Nested Repeated Measures Model Sufficiency and Estimation

A nested repeated measures model occurs in analysis of variance when a particular individual has the same number of subindividuals and each subindividual receives every pair of treatment levels. Let Y_i = (Y_illl,...,Y_imrc)’ be the vector of observations on the ith individual. It is assumed that the Y_i are independently normally distributed with mean μ_i and common covariance ∑＞0. The simplifying assumptions that all measurements have the same variance σ², every pair of measurements that comes from different subindividuals but the same individual has covariance σ²ρ₁; and every pair of measurements that comes from the same individual and same subindividual with (i) different columns and different rows treatments, (ii) the same column but different rows treatments, (iii) different columns but the same row treatments have σ²ρ₂,σ²ρ₃, σ²ρ₄ as their respective covariances is made. We assume that the design is given and use a coordinate-free approach to derive a complete sufficient statistic, minimum variance unbiased estimators (MVUE) and maximum likelihood estimators (MLE) for the nested repeated measures model.