In today’s world, we are faced with situations everyday where Statistics can be applied. In general, Statistics is the science of collecting, organizing, and analyzing numerical data. The techniques involved in Statistics are important for the work of many professions, thus the proper preparation and theoretical background of Statistics is valuable for many successful career paths. Marketing campaigns, the realm of gambling, professional sports, the world of business and economics, the political domain, education, and forecasting future occurrences are all areas which fundamentally rely on the use of Statistics. Statistics is a broad subject that branches off into several categories. In particular, Inferential Statistics contains two central topics: estimation theory and hypothesis testing. The goal of estimation theory is to arrive at an estimator of a parameter that can be implemented into one’s research. In order to achieve this estimator, statisticians must first determine a model that incorporates the process being studied. Once the model is determined, statisticians must find any limitations placed upon an estimator. These limitations can be found through the Cramer-Rao lower bound. Under smoothness conditions, the Cramer-Rao lower bound gives a formula for the lower bound on the variance of an unbiased estimator. Once the estimator is developed, it is tested against the limitations to see if it is valid relative to the model. Lastly, experiments are run using the estimator to test performance. From real data, statisticians are able to decide whether the estimator is incorrect, and in this case, they can go back and find a new estimator. It is important for an estimator to achieve a minimum average error (i.e. minimum variance unbiased estimator). This type of estimator is known to be an efficient estimator because the average error measure is the variance. Other performance measures for estimators include: bias and consistency. An estimator is said...

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