Bootstrap Confidence Intervals for Variance Components in the Unbalanced Random One –Way Model

The problem of constructing confidence intervals for the ratio of variance components in unbalanced random one-way model is to investigate. In this respect, various different exact methods, which are introduced, by Wald (1947), Bross (1950), Tukey (1951), Anderson, and Bancroft (1952) are used. However, these methods are rather difficult to compute since it involves the solution of non-linear equations. So several approximate methods, which are easier to compute, are discussed. These methods were introduced by Satterthwait (1946), Morigiti (1954), Bulmer (1957), Thomas and Hultiquist (1978), El-Bassiouni (1978), and El-Ganzouri (1986). In this search a Monte Carlo simulation study is conducted to examine the performance of the above mentioned methods. Recently, new alternative methods namely Bootstrap are also, given and included in comparison. Simulation depends on the sizes of the samples and that the basis of comparison are constructed based on the coverage values and average length. After damaged the results of the coverage values and average length are discussed. In design (1) the best method is (SAM) method, because it has appropriate coverage value with a small value of average length for all values of ρ, this method is from bootstrap methods. In design (2) and design (5) the best method is (Wald) method, because it has high coverage with a small value average length with all values of ρ, this method from exact methods. In design (3) and design (4) the best method is (Wald) for small values of ρ, ρ≤0.1and it is an exact method. However, for large values of ρ, ρ>0.1 the best method is (SAM) method and it is a bootstrap method. In applied case sugar -cane experiment the results of limits, the A.ADJ has the biggest upper limit and, (BB) method has the smallest upper limit. While the (A) method has the highest lower limit, where the bootstrap methods (BB), (Bt), and (SAM) have the smallest lower limits. However, for average length, the best method that has the smallest average length, and the best method is (BB) MAETHOD. In addition to. The another applied case wheat experiment the results of limits, the (Bt ) has the biggest upper limit  and Morigiti and Bulmer have the smallest upper limit .While ,Bross has the highest lower limit ,Where the And Ban has the smallest lower limit. However, for average length, the best method that has the smallest average length, and the best method is Bross method.

Keywords: Unbalanced One-Way Random Effect Model; Interval Estimation; Variance Components; Bootstrap Confidence Interval; Quadratic Estimators; Monte Carlo Simulation