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The large sample study of test statistics in a given hypotheses testing problem is commonly based on the following concept of asymptotic Bahadur efficiency [[#References|[a1]]], [[#References|[a2]]] (cf. also [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]]). Let  $  \Theta _ {0} $
and  $  \Theta _ {1} $
be the parametric sets corresponding to the null hypothesis and its alternative, respectively. Assume that large values of a test statistic (cf. [[Test statistics|Test statistics]])  $  T _ {n} = T _ {n} ( \mathbf x ) $
based on a random sample  $  \mathbf x = ( x _ {1} \dots x _ {n} ) $
give evidence against the null hypothesis. For a fixed  $  \theta \in \Theta _ {0} $
and a real number  $  t $, 
put  $  F _ {n} ( t \mid  \theta ) = {\mathsf P} _  \theta  ( T _ {n} < t ) $
and let  $  L _ {n} ( t \mid  \theta ) = 1 - F _ {n} ( t \mid  \theta ) $. 
The random quantity  $  L _ {n} ( T _ {n} ( \mathbf x ) \mid  \theta _ {0} ) $
is the  $  {\mathsf P} $-
value corresponding to the statistic  $  T $
when  $  \theta _ {0} $
is the true parametric value. For example, if  $  L _ {n} ( T _ {n} ( \mathbf x ) \mid  \theta _ {0} ) < \alpha $, 
the null hypothesis  $  \Theta _ {0} = \{ \theta _ {0} \} $
is rejected at the [[Significance level|significance level]]  $  \alpha $. 
If for  $  \eta \in \Theta _ {1} $
with  $  {\mathsf P} _  \eta  $-
probability one,

$$ 
\lim\limits 2n ^ {- 1 } { \mathop{\rm log} } L _ {n} ( T ( \mathbf x ) \mid  \theta ) = - d ( \eta \mid  \theta ) ,
$$

then  $  d ( \eta \mid  \theta ) $
is called the Bahadur exact slope of  $  T $. 
The larger the Bahadur exact slope, the faster the rate of decay of the  $  {\mathsf P} $-
value under the alternative. It is known that for any  $  T $, 
$  d ( \eta \mid  \theta ) \leq 2K ( \eta, \theta ) $, 
where  $  K ( \eta, \theta ) $
is the information number corresponding to  $  {\mathsf P} _  \eta  $
and  $  {\mathsf P} _  \theta  $. 
A test statistic  $  T $
is called Bahadur efficient at  $  \eta $
if

$$ 
e _ {T} ( \eta ) =  \inf  _  \theta  {
\frac{1}{2}
 } d ( \eta \mid  \theta ) =  \inf  _  \theta  K ( \eta, \theta ) .
$$

The concept of Bahadur efficiency allows one to compare two (sequences of) test statistics  $  T ^ {( 1 ) } $
and  $  T ^ {( 2 ) } $
from the following perspective. Let  $  N _ {i} $, 
$  i = 1,2 $, 
be the smallest sample size required to reject  $  \Theta _ {0} $
at the significance level  $  \alpha $
on the basis of a random sample  $  \mathbf x = ( x _ {1} , \dots ) $
when  $  \eta $
is the true parametric value. The ratio  $  { {N _ {2} } / {N _ {1} } } $
gives a measure of relative efficiency of  $  T ^ {( 1 ) } $
to  $  T ^ {( 2 ) } $. 
To reduce the number of arguments  $  \alpha $, 
$  \mathbf x $
and  $  \eta $, 
one usually considers the random variable which is the limit of this ratio, as  $  \alpha \rightarrow 0 $. 
In many situations this limit does not depend on  $  \mathbf x $, 
so it represents the efficiency of  $  T ^ {( 1 ) } $
against  $  T ^ {( 2 ) } $
at  $  \eta $
with the convenient formula

$$ 
{\lim\limits } _ {\alpha \rightarrow 0 } {
\frac{N _ {2} }{N _ {1} }
 } = {
\frac{d _ {1} ( \eta \mid  \theta _ {0} ) }{d _ {2} ( \eta \mid  \theta _ {0} ) }
 } ,
$$

where  $  d _ {1} $
and  $  d _ {2} $
are the corresponding Bahadur slopes.

To evaluate the exact slope, the following result ([[#References|[a2]]], Thm. 7.2) is commonly used. Assume that for any  $  \eta $
with  $  {\mathsf P} _  \eta  $-
probability one as  $  n \rightarrow \infty $, 
$  T _ {n} ( \mathbf x ) \rightarrow b ( \eta ) $
and the limit  $  g _  \theta  ( t ) = {\lim\limits } L _ {n} ( t \mid  \theta ) $
exists for  $  t $
taking values in an open interval and is a continuous function there. Then the exact slope of  $  T $
at  $  ( \eta, \theta ) $
has the form  $  d ( \eta \mid  \theta ) = g _  \theta  ( b ( \eta ) ) $. 
See [[#References|[a4]]] for generalizations of this formula.

The exact Bahadur slopes of many classical tests have been found. See [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.R. Bahadur,   "Rates of convergence of estimates and tests statistics"  ''Ann. Math. Stat.'' , '''38'''  (1967)  pp. 303–324</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.R. Bahadur,   "Some limit theorems in statistics" , ''Regional Conf. Ser. Applied Math.'' , SIAM  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Ya.Yu. Nikitin,   "Asymptotic efficiency of nonparametric tests" , Cambridge Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.J. Gleser,   "Large deviation indices and Bahadur exact slopes"  ''Statistics and Decision'' , '''1'''  (1984)  pp. 193–204</TD></TR></table>