In this previous post, I wrote about the asymptotic distribution of the Pearson statistic. Did you know that the Pearson statistic (and the related hypothesis test) is actually a special case of a general class of tests? In this post we describe the general test. The presentation follows that in Chapters 23 and 24 of Ferguson (1996) (Reference 1). I’m leaving out the proofs, which can be found in the reference.
(Warning: This post is going to be pretty abstract! Nevertheless, I think it’s worth a post since I don’t think the idea is well-known.)
Let’s define some quantities. Let be a sequence of random vectors whose distribution depends on a -dimensional parameter which lies in a parameter space . is assumed to be a non-empty open subset of , where . Next, assume that the are asymptotically normal, i.e. there exist and covariance matrix such that
Next, let be some covariance matrix, and define the quadratic form
We make 3 assumptions about and :
- is bicontinuous, i.e. .
- has a continuous first partial derivative, , of full rank .
- is continuous is and is uniformly bounded below, in the sense that there is a constant such that for all .
Definition. A minimum estimate is a value of , depending on , that minimizes . A sequence is a minimum sequence if
whatever the true value of . is going to be the statistic in our hypothesis test.
Let denote the true value of the parameter, and let , and denote , and respectively. The first theorem states that minimum sequences are asymptotically normal with a specific mean and covariance:
Theorem 1. For any minimum sequence , , where
The theorem above holds for any covariance matrix (that satisfies the assumption we make on it). We are interested in finding the matrix that makes the asymptotic covariance matrix above the smallest. The corollary below tells us what this is:
Corollary. If there is a non-singular such that , then . Moreover, for all .
It can be shown that we can take to be any generalized inverse of . If is non-singular, then we can take . For this choice of , the next theorem gives the asymptotic distribution of :
Theorem 2. , where is the rank of .
Application to Pearson’s
Whew! That was a lot. Let’s see how Pearson’s is a special case of this. (You might want to have the previous post open for reference.) For Pearson’s ,
- , the number of possible outcomes for each trial.
- , the vector of relative cell frequencies.
- , the vector of cell probabilities, written as a function of some -dimensional parameter , with .
- To make the expression equal to Pearson’s statistic, we have to take .
- It can be shown that is a generalized inverse of . Hence, Theorem 2 applies.
Other applications of general chi-square theory
This general chi-square theory has been used in recent years to construct hypothesis tests that work under various forms of privatized data: see References 2-4 (Reference 4 just came out this week!). There are probably other applications of this theory: if you know of any, please share!
- Ferguson, T. S. (1996). A Course in Large Sample Theory.
- Kifer, D., and Rogers, R. (2017). A New Class of Private Chi-Square Hypothesis Tests.
- Gaboardi, M., and Rogers, R. (2018). Local Private Hypothesis Testing: Chi-Square Tests.
- Friedberg, R., and Rogers, R. (2022). Privacy Aware Experimentation over Sensitive Groups: A General Chi Square Approach.