Let ( y 1 i , y 2 i ) , i 1 , , N , have a bivariate normal distribution with mean ( 1 , 2 ) and covariance parameters ( 11 , 12 , 22 ) and correlation coefficient Suppose that all N observations on y 1 are available but there are m N missing observations on y 2 Using the fact that the marginal dis...

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Let ( y 1 i , y 2 i ) , i = 1 , ,

Question:

Let $(y_{1 i}, y_{2 i}), i = 1, \dots, N$ , have a bivariate normal distribution with mean $(μ_{1}, μ_{2})$ and covariance parameters $(σ_{11}, σ_{12}, σ_{22})$ and correlation coefficient $ρ$ . Suppose that all $N$ observations on $y_{1}$ are available but there are $m < N$ missing observations on $y_{2}$ . Using the fact that the marginal distribution of $y_{j}$ is $N [μ_{j}, σ_{j j}]$ , and that conditionally $y_{2} ∣ y_{1} \sim N [μ_{2.1}, σ_{22.1}]$ , where $μ_{2.1} = μ_{2} +$ $σ_{12} / σ_{22} (y_{1} - μ_{1}), σ_{22.1} = (1 - ρ^{2}) σ_{22}$ , devise an EM algorithm for imputing the missing observations on $y_{1}$ .