Recent Advances in Entropy

Abstract

The main focus of the study is to compare the different entropy measures using exponential and size biased moment exponential distribution. Three different types of entropies have been compared. These entropies are Residual & Past entropies, α order entropies and α,β order entropies. These entropies have been compared mathematically, numerically and graphically. The generalized α order entropies and α,β order entropies for exponential distribution and SBM exponential distribution derived in chapter 3. Residual & Past entropies also derived in this chapter for both distributions. A new entropy has been derived using the hazard rate of the distribution. New entropy (HN1) produced the same result as Shannon’s 1948 entropy produces. This entropy fulfills the additive property of the Shannon entropy. HN1 is an alternate of the Shannon’s entropy. By extending the idea of hazard rate function, three new α order entropies have been derived. These entropies HN2, HN3 and HN4 produces the results same as Renyi’s (1961) entropy, Havrda & Charvat (1967) and Tsallis (1988) entropies respectively. This indicates that hazard rate is an information function. Another generalized α, β order entropy A1 has been derived. This is more generalized form of the entropy. Renyi (1961), Sunoj & Linu (2010), Rao et al. (2004) and Shannon (1948) entropies are the special cases of A1 entropy for different values of α and β. Residual and past entropies also derived for both distributions and compared. There are two new residual entropies A2, A4 and one past entropy A3 has been derived. These three entropies are also in generalized form. This will produce number of residual entropies and past entropies using different values of α, β. A numerical study has been conducted for the comparison. Awad & Alwaneh (1987) introduced relative loss for the comparison of entropies. In this study relative loss of all entropies have been derived and compared. The result of relative loss is negative for majority of α order entropies. This shows that entropy of SBME distribution is higher as compare to the exponential distribution except Awad et al. (1987) entropy measures. Awad et al. (1987) entropies show the positive result which implies that exponential distribution has higher entropy as compare to others. Residual & past entropy also shows the negative results of relative loss and indicates that SBME distribution has higher entropy as compared to others. Statistical interpretation of the entropy and relative loss is not so easy. Higher entropy of the distribution concludes that there is high randomness in the distribution. Graphical trend of the entropies shows the exponential decay for both distributions. The trend of new entropy A3 is exponentially increasing instead of the decreasing. The concluding remarks of the study is in favor of the new entropies. As new entropies produced the results same as old entropy measures. There are some important characteristics of the new entropies. In the hazard rate, denominator will have replaced another probability density function or distribution function, this will produce the comparison method of entropies. When the hazard rate replaced with the conditional probability density function, this is another comparison method for entropies. The derivation of new entropies is difficult if the hazard rate has complicated expression.