On identifiability and information-regularity in parametrized normal distributions

We describe methods to establish identifiability and information-regularity of parameters in normal distributions. Parameters are considered identifiable when they are determined uniquely by the probability distribution and they are information-regular when their Fisher information matrix is full rank. In normal distributions, information-regularity implies local identifiability, but the converse is not always true. Using the theory of holomorphic mappings, we show when the converse is true, allowing information-regularity to be established without having to explicitly compute the information matrix. Some examples are given.