Frechet - Cramer -Rao lower bound on variance is a land mark in the history of Statistics. Obtaining uniformly minimum variance unbiased estimator via lower bound involves the problem of construction of bound and study Kiefer bound is the best lower bound of this type. Its computation involves complications. Therefore, its applications were restricted. In this book Kiefer bound is computed for parameters and some parametric functions in various truncated families of distributions. This book talks about the attainment of Kiefer bound. The natural forms of truncated densities are introduced. The explicit forms of parametric functions, their uniformly minimum variance unbiased estimators and their variances which attain Kiefer bound (UMVUKBE) are obtained. Estimation of any parametric function in truncated families is considered. Expression for its estimator, Estimator of its variance etc. are provided. The magnitudes of Kiefer bound are compared with other bounds. Kiefer bound for Complete and censored samples are considered. References are provided. A brief introduction to Kiefer is provided in an Appendix.