Numerical Methods for Estimating Correlation Coefficient of Trivariate Gaussian

. Abstract. Given observed data, the fundamental task of statistical inference is to understand the underlying data-generating mechanism. This task usually entails several steps, including determining a good family of probability distributions that could have given rise to the observed data, and identifying the specific distribution from that family that best fits the data. The second step is usually called parameter estimation, where the parameters are what determines the specific distribution. In many instances, however, estimating parameters of a statistical model poses a significant challenge for statistical inference. Currently, there are many standard optimization methods used for estimating parameters, including numerical approximations such as the Newton-Raphson method. However, they may fail to find a correct set of maximum values of the function and draw incorrect conclusions, since their performance depends on both the geometry of the function and location of the starting point for the approximation. An alternative approach, used in the field of algebraic statistics, involves numerical approximations of the roots of the critical equations by the method of numerical algebraic geometry. This method is used to find all critical points of a function, before choosing the maximum value(s). In this paper, we focus on estimating correlation coe cients for multivariate normal random vectors when the mean is known. The bivariate case was solved in 2000 by Small, Wang and Yang, who emphasize the problem of multiple critical points of the likelihood function. The goal of this paper is to consider the first generalization of their work to the trivariate case, and o↵er a computational study using both numerical approaches to find the global maximum value of the likelihood function.