The sampling variances of estimators of variance cnmponents in the basic mixed linear model are derived for an approximate approach to the minimum variance quadratic unbiased estimation. The assumption made with the basic model is that the variance-covariance matrix for the ith random factor has the form Iσ* where I is an identity matrix of order equal to the number of levels for the factor and σ* is the variance component to be estimated. Assuming a two-way crossclassified model including fixed herd and random sire and residual effects in applied animal breeding and considering a certain unbalanced data structure, the efficiencies of estimators of variances of the two random effects by the approximate method, relative to the minimum variance quadratic unbiased estimators which are the best estimators, are numerically investigated within the parameter space of heritability which can be expressed as a function of the ratio of residual variance to sire variance and with choosing its prior values. The sampling variance of the estimator of sire variance with the approximate method is shown to be not minimized by choosing the prior value of heritability to be the true value, as is not the case with the minimum variance quadratic unbiased estimation. For a given data structure, with the approximate method, it is found that using a constant prior value of moderate magnitude for heritability gives the efficiencies of the estimator of sire variance higher than. 8 over the given range of the parameter.
We consider the linear measurement-error model in which both the dependent variable, y, and the independent variable, x, are subject to measurement errors. We propose a compromised estimator of the slope paramater. The estimated line lies between the two extremes: the regression line of y on x and that of x on y With normally distributed errors, we show that this estimator has lower moments. The approximate expectation, variance and mean spuared error are obtained by Taylor series expansion.
The theories of the Poisson process was applied to construct a pure theoretical model describing the diameter distribution of trees in selection forest stands. The author derived a diameter distribution function for selection forests under several assumptions on the cutting probability. The applicability of the derived function to the actual distribution was demonstrated using nine observed diameter distribution data from permanent sample plots of Ate (Thujopsis dolabrata SIEB. et ZUCC var. hondai MAKINO) selection forests at Noto district, Ishikawa. The derived function agreed well with the observations.