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''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} $[[#References|[8]]].

For other approaches to the definition of asymptotic efficiency of a test see [[#References|[2]]]–[[#References|[5]]]; sequential analogues of this concept are introduced in [[#References|[6]]]–[[#References|[7]]]. 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 [[#References|[9]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Stewart,   "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Bahadur,   "Rates of convergence of estimates and test statistics"  ''Ann. Math. Stat.'' , '''38''' :  2  (1967)  pp. 303–324</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Hodges,   E. Lehmann,   "The efficiency of some nonparametric competitors of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035070/e03507043.png" />-test"  ''Ann. Math. Stat.'' , '''27''' :  2  (1956)  pp. 324–335</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C.R. Rao,   "Linear statistical inference and its applications" , Wiley  (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Kallenberg,   "Chernoff efficiency and deficiency"  ''Ann. Statist.'' , '''10''' :  2  (1982)  pp. 583–594</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Berk,   L. Brown,   "Sequential Bahadur efficiency"  ''Ann. Statist.'' , '''6''' :  3  (1978)  pp. 567–581</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Berk,   "Asymptotic efficiencies of sequential tests"  ''Ann. Statist.'' , '''4''' :  5  (1976)  pp. 891–911</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Wieland,   "A condition under which the Pitman and Bahadur approaches to efficiency coincide"  ''Ann. Statist.'' , '''4''' :  5  (1976)  pp. 1003–1011</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Groeneboom,   J. Oosterhoff,   "Bahadur efficiency and small-sample efficiency"  ''Internat. Stat. Rev.'' , '''49''' :  2  (1981)  pp. 127–141</TD></TR></table>

====Comments====
Reference [[#References|[a1]]] (and other work) suggest that, in the practically important case of small sample situations, the Pitman approach yields, in general, better approximations than the Bahadur approach does.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Groeneboom,   J. Oosterhoff,   "Bahadur efficiencies and probabilities of large deviations"  ''Stat. Neerlandica'' , '''31'''  (1977)  pp. 1–24</TD></TR></table>