Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case

In this paper, we assume the existence and consistency of the maximum likelihood estimate (MLE) in the independent not identically distributed (i.n.i.d.) case and we establish its asymptotic normality. The regularity conditions employed do not involve the third order derivatives of the underlying probability density functions (p.d.f.'s). This research was supported by the National Science Foundation, Grant GR-20036, and the Office of Scientific Research and Development of the Greek Government.