Fractional cumulative residual Kullback-Leibler information based on Tsallis entropy

Abstract

• Traditional cumulative residual Kullback Leibler information is extended to fractional orders by combining it with Tsallis entropy, called fractional CRKL. • Some properties of the proposed measurement are studied and proved. • The proposed CRKL can be estimated by generalized Fisher information. • Discrete fractional CRKL is defined for calculation. • It is applied to financial time series to detect the dissimilarities between different stock indices and to identify the significant events in specific periods. The cumulative residual Kullback-Leibler information was recently proposed as a suitable generalization of Kullback-Leibler information to the survival function. In this paper, we extend the traditional cumulative residual Kullback-Leibler information to fractional orders by combining it with Tsallis entropy, called fractional CRKL. Some properties of the proposed measure are studied and proved. It can be estimated by generalized Fisher information. In addition, we also define discrete fractional CRKL for calculation. Some distributions are enumerated to verify the validity of the new measure. Finally, it is applied to financial time series to detect the dissimilarities between different stock indices and to identify the significant events in specific periods.