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In reliability analysis, an investigator is often interested in the assessment of a specific risk in presence of other risk factors. It is well known as the competing risks problem in the statistical literature. In this paper, we consider the analysis of incomplete data in the presence of competing risks among several groups. We mainly consider the latent failure times model formulation, and it is assumed that the lifetime distributions of the different latent failure times of a particular group follow Weibull distributions with different scale parameters, but the same shape parameter. Maximum likelihood estimators of the different parameters are obtained using a simple iterative procedure, and also by EM algorithm. Asymptotic distributions of the maximum likelihood estimators of the different parameters are obtained, and based on the asymptotic distributions, asymptotic confidence intervals are also proposed. Testing equality of the parameters among several groups is performed. One data set has been analysed for illustrative purposes.