The article by R. A. Fisher, titled "On the Mathematical Foundations of Theoretical Statistics," discusses the theoretical aspects of statistical methods and their applications. Fisher addresses the neglect of theoretical statistics, highlighting the need for precise definitions and clear formulations of statistical problems. He outlines three main types of statistical problems: specification, estimation, and distribution. Fisher emphasizes the importance of defining the task of the statistician, which is to reduce a large amount of data to a few quantities that represent the whole or most of the relevant information. Fisher introduces several key concepts, including consistency, efficiency, and sufficiency. Consistency is defined as the criterion that a statistic should satisfy when calculated from the whole population. Efficiency is measured by the ratio of the intrinsic accuracy of a statistic to that of the most efficient statistic possible. Sufficiency is the criterion that a statistic should meet when no other statistic derived from the same sample provides additional information about the parameter being estimated. Fisher also discusses the method of maximum likelihood, which involves choosing the set of parameter values that maximize the likelihood function. He contrasts this method with Bayes' theorem and inverse probability, criticizing their arbitrary assumptions and lack of rigorous mathematical foundation. Fisher provides examples to illustrate the application of these concepts, such as the correction for grouped data and the method of moments. The article concludes with a detailed discussion of the method of maximum likelihood, emphasizing its practical utility and mathematical rigor. Fisher argues that this method provides a more reliable approach to estimating parameters from samples compared to older methods based on inverse probability.The article by R. A. Fisher, titled "On the Mathematical Foundations of Theoretical Statistics," discusses the theoretical aspects of statistical methods and their applications. Fisher addresses the neglect of theoretical statistics, highlighting the need for precise definitions and clear formulations of statistical problems. He outlines three main types of statistical problems: specification, estimation, and distribution. Fisher emphasizes the importance of defining the task of the statistician, which is to reduce a large amount of data to a few quantities that represent the whole or most of the relevant information. Fisher introduces several key concepts, including consistency, efficiency, and sufficiency. Consistency is defined as the criterion that a statistic should satisfy when calculated from the whole population. Efficiency is measured by the ratio of the intrinsic accuracy of a statistic to that of the most efficient statistic possible. Sufficiency is the criterion that a statistic should meet when no other statistic derived from the same sample provides additional information about the parameter being estimated. Fisher also discusses the method of maximum likelihood, which involves choosing the set of parameter values that maximize the likelihood function. He contrasts this method with Bayes' theorem and inverse probability, criticizing their arbitrary assumptions and lack of rigorous mathematical foundation. Fisher provides examples to illustrate the application of these concepts, such as the correction for grouped data and the method of moments. The article concludes with a detailed discussion of the method of maximum likelihood, emphasizing its practical utility and mathematical rigor. Fisher argues that this method provides a more reliable approach to estimating parameters from samples compared to older methods based on inverse probability.