Maximum Likelihood Estimation of Bivariate Normal Correlation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Multivariate Estimation.

Your Task
Let $(X_1, Y_1), \dots, (X_n, Y_n)$ be i.i.d. draws from a bivariate normal distribution with zero means ($\mu_X = \mu_Y = 0$), known unit variances ($\sigma_X^2 = \sigma_Y^2 = 1$), and unknown correlation coefficient $\rho \in (-1, 1)$.
(1) Write down the exact likelihood function $L(\rho)$.
(2) Derive the score function and the cubic polynomial equation for the MLE $\hat{\rho}$.
(3) Compute the Fisher Information $\mathcal{I}(\rho)$ and the asymptotic variance of $\hat{\rho}$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.