Pitman Efficiency

Abstract

Abstract The Pitman efficiency is an index for comparing test procedures or estimators. It is especially important for comparing procedures in large samples. If procedure P1 requires n1 observations to attain a certain power of a test, or a specified mean squared error, and procedure P2 requires n2 observations to achieve the same precision, the Pitman relative efficiency of P1 against P2 is n2/n1. Accordingly, if the mean squared error of an estimator E1 is v1/n + o(1/n), and if that of another estimator, E2, is v2/n + o(1/n), then the asymptotic (large sample) relative efficiency of E1 against E2 is v2/v1. Under smoothness regularity conditions, the variance of an unbiased estimator, in the one‐parameter case, is bounded below by the Cramer–Rao (CR) lower bound. Accordingly, the ratio of the CR‐lower bound to the variance of the estimator is defined as the efficiency index of the unbiased estimator. This is the Pitman relative efficiency of the estimator against the best possible one. Moreover, the CR‐lower bound contains the Fisher information function as a factor, and the inverse of the Fisher information function is the asymptotic variance of the maximum likelihood estimator (bf MLE). Therefore, the efficiency of an unbiased estimator is measured relative to the asymptotic variance of the maximum likelihood estimator. The Pitman relative efficiency of consistent and asymptotically normal test statistics is defined as the ratio of their corresponding efficacy ratios . The asymptotic efficacy relates the rate of change of the mean function of the statistic to its asymptotic standard deviation. For further information, see the article.