Overview

In statistics, efficiency describes how well a procedure — usually an estimator or a hypothesis test — extracts information from data. Informally, a more efficient method achieves a specified level of accuracy or power with fewer observations than a less efficient one. Efficiency is therefore a comparative, often quantitative, measure used when choosing among competing estimators, experimental designs, or testing methods.

Measures and interpretations

There are several ways to quantify efficiency. For point estimation, common measures include variance, mean squared error (MSE), and asymptotic variance. An estimator that attains the smallest possible variance under given conditions (for example, the Cramér–Rao lower bound) is called efficient in that sense. Relative efficiency compares two procedures by taking a ratio of their variances or the sample sizes required to reach equal performance. For hypothesis tests, notions such as Pitman efficiency (based on local power comparisons) and Bahadur efficiency (based on exponential rates of tail probabilities) are used to compare sensitivity to alternatives.

Theory and historical development

The formal study of efficiency grew from foundational work on likelihood, information, and bounds for estimation: Fisher's ideas about information, and later results like the Cramér–Rao inequality, give theoretical benchmarks for variance. Asymptotic theory clarified how estimators behave for large samples and introduced asymptotic efficiency as a key criterion. Multiple definitions coexist because efficiency can be defined with respect to different loss functions, sample sizes, or limiting regimes.

Practical considerations and examples

In practice, efficiency guides method choice but must be balanced with other properties. A maximum likelihood estimator may be asymptotically efficient under a correct model, yet perform poorly if the model is misspecified or if robustness is needed. Nonparametric or robust estimators can sacrifice efficiency under idealized models but gain resilience to outliers or deviations. For instance, the sample mean is highly efficient for estimating a normal mean under the usual assumptions, while the sample median is less efficient there but more robust with heavy-tailed data.

Uses, comparisons, and trade-offs

Efficiency is widely used to compare designs and tests, to determine required sample sizes, and to select estimators for applied work. When reporting efficiency one should state the criterion (variance, MSE, asymptotic variance, power) and the reference procedure or bound. Computational cost is a distinct concern: a statistically efficient method that is computationally infeasible may be less desirable than a slightly less efficient but much faster alternative.

For introductions and comparisons of different statistical procedures see comparisons of procedures, and for basic definitions of estimators and their properties see estimators and inference. Understanding efficiency helps balance precision, robustness, and computational practicality when designing studies or choosing methods.