# Efficiency, asymptotic

of a test

A concept that makes it possible in the case of large samples to make a quantitative comparison of two distinct statistical tests for a certain statistical hypothesis. The need to measure the efficiency of tests arose in the 1930s and -forties when simple (from the computational point of view) but "inefficient" rank procedures made their appearance.

There are several distinct approaches to the definition of the asymptotic efficiency of a test. Suppose that a distribution of observations is defined by a real parameter $\theta$ and that it is required to verify the hypothesis $H _ {0}$: $\theta = \theta _ {0}$ against the alternative $H _ {1}$: $\theta \neq \theta _ {0}$. Suppose also that for a certain test with significance level $\alpha$ there are $N _ {1}$ observations needed to achieve a power $\beta$ against the given alternative $\theta$ and that another test of the same level needs for this purpose $N _ {2}$ observations. Then one can define the relative efficiency of the first test with respect to the second by the formula $e _ {12} = N _ {2} / N _ {1}$. The concept of relative efficiency gives exhaustive information for the comparison of tests, but proves to be inconvenient for applications, since $e _ {12}$ is a function of the three arguments $\alpha$, $\beta$ and $\theta$ and, as a rule, does not lend itself to computation in explicit form. To overcome this difficulty one uses a passage to a limit.

The quantity $\lim\limits _ {\theta \rightarrow \theta _ {0} } e _ {12} ( \alpha , \beta , \theta )$, for fixed $\alpha$ and $\beta$( if the limit exists), is called the asymptotic relative efficiency in the sense of Pitman. Similarly one defines the asymptotic relative efficiency in the sense of Bahadur, where for fixed $\beta$, $\theta$ the limit is taken as $\alpha$ tends to zero, and the asymptotic relative efficiency in the sense of Hodges and Lehmann, when for fixed $\alpha$ and $\theta$ one computes the limit as $\beta \rightarrow 1$.

Each of these definitions has its own merits and shortfalls. For example, the Pitman efficiency is, as a rule, easier to calculate than the Bahadur one (the calculation of the latter involves the non-trivial problem of studying the asymptotic probability of large deviations of test statistics); however, in a number of cases it turns out to be a less sensitive tool for the comparison of two tests.

Suppose, for example, that the observations are distributed according to the normal law with average $\theta$ and variance 1 and that the hypothesis $H _ {0}$: $\theta = 0$ is to be verified against the alternative $H _ {1}$: $\theta > 0$. Suppose also that one considers a significance test based on a sample mean $\overline{X}\;$ and Student ratio $t$. Since the $t$- test does not use information on the variance, the optimal test must be that based on $\overline{X}\;$. However, from the point of view of Pitman efficiency these tests are equivalent. On the other hand, the Bahadur efficiency of the $t$- test in relation to $\overline{X}\;$ is strictly less than 1 for any $\theta > 0$.

In more complicated cases the Pitman efficiency may depend on $\alpha$ or $\beta$ and its calculation becomes very tedious. Then one calculates its limiting value as $\beta \rightarrow 1$ or $\alpha \rightarrow 0$. The latter usually is the same as the limiting value of the Bahadur efficiency as $\theta \rightarrow \theta _ {0}$.

For other approaches to the definition of asymptotic efficiency of a test see ; sequential analogues of this concept are introduced in . The choice of one definition or another must be based on which of them gives a more accurate approximation to the relative efficiency $e _ {12}$; however, at present (1988) little is known in this direction .

How to Cite This Entry:
Efficiency, asymptotic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Efficiency,_asymptotic&oldid=46791
This article was adapted from an original article by Ya.Yu. Nikitin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article