Nonparametric Estimation of the Ratio of Variance Components

Consider the one-way random effects model Z _ij = µ + α _i + ε _ij , i = 1,2,…, m, j = 1,2,…, n, where α _i and ε _ij are mutually independent and distributed with mean zero and variance σ_α2 and σ_ε2, respectively. An estimation procedure is developed for estimating $$\rho ^2 = \sigma _\alpha ^2 /\sigma _\varepsilon ^2$$, the ratio of variance components, using a method similar to that proposed by Shorack (1969) for estimating the ratio of two scale parameters. An extensive Monte Carlo study is carried out in order to investigate the performance of the proposed method when compared with the other two methods provided by Groggel, Wackerly and Rao (1987) and Bhattacharyya (1977). The study suggests that the proposed method is superior to those two alternative methods for certain ranges of the model parameters. Finally, we propose a small-sample adjustment which improves the performance of the estimation method.