the Fisher information into results in Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér–Rao lower bound for all values of and . In other words, the sample mean is the (necessarily unique) efficient estimator, and thus also the minimum variance unbiased estimator (MVUE), in addition to being the maximum likelihood estimator. Maximum of a uniform distribution One of the simplest non-trivial examples of estimation is the estimation of the maximum of a uniform distribution. It is used as a hands-on classroom exercise and to illustrate basic principles of estimation theory. Further, in the case of estimation based on a single sample, it demonstrates philosophical issues and possible misunderstandings in the use of maximum likelihood estimators and likelihood functions. Given a discrete uniform distribution with unknown maximum, the UMVU estimator for the maximum is given by where m is the sample maximum and k is the sample size, sampling without replacement. This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II. The formula may be understood intuitively as; "The sample maximum plus the average gap between observations in the sample", the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum. This has a variance of so a standard deviation of approximately , the (population) average size of a gap between samples; compare above. This can be seen as a very simple case of maximum spacing estimation. The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased. Applications Numerous fields require the use of estimation theory. Some of these fields include: Interpretation of scientific experiments Signal processing Clinical trials Opinion polls Quality control Telecommunications Project management Software engineering Control theory (in particular Adaptive control) Network intrusion detection system Orbit determination Measured data are likely to be subject to noise or uncertainty and it is through statistical probability that optimal solutions are sought to extract as much information from the data as possible. See also Best linear unbiased estimator (BLUE) Completeness (statistics) Detection theory Efficiency (statistics) Expectation-maximization algorithm (EM algorithm) Fermi problem Grey box model Information theory Least-squares spectral analysis Matched filter Maximum entropy spectral estimation Nuisance parameter Parametric equation Pareto principle Rule of