Essay: On the Mathematical Foundations of Theoretical Statistics

Overview
Ronald A. Fisher's 1922 essay "On the Mathematical Foundations of Theoretical Statistics" set out a sharply articulated framework for statistical inference that reshaped the field. It argued for treating statistical problems through parametric models and for grounding estimation and hypothesis procedures in properties that could be evaluated mathematically. The essay moved statistical thought away from ad hoc calculational recipes toward a coherent set of principles centered on information, likelihood, and the behavior of estimators.

Core ideas
Central to the exposition is the likelihood function as the primary vehicle for learning about parameters from data. Fisher presented likelihood not merely as a computational device but as the quantity whose maximization yields estimators with attractive large-sample and practical properties. He introduced and emphasized three desiderata for estimators: consistency, meaning convergence to the true parameter as sample size grows; efficiency, measured by a small variance relative to other unbiased estimators; and sufficiency, the idea that a statistic can capture all the information in the data about a parameter so that no further reduction is possible without loss.

Technical developments
The essay develops formal definitions and relationships that underpin those desiderata. Fisher formulated the score function as the derivative of the log-likelihood and used its variance to motivate a measure of information about a parameter contained in the sample. This notion of information links directly to how well parameters can be estimated and to asymptotic normality of estimators. Fisher showed that maximum likelihood estimators (MLEs) often attain desirable asymptotic properties: they are consistent, asymptotically normal, and achieve the smallest attainable variance among a broad class of estimators, establishing a concrete sense of efficiency.

On sufficiency and reduction
Fisher's treatment of sufficiency articulated why low-dimensional summaries can be preferred: a sufficient statistic condenses the evidence about a parameter without discarding relevancy. He argued that working with sufficient statistics simplifies inference and clarifies the role of models, since any inference should depend only on the information retained by the sufficient statistic. The exposition gave practitioners both a guiding principle and practical motivation for seeking reductions that preserve parameter information.

Methodological implications
The essay critiques competing methods such as method-of-moments and purely descriptive procedures when they lack principled optimality. By elevating likelihood-based reasoning and by comparing estimators through variance and information, it provided a clear route to judging estimators and designing experiments. Fisher also sketched how sample-size considerations and experimental design could be informed by information measures, foreshadowing ideas that would become central to experimental planning and power analysis.

Philosophical stance and limitations
Fisher favored a frequentist viewpoint grounded in long-run behavior and asymptotics rather than in subjective probability assignments to parameters. This led to emphasis on properties like consistency and efficiency rather than direct probabilistic statements about parameter values. Some concepts were left informal or developed heuristically, and later work by Neyman, Pearson, Cramér, Rao and others formalized and extended many of the bounds, decision-theoretic frameworks, and optimality results that Fisher had sketched.

Legacy
The essay catalyzed modern theoretical statistics by introducing core concepts and a program for rigorous evaluation of inferential methods. Maximum likelihood, sufficiency, the score and information, and the prioritization of consistency and efficiency became pillars of the field. Debates and refinements following Fisher's proposals drove much of 20th-century statistical theory, but the essay's enduring contribution remains its clear insistence that statistical procedures should be judged by mathematical criteria and that likelihood-based methods provide a principled path to inference.