It is a common way to formulate the detection problem into composite hypothesis testing.
Detection of unknown sparse signal mixed with noise is a long-standing problem of composite hypothesis testing that arises in many scientific applications such as signal processing , wireless sensor networks , and remote sensing systems.
Thereby, the problem addressed could be formulated as a binary composite hypothesis test.
In addition, it provides the prerequisite that we could take N-P test as the UMP test when detecting the sparse signal in a composite hypothesis framework.
Among their topics are combining information across genome-wide linkage scans, heterogeneity in the meta-analysis of quantitative trait linkage studies, an approach to composite hypothesis
testing built on intersection-union tests and Bayesian posterior probabilities, significance testing for small microarray experiments, combining genomic data in human studies, and a misclassification model for inferring transcriptional regulatory networks.
The answer to this question provides guidance for determining when a composite hypothesis (i.e., a composition of single hypotheses) is warranted.
Guideline 1: A joint test of a composite hypothesis ought to be used if an inference or conclusion requires multiple hypotheses to be simultaneously true.
If an independent variable is considered to be simultaneously related to a number of dependent variables, then a joint test of a composite hypothesis is warranted.
Rejecting a joint test of a composite hypothesis does not tell us which specific alternative case is warranted.
This is actually a composite hypothesis
testable with the Wald statistic (Rao 1973) that involves two or more components or sub-hypotheses in the sense that a number of group equalities are tested.
In contrast, a composite hypothesis
provides a range of values of the parameter for testing.
Nor would the experiment allow us to distinguish between aspects of the composite hypothesis
: With either demand function, no lag occurs if expectations are rational.