Characteristics of Convergence Curve of the PE Estimate of Variance Ratio in the Balanced Case

摘要

The pseudoexpectation (PE) methods are an iterative procedure for estimating variance components in the general mixed linear model. The computational loads in the PE methods are relatively small and the estimates converge faster, although the methodsare an approximation to the restricted maximum likelihood (REML) procedure. In the field of animal breeding, it has been considered that the PE methods can be used in estimating from large data sets including no or weak influence of selection, or in obtaining initial values to use in the REML estimation. This paper reports some basic characteristics of the convergence curve of the PE variance ratio estimate in the case of balanced data. In the specific balanced case, the PE and REML estimates are identical, but their convergence curves are different. The ratios of the difference between the consecutive estimates in the z'th and the i— 7th rounds to that in the i + 7th and the ith rounds vary, for instance, in the REML estimation using the expectation-maximization algorithm, while those with the PE methods take a constant value regardless of initial values and rounds of iteration. Hence, the PE curve of convergence can be exactly represented using the result for a geometric series. Consequently, the PE curve is equivalent to the curve assumed underlyingly in the extrapolation techniques, or the techniques of directly predicting the final value such as the Aitken extrapolation and the common intercept approach. In the unbalanced case, the property ofthis kind of the PE curve is no longer retained, and the degrees of discrepancies among the PE and the extrapolation estimates would be dependent on the structure of the data to be analyzed. When one wishes to accelerate convergence in the REML analysisof unbalanced data, it is important to notice that the PE and the extrapolation techniques each have the advantages and disadvantages for the purpose.