Estimating the Selection Gradient of a Function-valued Trait

Abstract

Kirkpatrick and Heckman initiated the study of function-valued traits in 1989. How to estimate the selection gradient of a function-valued trait is a major question asked by evolutionary biologists. In this dissertation, we give an explicit expansion of the selection gradient and construct estimators based on two different samples: one consisting of independent organisms (the independent case), and the other consisting of independent families of equally related organisms (the dependent case). In the independent case we first construct and prove the joint consistency of sieve estimators of the mean and covariance functions of a Gaussian process, based on previous developments by Beder. From this we prove the consistency of the estimator of the selection gradient. This is supported by simulations. Using this estimator of the selection gradient, the estimated between-generation change in the mean phenotype is shown in simulations to be consistent. In the dependent case we are able to estimate both the phenotypic and the genetic covariance functions. Simulations indicate consistency of these estimators, but appear not to support the consistency of the estimator of the selection gradient, nor of the estimator of the between-generation change in the mean phenotype. A probable source of this problem is identified.