Bahadur efficiency

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,

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

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 ([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