Cumulative Residual Fisher Information Based on Finite and Infinite Mixture Models

In this work, we extend the concept of Cumulative Residual Fisher (CRF) information through finite and infinite mixture survival functions. We first examine this information measure for infinite mixture survival functions and introduce the Jensen-cumulative residual Fisher (Jensen-CRF) information for these infinite mixture models. We show that this Jensen-type information measure can be expressed via mixture of an infinite number of relative cumulative residual information measures. We then propose the generalized cumulative residual Fisher (generalized CRF) information, emphasizing its convexity property, supported by a key lemma ensuring this trait. Next, we consider the q-cumulative residual Fisher (q-CRF) information and investigate its convexity property through a significant lemma. Further, we define the Jensen-q-cumulative residual Fisher (Jensen-q-CRF) divergence and relative q-cumulative residual Fisher (relative q-CRF) information in the case of a finite mixture model. Finally, we apply the Jensen-q-CRF divergence to image processing. Our numerical results related to image quality assessment show that the Jensen-q-CRF divergence measure effectively quantifies image similarity.