# Composite Hypothesis Test

Hypothesis tests > Composite Hypothesis Test

## What is a Composite Hypothesis Test?

A composite hypothesis test contains more than one parameter and more than one model. In a simple hypothesis test, the probability density functions for both the null hypothesis (H0) and alternate hypothesis (H1) are known. In academic and hypothetical situations, the simple hypothesis test works for most cases. However, in real life it’s much more challenging to specify all of the pdfs for a particular situation.

## Approaches to Composite Hypothesis Testing

The parameters in a composite hypothesis test are not completely known.

As the composite test involves one or more unknown parameters, so H0 or H1 may not be completely stated. It can be challenging to choose between the two hypotheses if you have incomplete information. Approaches to making that decision include:

1. Bayesian approach: the unknown parameter is assigned a prior PDF.
2. Generalized likelihood ratio test approach: the unknown parameter is estimated and placed into a likelihood ratio test.

## Composite Null Hypothesis

In real life, null hypotheses are usually composite unless the problem is very simple. An example of a composite hypothesis, which has multiple possible values, is:
H0: μ ≥ 100

## References

Ghobadzadeh, A. et. al. Separating Function Estimation Tests: A New Perspective onBinary Composite Hypothesis Testing. Retrieved August 19, 2019 from: http://post.queensu.ca/~gazor/T-SP-13550-2012.pdf
Lindsey, J. Parametric Statistical Inference. Retrieved August 19, 2019 from: https://books.google.com/books?id=YnsQ-NnMxJ8C
Nowak, R. (2010). Lecture 10: Composite Hypothesis Testing. Retrieved August 19, 2019 from: http://nowak.ece.wisc.edu/ece830/ece830_fall11_lecture10.pdf
Lecture 29 : Bayesian Composite Hypothesis Testing. Retrieved August 19, 2019 from: https://nptel.ac.in/courses/117103018/module8/lec29/1.html

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