Mathematics · Estimation Theory

Let (X 1 , Y 1 ), ..., (X n , Y n ), n ≥ 4 be a random sample from the bivariate normal distribution with E(X 1 ) = μ 1 , E(Y 1 ) = μ 2 , Var(X 1 ) = Var(Y 1 ) = σ 2 and the correlation coefficient between X 1 and Y 1 equal to ρ. All four parameters μ 1 , μ 2 , σ 2 and ρ are unknown. Let S xx = ∑ i=1 n (X i - X̄) 2 , S yy = ∑ i=1 n (Y i - Ȳ) 2 , and S xy = ∑ i=1 n (X i - X̄)(Y i - Ȳ). Which of the following is the maximum likelihood estimator of ρ?