Properties of the maximum qq-likelihood estimator for independent random variables

In the last two decades, the nonextensive statistics proposed by Tsallis have been extensively discussed in terms of the Tsallis entropy, which is a generalization of the Boltzmann–Gibbs–Shannon entropy. Within the nonextensive framework, many kinds of generalizations have been made. Suyari [H. Suyari, IEEE Trans. on Inf. Theory, 51 (2005) 753] has proposed the generalized likelihood called the qq-likelihood, in which a traditional product operator is replaced by the qq-product. We study properties of the maximum qq-likelihood estimator (MqLE) which is a generalization of the conventional maximum likelihood estimator with the use of the qq-likelihood. We discuss MqLE from the viewpoint of the minimization of the divergences in the nonextensive statistics. It has been shown that optimum parameters determined by the MqLE are nearly in agreement with those yielding the minimum distance of the divergence proposed by Rajagopal [A.K. Rajagopal, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and Its Applications, Springer, 2000, p. 99]. We here show that the consistency of MqLE is achieved along with the non-negativity of Rajagopal’s divergence. The asymptotic Gaussianity and robustness of the MqLE for independent random variables are discussed both by analytical methods and simulations.