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 (H_{0}) and alternate hypothesis (H_{1}) 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

As the composite test involves one or more unknown parameters, so H_{0}or H

_{1}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:

**Bayesian approach**: the unknown parameter is assigned a prior PDF.**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:

H_{0}: 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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