Optimal designs for some stochastic processes whose covariance is a function of the mean

This paper considers optimal experimental designs for models with correlated observations through a covariance function depending on the magnitude of the responses. This suggests the use of stochastic processes whose covariance structure is a function of the mean. Covariance functions must be positive definite. This fact is nontrivial in this context and constitutes one of the challenges of the present paper. We show that there exists a huge class of functions that, composed with the mean of the process in some way, preserves positive definiteness and can be used for the purposes of modeling and computing optimal designs in more realistic situations. We offer some examples for an easy construction of such covariances and then study the problem of locally D-optimal designs through an illustrative example as well as a real radiation retention model in the human body. Copyright Sociedad de Estadística e Investigación Operativa 2013