This paper is conserned with the statistical problem of estimating the failure rates of a populations exposed to several simultaneous hazards or risks. Two distinct models are disscussed: (a) a model that is appropriate when the risk-specific hazard rates are linear functions of time. The paper is divided into four parts. The first section gives the background needed for the development of these two competing-risk models. The second section treats the constant case and the third sections treats the linear case. In the last section a numerical example is presented and the parameteer-estimation problem illustrated. For both models the method of finding the maximum-likelihood estimators is developed. The linear-model situation includes a discussion of how a nonlinear programming procedure can be used to obtain the parameter estimates. The paper ends with an investigation of the statistical properties of the estimated parameter using a suitable simulation technique.