The objective of this study was to investigate genetic merit of return over feed (ROF), which is a herd profit index defined by CanWest Dairy Herd Improvement as a difference between milk income and feed cost. A multiple-trait (MT) model and random regression model (RRM) were used. The traits analyzed in MT were rearing cost and ROF of the first three lactations. In RRM, a cumulative ROF was fitted as function of age and rearing cost was treated as a correlated trait. Variance components were estimated within a Bayesian framework by Gibbs sampling using a subsample of data. Breeding values were then estimated for 3 041 078 animals using records of 1 951 893 cows. Estimates of heritability for rearing cost from MT and RRM were 0.23 and 0.22, respectively. ROF per lactation and cumulative ROF were negatively correlated with rearing cost. Estimates of heritability of ROF through the first, second and third lactation from MT were 0.27, 0.10 and 0.08, respectively. Estimates of heritability of ROF from RRM increased with age and ranged from 0.08 through 0.31. Estimated breeding values (EBVs) for ROF from MT and RRM were moderately correlated with official EBV for production traits and the Canadian selection index (Lifetime Profit Index). Herd life EBV had −0.07 and 0.19 correlations with EBVs for ROF from MT and RRM, respectively. From both MT and RRM, small favorable correlations were reported between EBVs for ROF and for bone quality and angularity, whereas low unfavorable correlations were reported with EBV for udder depth, front end and chest width. Majority of correlations between EBVs for ROF and for reproduction traits were near 0, with the exception of EBV for gestation length, calf size and calving ease, where small favorable correlations were reported. The ROF is a good indicator of cow profitability despite the fact that it is a simplified profit index that does not account for animal-specific health and reproductive cost. However, because ROF does not account for differences in heritabilities between components of profit, ROF is not recommended to be used for direct selection for profit.

The quality of kinematic navigation and positioning depends on the quality of the model describing the vehicle movements and the reliability of the observations. An adaptive Kalman filtering is introduced. Three kinds of adaptive factors based on the discrepancy between the geometrical positions and the kinematic model predictions and a variance component ratio between model predictions and observations are described. A new exponential adaptive factor is established. The theoretical curves of the adaptive factors are drawn and a practical example is given. The errors of four adaptive filtering results and the corresponding curves of the adaptive factors are also drawn. It is shown, by comparison and analysis, that all of the four adaptive factors can control the influences of the vehicle disturbances in movements on the navigation results. The results derived by the adaptive factor constructed by the variance component ratio are slightly better than those derived by other adaptive factors.

Generalised k-statistics associated with multi-indexed arrays of random variables satisfying a generalised form of exchangeability are studied. By showing that they form multi-indexed reversed martingales and that the associated family of σ-fields possesses certain conditional independence properties, conditions for the a.s. convergence of generalised k-statistics are obtained. When the arrays of random variables are sums of independent arrays of independent effects, as is the case with the standard random effects anova models, the limits are identified as the associated generalixed cumulants.