View source for Intermediate efficiency

''Kallenberg efficiency''

A concept used to compare the performance of statistical tests (cf. also [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]]). Write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200501.png" /> for the sample size required to attain with a level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200502.png" /> test a prescribed power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200503.png" /> at an alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200504.png" />. If one has two tests with corresponding numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200506.png" />, respectively, the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200507.png" /> is called the relative efficiency of test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200508.png" /> with respect to test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i1200509.png" />. If the relative efficiency equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005010.png" />, test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005011.png" /> needs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005012.png" /> times as much observations to perform equally well as test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005013.png" /> and hence test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005014.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005015.png" /> times as efficient as test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005016.png" /> (cf. also [[Efficient test|Efficient test]]).

In general, the relative efficiency is hard to compute and, if it can be computed, hard to evaluate, as it depends on three arguments: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005019.png" />. (Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005020.png" /> is not restricted to be a Euclidean parameter; it can also be an abstract parameter, as for instance the distribution function.) Therefore, an asymptotic approach, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005021.png" /> tends to infinity, is welcome to simplify both the computation and interpretation, thus hoping that the limit gives a sufficiently good approximation of the far more complicated finite-sample case.

When sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005022.png" /> to infinity, two guiding principles are:

a) to  "decrease the significance probability as N increases" , i.e. to send <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005024.png" />; or

b) to  "move the alternative hypothesis steadily closer to the null hypothesis" , i.e. to send <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005025.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005026.png" />. Both principles are attractive: with more observations it seems reasonable to have a stronger requirement on the level and, on the other hand, for alternatives far away from the null hypothesis there is no need for statistical methods, since they are obviously different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005027.png" />.

In Pitman's asymptotic efficiency concept, method b) is used, while one deals with fixed levels, thus ignoring principle a). In Bahadur's asymptotic efficiency concept, method a) is actually used, while one considers fixed alternatives, thereby ignoring principle b). (Cf. also [[Bahadur efficiency|Bahadur efficiency]]; [[Efficiency, asymptotic|Efficiency, asymptotic]].) Intermediate or Kallenberg efficiency applies both attractive principles simultaneously.

As a consequence of Bahadur's approach, in typical cases the level of significance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005028.png" /> required to attain a fixed power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005029.png" /> at a fixed alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005030.png" /> tends to zero at an exponential rate as the number of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005031.png" /> tends to infinity. There remains a whole range of sequences of levels  "intermediate"  between these two extremes of very fast convergence to zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005032.png" /> and the fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005033.png" /> in the case of Pitman efficiency. The efficiency concept introduced by W.C.M. Kallenberg [[#References|[a11]]] deals with this intermediate range and is therefore called intermediate efficiency, or, for short, i-efficiency.

A related approach is applied by P. Groeneboom [[#References|[a5]]], Sect. 3.4, studying very precisely the behaviour of several tests for the multivariate linear hypothesis from an  "intermediate"  point of view. Other efficiency concepts with an  "intermediate"  flavour can be found in [[#References|[a6]]], [[#References|[a18]]] and [[#References|[a2]]].

Instead of applying principles a) and b) simultaneously, in a lot of papers they are applied one after the other. For an excellent treatment in the case of non-parametric tests see [[#References|[a16]]], where also many further references can be found (cf. also [[Non-parametric test|Non-parametric test]]). General results on limiting equivalence of local and non-local measures of efficiency are presented in [[#References|[a4]]], [[#References|[a14]]] and [[#References|[a13]]].

The definition of intermediate or Kallenberg efficiency is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005034.png" /> be a sequence of independent, identically distributed random variables with distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005035.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005036.png" /> in the parameter space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005037.png" />. The hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005038.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005039.png" /> has to be tested against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005040.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005043.png" /> are given subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005044.png" />. For a family of tests <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005045.png" />, denote the power at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005046.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005048.png" /> is the available number of observations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005049.png" /> is the level of the test (cf. also [[Significance level|Significance level]]). Suppose one has two families of tests, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005051.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005052.png" /> be a sequence of levels with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

thus ensuring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005054.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005055.png" />, but not exponentially fast. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005056.png" /> be a sequence of alternatives tending to the null hypothesis, in the sense that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005060.png" /> denotes the [[Hellinger distance|Hellinger distance]] between the probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005062.png" />. This ensures that the alternatives tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005063.png" />, but in a slower way than contiguous alternatives, cf. [[#References|[a17]]]. Typically, for Euclidean parameters, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005064.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005065.png" /> and hence in such cases formula (a3) concerns convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005066.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005067.png" /> at a rate slower than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005068.png" />. (The latter is the usual rate for contiguous alternatives.)

Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005069.png" /> as the smallest number of observations needed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005070.png" /> to perform as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005071.png" /> in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005072.png" />, the power at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005073.png" /> of the level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005074.png" /> test of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005075.png" /> based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005076.png" /> observations, is, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005077.png" />, at least as large as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005078.png" />, the power at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005079.png" /> of the level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005080.png" /> test of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005081.png" /> based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005082.png" /> observations. If the sequence of levels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005083.png" /> satisfies next to (a1) also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005085.png" /></td> </tr></table>

exists and does not depend on the special sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005087.png" /> under consideration, one says that the intermediate or Kallenberg efficiency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005088.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005089.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005090.png" />. If (a4) is replaced by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005091.png" /></td> </tr></table>

one speaks of weak intermediate or weak Kallenberg efficiency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005092.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005093.png" /> and one uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005094.png" />. Otherwise, that is, if all sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005095.png" /> satisfying (a1) are under consideration, one speaks of strong intermediate or strong Kallenberg efficiency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005096.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005097.png" />, with notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005098.png" />. Note that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i12005099.png" /></td> </tr></table>

So, the whole intermediate range of levels between the Pitman and Bahadur case is built up with three increasing ranges. For example, if an i-efficiency result can be proved only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050100.png" /> at a lower rate than powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050101.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050102.png" />, one speaks of a weak i-efficiency result. The several types of i-efficiency correspond with the existence of several types of moderate and Cramér-type large deviation theorems.

To compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050103.png" /> under the null hypothesis, one needs a moderate deviation result (see [[#References|[a7]]] and references therein for results of this type), since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050104.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050105.png" />. Under the alternatives a kind of [[Law of large numbers|law of large numbers]] is involved. The precise computation is described in [[#References|[a11]]], Lemma 2.1; Corol. 2.2, where also many examples are presented.

In many testing problems, likelihood-ratio tests (cf. also [[Likelihood-ratio test|Likelihood-ratio test]]) are asymptotically optimal (cf. also [[Asymptotic optimality|Asymptotic optimality]]) when comparison is made in a non-local way, cf. [[#References|[a1]]], [[#References|[a3]]], [[#References|[a10]]]. On the other hand, likelihood ratio tests usually are not asymptotically optimal with respect to criteria based on the local performance of tests. It turns out that in exponential families, likelihood ratio tests have strong i-efficiency greater than or equal to one with respect to every other test, thus being optimal according to the criterion of i-efficiency.

Locally most powerful tests are often Pitman efficient. On the other hand, locally most powerful tests are far from optimal from a non-local point of view. It turns out that in curved exponential families locally most powerful tests have strong i-efficiency greater than or equal to one with respect to every other test, thus being optimal according to the criterion of i-efficiency.

Optimality, in the sense of weak i-efficiency, of certain goodness-of-fit tests (cf. also [[Goodness-of-fit test|Goodness-of-fit test]]) in the case of censored data is shown in [[#References|[a15]]], while i-efficiency of decomposable statistics in a multinomial scheme is analyzed in [[#References|[a9]]]. For a generalization of the concept see [[#References|[a8]]], where it is shown that data-driven Neyman tests are asymptotically optimal.

Application of an intermediate approach in estimation theory can be found in [[#References|[a12]]]. This is based on the probability that a [[Statistical estimator|statistical estimator]] deviates by more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050106.png" /> from its target <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050107.png" />, for instance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050108.png" /> for the estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050109.png" />. The intermediate range concerns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050111.png" />. Under certain regularity conditions, there is an asymptotic lower bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120050/i120050112.png" />, similar to the (Fisher) information bound in the local theory. An estimator is called optimal in the intermediate sense if it attains this lower bound.

====References====
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