Kernel Estimation of Tsallis Entropy and its Generalization for Length-biased Data

A new generalized Shannon entropy is Tsallis entropy. The Shannon entropy is additive and the Tsallis entropy is, however, non-additive. Due to the flexibility of the Tsallis entropy compared to the Shannon entropy, the non-additive entropy measures find their justification in many areas. In this paper, we propose two non- parametric kernel estimators for the Tsallis entropy and two non-parametric kernel estimators for the residual Tsallis entropy for the length-biased data. We investigate some asymptotic properties for these estimators such as the consistency and asymptotic normality. We obtain the bias, variance and the mean integrated squared error (MISE) of estimators. We also compare the behaviour of proposed estimators using the Monte Carlo simulation and plot some figures to see how close the fitted distribution is to the histogram of the data. In the end, we use a real dataset to show the performance of the proposed estimators.