Consistent test
consistent statistical test
A statistical test that reliably distinguishes a hypothesis to be tested from an alternative by increasing the number of observations to infinity.
Let be a sequence of independent identically-distributed random variables taking values in a sample space
,
, and suppose one is testing the hypothesis
:
against the alternative
:
, with an error of the first kind (see Significance level) being given in advance and equal to
(
). Suppose that the first
observations
are used to construct a statistical test of level
for testing
against
, and let
,
, be its power function (cf. Power function of a test), which gives for every
the probability that this test rejects
when the random variable
is subject to the law
. Of course
for all
. By increasing the number of observations without limit it is possible to construct a sequence of statistical tests of a prescribed level
intended to test
against
; the corresponding sequence of power functions
satisfies the condition
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If under these conditions the sequence of power functions is such that, for any fixed
,
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then one says that a consistent sequence of statistical tests of level has been constructed for testing
against
. With a certain amount of license, one says that a consistent test has been constructed. Since
,
(which is the restriction of
,
, to
), is the power of the statistical test constructed from the observations
, the property of consistency of a sequence of statistical tests can be expressed as follows: The corresponding powers
,
, converge on
to the function identically equal to 1 on
.
Example. Let be independent identically-distributed random variables whose distribution function belongs to the family
of all continuous distribution functions on
, and let
be a vector of positive probabilities such that
. Further, let
be any distribution function of
. Then
and
uniquely determine a partition of the real axis into
intervals
, where
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In other words, the end points of the intervals are quantiles of the distribution function . These intervals determine a partition of
into two disjoint sets
and
as follows: A distribution function
of
belongs to
if and only if
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and otherwise . Now let
be the vector of counts obtained as a result of grouping the first
random variables
(
) into the intervals
. Then to test the hypothesis
that the distribution function of the
belongs to the set
against the alternative
that it belongs to the set
, one can make use of the "chi-squared" test based on the statistic
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According to this, with significance level (
), the hypothesis
must be rejected whenever
, where
is the upper
-quantile of the "chi-squared" distribution with
degrees of freedom. From the general theory of tests of "chi-squared" type it follows that when
is correct,
![]() |
which also shows the consistency of the "chi-squared" test for testing against
. But if one takes an arbitrary non-empty subset of
and considers the problem of testing against the alternative
, then it is clear that the "chi-squared" sequence of tests based on the statistics
is not consistent, since
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and, in particular,
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References
[1] | S.S. Wilks, "Mathematical statistics" , Wiley (1962) |
[2] | E. Lehman, "Testing statistical hypotheses" , Wiley (1959) |
Consistent test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Consistent_test&oldid=12096