In many situations, we seek to check consistency of the
observations with some preconceived model. Alternative modelsare usually difficult to describe parametrically since
inconsistency may be beyond our modeling capabilities. We needa test that accepts consistency of observations with a model or
rejects the model without pronouncing a more favoredalternative. Assuming we know (or presume to know) the
probability distribution of the observations under
, the models are
:
:
Null hypothesis testing seeks to determine
if the observations are consistent with this description. Thebest procedure for consistency testing amounts to determining
whether the observations lie in a highly probable region asdefined by the null probability distribution. However, no one
region defines a probability that is less than unity. We mustrestrict the size of the region so that it best represents those
observations maximally consistent with the model whilesatisfying a performance criterion. Letting
be a false-alarm probability established by us, we
define the decision region
to satisfy
and
Usually, this region is located about the mean, but
may not be symmetrically centered if the probability density isskewed. Our null hypothesis test for model consistency becomes
Consider the problem of determining whether the sequence
,
, is white and Gaussian with zero mean and unit
variance. Stated this way, the alternative modelis not
provided: is this model correct or not? We could estimate theprobability density function of the observations and test the
estimate for consistency. Here we take the null-hypothesistesting approach of converting this problem into a
one-dimensional one by considering the statistic
, which has a
. Because this probability distribution is unimodal,
the decision region can be safely assumed to be an interval
.
This one-dimensional result for
the consistency test may extend to the multi-dimensional casein the obvious way.
In this case, we can find an analytic
solution to the problem of determining the decision region.Letting
denote the width of the interval, we seek the
solution of the constrained optimization problem
We convert the constrained problem into an
unconstrained one using Lagrange multipliers.
Evaluation of the derivative of this quantity with respect to
yields the result
: to minimize the interval's width, the probability
density function's values at the interval's endpoints must beequal. Finding these endpoints to satisfy the constraints
amounts to searching the probability distribution at suchpoints for increasing values of
until the required probability is contained within.
For
and
, the optimal decision region for the
distribution is
.
demonstrates ten
testing trials for observations that fit the model and forobservations that don't.
Ten trials of testing a 100-element sequence for
consistency with a white, Gaussian model -
- for three situations. In the first (shown by
the circles), the observations do conform to the model.In the second (boxes), the observations are zero-mean
Gaussian but with variance two. Finally, the thirdexample (crosses) has white observations with a density
closely resembling the Gaussian: a hyperbolic secantdensity having zero mean and unit variance. The sum of
squared observations for each example are shown with theoptimal
interval displayed. Note how dramatically the
test statistic departs from the decision interval whenparameters disagree.
