For a one-parameter exponential family of distributions, a method to find the uniformly minimum variance unbiased (UMVU) estimator based on the complete sufficient statistic is given in Jani and Dave [1] by change of the expression of the unbiasedness condition. But, it heavily depends on the concrete form of the distribution of the statistic in obtaining indeed the UMVU estimator. In this paper, from the different point of view, the construction of the UMVU estimator for a one-parameter exponential families of distributions and certain two-parameter family of distributions is discussed. Some examples are also given.