7] discussed entropy as ΞHC (y) = 1 21−ς − 1 [∫ +∞ −∞ [g(y)]ςdy − 1 ] ς > 0, ς ̸= 0 (3) [9] investigated the entropy is ΞAR(y) = 1 2ς−1 − 1 [∫ +∞ −∞ [g(y)]ςdy ]ς − 1 ς > 0, ς ̸= 0 (4) [10] used entropy as J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 3 of 17 ΞSM (y) = 1 21−ς − 1 [ exp { (ς − 1) ∫ +∞ −∞ g(y) ln g(y)dy } − 1 ] ς > 0, ς ̸= 1 (5) [11] derived generalized entropy as ΞA1(y) = − ∫ +∞ −∞ g(y) ln [ g(y) δ ] dy; δ = Sup[g(y)] (6) [11] produced generalization of Renyi Entropy, ΞA2(y) = 1 ς − 1 ln [∫ +∞ −∞ [g(y)]ς δς−1 dy ] ς > 0, ς ̸= 1 (7) [7] derived the generalized of Havrda and Charvat Entropy, ΞA3(y) = 1 21−ς − 1 [∫ +∞ −∞ [g(y)]ς δς−1 dy − 1 ] ς > 0, ς ̸= 1 (8) [12] defined the generalized entropy as ΞT (y) = 1 ς − 1 [ 1− ∫ +∞ −∞ [g(y)]ςdy ] ς > 0, ς ̸= 1 (9) The aim of present study is to derive the entropy measures for Exponetiated Exponen- tial distribution and truncated Exponetiated Exponential distribution, and compared the relative loss of entropy under different parametric values. 3. Exponentiated Exponential Distribution (EED) and Truncated Exponentiated Exponential Distribution (TEED): Let ’ y ’ be a r.v has EED with β and γ are parameters having pdf and cdf as g(y;β, γ) = βγ ( 1− e−γy )β−1 e−γy (10) G(y;β, γ) = ( 1− e−γy )β (11) Let ’ z ’ be a r.v of TEED having cdf and pdf can be defined as: G(z; t, β, γ) = (1− e−γz) β (1− e−γt)β (12) and g(z; t, β, γ) = βγ (1− e−γz) β−1 e−γz (1− e−γt)β (13) J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 4 of 17 Suppose Ξn(y) and Ξn(z) are two corresponding entropies, the relative loss of entropy by using ’ y ’ otherthan ’ z ’ can be defined as: LΞn(y) = Ξn(y)− Ξn(z) Ξn(y) (14) 3.1 Entropies of EED and TEED: In this section, by following section 2.1, different entropies measures of EED and TEED are derived. Then, using these two distributions, the relative loss of entropy is obtained by using ’ y ’ instead of ’ z ’. (i) Shannon Entropy of EED and TEED is given as: Ξs(y) = − ∫ ∞ 0 g(y) ln g(y)dy ΞS(y) = − ∫ ∞ 0 βγ ( 1− e−γz )β−1 e−γz ln [ βγ ( 1− e−γz )β−1 e−γz ] dy Making transformation e−γz = w, finally Shannon entropy becomes, ΞS(y) = − log(βγ)− φ(1) + (1− β)φ(β) + βφ(β + 1) (15) Where φ(□) is a digamma function. The Shannon Entropy of TEED is as ΞS(z) = − ∫ ∞ 0 βγ (1− e−γz) β−1 e−γz (1− e−γt)β ln { βγ (1− e−γz) β−1 e−γz (1− e−γt)β } dy (16) After making transformation and a little simplification, one can get ΞS(z) = − lnβγ (1− e−γt)β − β (1− e−γt)β φ(β)− β (1− e−γt)β φ(β + 1) (17) Therefore, the final expression of Shannon entropy of TEED is ΞS(z) = − 1 (1− e−γt)β [− lnβ − βφ(β)− βφ(β + 1)− lnG(z; t, β, γ)] (18) (ii) The Renyi Entropy of ’ y ’ and ’ z ’ is as follow ΞR(y) = 1 1− ς log ∫ ∞ 0 [g(y)]ςdy After simplification, we get the Renyi Entropy for EED as J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 5 of 17 ΞR(y) = − log βγ + log β + logB(ς, ς(β − 1) + 1) 1− ς (19) the Renyi Entropy for TEED as ΞR(z) = log ( βγ (1− e−γt)β ) + log β + logB(ς, ς(β − 1) + 1) 1− ς (20) (iv) Tsallis Entropy of EED and TEED is given as: ΞT (y) = 1 1− ς [ 1− βγς−1B(ς, ς(β − 1) + 1) ] (21) ΞT (z) = 1 1− ς [ 1− βγς−1 (1− e−γt)β B(ς, ς(β − 1) + 1) ] (22) (v) The Havrda and Charvat Entropy of EED and TEED is given as: ΞHC(y) = 1 21−ς − 1 [ βςγς−1B(ς, ς(β − 1) + 1)− 1 ] (23) ΞHC(z) = 1 21−ς − 1 [ βςγς−1 (1− e−γt)βς B(ς, ς(β − 1) + 1)− 1 ] (24) (vi) Arimoto’s Entropy of EED and TEED is given as: ΞAR(y) = 1 (21−ς − 1) [{ (βςγ) 1 ς ς B ( 1 ς , (β − 1) ς + 1 )} − 1 ] (25) ΞAR(z) = 1 (21−ς − 1) [{ (βγ) 1 ς γ (1− e−γt) β ς B ( 1 ς , (β − 1) ς + 1 )}ς − 1 ] (26) (vii) Sharma and Mittal’s entropy of EED and TEED is given as: ΞSM (y) = 1 (21−ς − 1) [exp(ς − 1){− log(βγ)− φ(1) + (1− β)φ(β) + βφ(β + 1)} − 1] (27) ΞSM (z) = 1 (21−ς − 1) [exp(ς − 1){− log(βγ)− βφ(β) + βφ(β + 1) + lnG(z; t, β, γ)} − 1] (28) J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 6 of 17 (viii) Awad’s et al. entropy of EED and TEED is given as: ΞA1(y) = log(δ)− log(βγ) + φ(1)− (1− β)φ(β) + βφ(β + 1) (29) ΞA1(z) = log(δ) + 1 (1− e−γt)β [log(βγ) + βφ(β) + βφ(β + 1)− lnG(z; t, β, γ)] (30) (ix) Awad’s entropy of EED and TEED is given as: ΞA2(y) = 1 ς − 1 ln [ 1 δς − 1 { log(βγ) + log β + logB(ς, ς(β − 1) + 1) 1− ς }] (31) ΞA2(z) = 1 ς − 1 ln [ 1 δς − 1 { log ( βγ (1− e−γt)β ) + log β + logB(ς, ς(β − 1) + 1) 1− ς }] (32) (x) Awad’s Entropy of EED and TEED is given as: ΞA3(y) = 1 2ς−1 − 1 [ 1 δς − 1 { log(βγ) + log β + logB(ς, ς(β − 1) + 1) 1− ς }] (33) ΞA3(z) = 1 2ς−1 − 1 [ 1 δς − 1 { log ( βγ (1− e−γt)β ) + log β + logB(ς, ς(β − 1) + 1) 1− ς }] (34) 4. Results and Discussion In this section, the results of entropies using different parametric values of (β, γ, ς and t) are discussed. In table (1) and (2), the combination of different parametric values are to be chosen as (β =), (γ =) and (ς =). On the basis of these combinations of parametric values, the relative loss function is calculated. In table (3) and (4) illustrate that relative loss of all entropies for Exponnetiated Exponential and Trucated Expoenentiated Exponential distribution. The result reveal that with an increasing in the value of shape parameter t, the entropy measures show a decreasing behavior and same in the same of parameter t. Table (3) and (4) shows that as ’ t ’ increases, the Shannon entropy decreases, whereas with the increase of truncated parameter ’ t ’ decreases Renyi, Tsalli, Havdra and increases. In table (1) and (2), shows clear findings but requires deeper exploration of entropy behavior changes across different parameter adjustments. The conclusion needs to show how parameter t makes Shannon entropy grow since the study does not explain this pat- tern clearly. An examination of the parameter t and its effects on the PDF reveals how changes in t affect the distribution shape to influence uncertainty as measured by Shannon entropy. Examining this behavior in specific settings such as reliability tests and survival experiments will provide useful practical interpretation for the results. The additional J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 7 of 17 analysis would improve our interpretation of outcomes while showing exactly how t affects the entropy level. Table. 1 Entropy values for Exponentiated Exponential distribution using γ = 0.5, ς = 0.5 t ΞS(y) ΞR(y) ΞT (y) ΞHC(y) ΞAR(y) ΞSM (y) ΞA1(y) ΞA2(y) ΞA3(y) 0.2 1.657 1.963 2.643 2.924 3.745 4.856 3.928 4.665 5.093 0.4 1.342 1.874 2.316 2.782 3.548 4.623 3.109 4.762 5.006 0.6 1.137 1.509 2.078 2.663 3.337 4.098 3.095 3.817 4.298 0.8 0.793 1.148 1.940 2.516 3.286 3.827 3.100 2.629 2.647 1.0 0.649 0.746 1.877 1.732 2.647 2.573 2.645 2.109 2.194 1.5 0.385 0.274 1.574 1.504 2.554 2.674 2.193 1.846 2.167 2.0 0.174 0.203 1.483 1.184 2.108 2.105 1.746 1.239 1.738 Table. 2 Entropy values for truncated Exponentiated Exponential distribution using β = 0.5, γ = 0.5, ς = 0.5 t ΞS(z) ΞR(z) ΞT (z) ΞHC(z) ΞAR(z) ΞSM (z) ΞA1(z) ΞA2(z) ΞA3(z) 0.2 2.115 2.376 3.147 3.667 2.873 4.095 4.868 3.017 3.029 0.4 2.120 2.101 2.885 2.920 2.665 4.092 4.782 2.774 2.894 0.6 1.742 1.939 2.638 2.830 2.483 3.759 4.692 2.648 2.777 0.8 1.093 1.254 2.440 2.755 2.387 3.284 4.553 2.371 2.474 1.0 0.877 1.093 2.143 2.163 2.194 3.119 4.284 2.531 2.389 1.5 0.289 1.035 1.873 1.934 2.003 2.648 3.298 1.783 1.586 2.0 0.093 0.727 1.443 1.932 1.903 2.465 3.266 1.367 1.382 Table. 3 Entropy values for Exponentiated Exponential distribution using γ = 1, ς = 0.5 t ΞS(y) ΞR(y) ΞT (y) ΞHC(y) ΞAR(y) ΞSM (y) ΞA1(y) ΞA2(y) ΞA3(y) 0.2 1.657 1.966 2.678 2.967 3.775 4.886 3.968 4.625 5.396 0.4 1.342 1.874 2.416 2.682 3.548 4.623 3.759 4.592 5.106 0.6 1.137 1.489 2.383 2.563 3.437 4.098 3.355 3.817 4.598 0.8 0.593 1.133 2.001 2.416 3.386 3.827 3.180 2.399 2.511 1.0 0.449 0.564 1.977 1.432 2.447 2.573 2.954 2.299 2.135 1.5 0.225 0.255 1.774 1.274 2.354 2.674 2.138 1.743 2.163 2.0 0.134 0.225 1.383 1.214 2.158 2.105 1.746 1.295 1.908 J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 8 of 17 Table. 4 Entropy values for truncated Exponentiated Exponential distribution using β = 1, γ = 0.5, ς = 0.5 t ΞS(z) ΞR(z) ΞT (z) ΞHC(z) ΞAR(z) ΞSM (z) ΞA1(z) ΞA2(z) ΞA3(z) 0.2 2.875 2.576 3.147 3.467 2.874 4.198 4.848 3.298 3.328 0.4 2.120 2.101 2.885 2.920 2.835 4.000 4.782 2.981 2.894 0.6 1.772 1.939 2.638 2.995 2.383 3.399 4.392 2.859 2.782 0.8 1.871 1.254 2.440 2.834 2.297 3.194 4.153 1.671 2.445 1.0 0.763 1.093 2.143 2.163 s 2.138 3.023 4.004 1.531 2.348 1.5 0.588 1.035 1.873 1.934 2.107 2.578 3.549 1.295 1.993 2.0 0.210 0.727 1.443 1.932 1.903 2.480 3.103 1.609 1.758 Table. 5 Relative loss for β = 0.4, γ = 0.3, ς = 0.5 t ΞS(y) ΞR(y) ΞT (y) ΞHC(y) ΞAR(y) ΞSM (y) ΞA1(y) ΞA2(y) ΞA3(y) 0.2 -1.927 1.827 1.109 1.763 1.393 1.983 1.837 1.368 1.537 0.4 -1.812 1.667 0.983 1.348 1.225 1.683 1.636 1.183 1.793 0.6 -1.766 1.553 0.788 1.391 1.193 1.356 1.368 1.109 1.348 0.8 -1.598 1.428 0.639 1.164 0.683 1.227 1.311 0.924 1.039 1.0 -1.172 1.198 0.535 1.093 0.663 1.184 0.667 0.832 0.799 1.5 -0.411 0.873 0.283 0.673 0.274 0.378 0.394 0.735 0.676 2.0 -0.298 0.374 0.016 0.336 0.738 0.274 0.297 0.338 0.448 Table. 6 Relative loss for β = 0.8, γ = 0.7, ς = 0.25 t ΞS(z) ΞR(z) ΞT (z) ΞHC(z) ΞAR(z) ΞSM (z) ΞA1(z) ΞA2(z) ΞA3(z) 0.2 -1.209 1.387 1.092 1.009 1.293 2.784 2.564 2.831 2.009 0.4 -1.109 1.293 1.002 0.927 1.109 1.937 2.106 2.645 1.998 0.6 -1.100 1.104 0.919 0.683 1.038 1.648 1.834 2.194 1.646 0.8 -0.937 0.928 0.783 0.554 0.839 1.467 1.749 2.177 1.344 1.0 -0.910 0.904 0.615 0.309 0.378 1.392 1.664 1.748 0.978 1.5 -0.745 0.778 0.567 0.298 0.239 0.923 1.548 1.478 0.347 2.0 -0.329 0.276 0.275 0.222 0.196 0.451 1.063 1.390 0.227 Shannon Entropy of EED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 9 of 17 Shannon Entropy of TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 10 of 17 Figure. 1 Shannon Entropy of EED and TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 11 of 17 Figure. 2 Renyi Entropy of EED and TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 12 of 17 J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 13 of 17 Figure. 3 Tsalli Entropy of EED and TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 14 of 17 Figure. 4 Havrda and Charvat Entropy of EED and TEED Figure. 5 Arimoto’s Entropy of EED and TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 15 of 17 Figure. 6 Sharma and Mittal’s entropy of EED and TEED Figure. 7 Awad’s et al. entropy of EED and TEED J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 16 of 17 Figure. 8 Awad’s entropy of EED and TEED Figure. 9 Awad’s entropy of EED and TEED 5. Conclusion This research examines most entropy measures for EE and TEE models to show how well they represent distribution uncertainty. Our analysis uses Shannon entropy plus Rényi entropy and similar tools to inspect model parameter changes versus entropy results. Our tests show entropy measures behave correctly to detect how these step-like distributions work and how they handle specific data patterns. As the parameter values grow the model produces more uncertain results. Because its range is limited the truncated distribution shows distinct entropy behavior which expands our abilities to represent real-world mea- surements with specified upper and lower boundaries. These results benefit fields that require accurate modeling of uncertain data since reliability studies and information the- ory rely on precise knowledge of data uncertainty. Our research adds essential knowledge J. G. Dar, I. Sajjad, S. Tamboli / Eur. J. Pure Appl. Math, 18 (2) (2025), 5796 17 of 17 about entropy measurements for popular statistical distributions and creates new direc- tions for engineering biology machine learning and other related studies. Conflict of Interest: The authors declare that they have no Conflict of interest References [1] Claude E Shannon. A mathematical theory of communication. The Bell system technical journal, 27(3):379–423, 1948. [2] Alfonso Delgado-Bonal and Javier Martín-Torres. Human vision is determined based on information theory. Scientific reports, 6(1):36038, 2016. [3] Anthony Zador. Spikes: Exploring the neural code. Science, 277(5327):772–773, 1997. [4] Vaishali Manish Joshi and Javid Dar. Some results on mathai-haubold fuzzy entropy. 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