A case-cohort design was recently proposed [Prentice (1986)] as a means of reducing cost in large epidemiologic cohort studies. A "pseudolikelihood" procedure was described for relative risk regression parameter estimation. This procedure involves covariate data only on subjects who develop disease and on a random subset of the entire cohort. In contrast, the usual partial likelihood estimation procedure requires covariate histories on the entire cohort. Accordingly, a case-cohort design may affect cost saving, particularly with large cohorts and infrequent disease occurrence. Asymptotic distribution theory for such pseudolikelihood estimators, along with that for corresponding cumulative failure rate estimators, are presented here. Certain asymptotic efficiency expressions relative to full-cohort estimators are developed and tabulated in situations of relevance to the design of large-scale disease prevention trials. The theoretical developments make use of martingale convergence results and finite population convergence results.