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

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 on Binary 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