Weighted maximum likelihood estimation for individual growth models

We apply a class of stochastic differential equations to model individual growth in a randomly fluctuating environment using cattle weight data. We have used maximum likelihood theory to estimate the parameters. However, for cattle data, it is often not feasible to obtain animal's observations at equally spaced ages nor even at the same ages for different animals and there is typically a small number of observations at older ages. For these reasons, maximum likelihood estimates can be quite inaccurate, being interesting to consider in the likelihood function a weight function associated to the elapsed times between two consecutive observations of each animal, which results in the weighted maximum likelihood method. We compare the results obtained from both methods in several data structures and conclude that the weighted maximum likelihood improves the estimation when observations at older ages are scarce and the observation instants are unequally spaced, whereas the maximum likelihood estimates are recommended when animals are weighted at equally spaced ages. For unequally spaced observations, a bootstrap estimation method was also applied to correct the bias of the maximum likelihood estimates; it revealed to be a more precise alternative, except when the available data only has young animals.

Automated summary

We used a class of stochastic differential equations to model individual growth in a randomly fluctuating environment using cattle weight data. Maximum likelihood theory was applied to estimate the parameters, but for cattle data, it is often difficult to obtain observations at equally spaced ages or even at the same ages for different animals, leading to inaccurate maximum likelihood estimates. To improve the estimation, we introduced a weight function associated with the elapsed times between observations of each animal in the likelihood function, resulting in the weighted maximum likelihood method. Comparing the results from both methods in different data structures, we found that the weighted maximum likelihood method performs better when there are few observations at older ages and the observation instants are unequally spaced. For unequally spaced observations, a bootstrap estimation method was also applied and proved to be a more precise alternative, except when the available data only includes young animals.