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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 [a1], [a2] (cf. also 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) $T _ {n} = T _ {n} ( \mathbf x )$ 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 $\alpha$. If for $\eta \in \Theta _ {1}$ with ${\mathsf P} _ \eta$- probability one,

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

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 ([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 [a4] for generalizations of this formula.

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

#### References

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