Tag: chi-squared

Sampling from the non-central chi-squared distribution

The non-central chi-squared distribution is a generalisation of the regular chi-squared distribution. The chi-squared distribution turns up in many statistical tests as the (approximate) distribution of a test statistic under the null hypothesis. Under alternative hypotheses, those same statistics often have approximate non-central chi-squared distributions.

This means that the non-central chi-squared distribution is often used to study the power of said statistical tests. In this post I give the definition of the non-central chi-squared distribution, discuss an important invariance property and show how to efficiently sample from this distribution.

Definition

Let Z be a normally distributed random vector with mean 0 and covariance I_n. Given a vector \mu \in \mathbb{R}^n, the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter \Vert \mu\Vert_2^2 is the distribution of the quantity

\Vert Z+\mu \Vert_2^2 = \sum\limits_{i=1}^n (Z_i+\mu_i)^2.

This distribution is denoted by \chi^2_n(\Vert \mu \Vert_2^2). As this notation suggests, the distribution of \Vert Z+\mu \Vert_2^2 depends only on \Vert \mu \Vert_2^2, the norm of \mu. The first few times I heard this fact, I had no idea why it would be true (and even found it a little spooky). But, as we will see below, the result is actually a simply consequence of the fact that standard normal vectors are invariant under rotations.

Rotational invariance

Suppose that we have two vectors \mu, \nu \in \mathbb{R}^n such that \Vert \mu\Vert_2^2 = \Vert \nu \Vert_2^2. We wish to show that if Z \sim \mathcal{N}(0,I_n), then

\Vert Z+\mu \Vert_2^2 has the same distribution as \Vert Z + \nu \Vert_2^2.

Since \mu and \nu have the same norm there exists an orthogonal matrix U \in \mathbb{R}^{n \times n} such that U\mu = \nu. Since U is orthogonal and Z \sim \mathcal{N}(0,I_n), we have Z'=UZ \sim \mathcal{N}(U0,UU^T) = \mathcal{N}(0,I_n). Furthermore, since U is orthogonal, U preserves the norm \Vert \cdot \Vert_2^2. This is because, for all x \in \mathbb{R}^n,

\Vert Ux\Vert_2^2 = (Ux)^TUx = x^TU^TUx=x^Tx=\Vert x\Vert_2^2.

Putting all these pieces together we have

\Vert Z+\mu \Vert_2^2 = \Vert U(Z+\mu)\Vert_2^2 = \Vert UZ + U\mu \Vert_2^2 = \Vert Z'+\nu \Vert_2^2.

Since Z and Z' have the same distribution, we can conclude that \Vert Z'+\nu \Vert_2^2 has the same distribution as \Vert Z + \nu \Vert. Since \Vert Z + \mu \Vert_2^2 = \Vert Z'+\nu \Vert_2^2, we are done.

Sampling

Above we showed that the distribution of the non-central chi-squared distribution, \chi^2_n(\Vert \mu\Vert_2^2) depends only on the norm of the vector \mu. We will now use this to provide an algorithm that can efficiently generate samples from \chi^2_n(\Vert \mu \Vert_2^2).

A naive way to sample from \chi^2_n(\Vert \mu \Vert_2^2) would be to sample n independent standard normal random variables Z_i and then return \sum_{i=1}^n (Z_i+\mu_i)^2. But for large values of n this would be very slow as we have to simulate n auxiliary random variables Z_i for each sample from \chi^2_n(\Vert \mu \Vert_2^2). This approach would not scale well if we needed many samples.

An alternative approach uses the rotation invariance described above. The distribution \chi^2_n(\Vert \mu \Vert_2^2) depends only on \Vert \mu \Vert_2^2 and not directly on \mu. Thus, given \mu, we could instead work with \nu = \Vert \mu \Vert_2 e_1 where e_1 is the vector with a 1 in the first coordinate and 0s in all other coordinates. If we use \nu instead of \mu, we have

\sum\limits_{i=1}^n (Z_i+\nu_i)^2 = (Z_1+\Vert \mu \Vert_2)^2 + \sum\limits_{i=2}^{n}Z_i^2.

The sum \sum_{i=2}^n Z_i^2 follows the regular chi-squared distribution with n-1 degrees of freedom and is independent of Z_1. The regular chi-squared distribution is a special case of the gamma distribution and can be effectively sampled with rejection sampling for large shape parameter (see here).

The shape parameter for \sum_{i=2}^n Z_i^2 is \frac{n-1}{2}, so for large values of n we can efficiently sample a value Y that follows that same distribution as \sum_{i=2}^n Z_i^2 \sim \chi^2_{n-1}. Finally to get a sample from \chi^2_n(\Vert \mu \Vert_2^2) we independently sample Z_1, and then return the sum (Z_1+\Vert \mu\Vert_2)^2 +Y.

Conclusion

In this post, we saw that the rotational invariance of the standard normal distribution gives a similar invariance for the non-central chi-squared distribution.

This invariance allowed us to efficiently sample from the non-central chi-squared distribution. The sampling procedure worked by reducing the problem to sampling from the regular chi-squared distribution.

The same invariance property is also used to calculate the cumulative distribution function and density of the non-central chi-squared distribution. Although the resulting formulas are not for the faint of heart.