F. Variance Component Models

2. In the context of a balanced one-way random effect model where the εij’s are N = nk i.i.d. N(0, σ), εij − εi. and εi′. − ε.. are independent for all choices of i, j, and i′. Proof: It suffices to show that cov(εij − εi., εi′.− ε..) = 0 for all i, i′, and j due to normality. case 1 : i = i′. cov(εij − εi., εi. − ε..) = cov(εij, εi.) − cov(εij, ε..) − cov(εi., εi.) + cov(εi., ε..) = σ/n− σ/nk − σ/n+ nσ/nk = 0. case 2 : i 6= i′. cov(εij − εi., εi′. − ε..) = cov(εij, εi′.) − cov(εij, ε..) − cov(εi., εi′.) + cov(εi., ε..) = 0− σ/nk − 0 + nσ/(nk · n) = 0