Noncentral Chi-Square Poisson Mixture

Mixture Representation

The noncentral chi-square distribution arises when standard normal variables with non-zero means are squared and summed. Remarkably, this complicated distribution can be exactly resolved into an infinite mixture of ordinary (central) chi-square distributions, weighted by Poisson probabilities.

A Worked Example

Suppose we have degrees of freedom ν = 2 and noncentrality λ = 2. Let's calculate the contribution of the component index K = 1 when probed at x = 2.

First, the Poisson rate parameter is half the noncentrality parameter: λ_p = 2 / 2 = 1.
The Poisson component weight for K = 1 is e^-1 (1¹ / 1!) = e^-1 ≈ 0.367879441171.
The component degrees of freedom for K = 1 is ν + 2K = 2 + 2(1) = 4.
The component density is the central chi-square density with 4 degrees of freedom evaluated at x = 2. Using the PDF formula: (2)^{4/2 - 1} e^-2/2 / (2^4/2 Γ(4/2)) = 2 e^-1 / (4 × 1) = e^-1 / 2 ≈ 0.183939720586.
Therefore, this specific component K = 1 contributes 0.367879441171 × 0.183939720586 ≈ 0.067667641618 to the total noncentral density at x = 2. The total noncentral density aggregates components across all K = 0, 1, 2, ..., and approximately equals 0.154254161277.

Interactive Experiment

Open the Noncentral Chi-Square Poisson Mixture module.

Set Degrees of freedom to 2 and Noncentrality to 2.
Set Probe x to 2 and Mixture index to 1. Notice how the Component density reads approximately 0.183939720586.
Change the Mixture index to 0. The Poisson component weight jumps to 0.367879441171, but the component degrees drops to 2, and the component density drops to 0.183939720586.
Adjust Noncentrality to 8. You will see the total noncentral density curve shift dramatically rightward as the mean grows to ν + λ = 10.

Limitations

For large noncentrality parameters, the Poisson weights shift further to the right. The infinite sum over K must be truncated in practice; the interactive module here aggregates the first 150 components or truncates when weights fall below machine epsilon, which suffices for the finite domain λ ≤ 8 explored. Additionally, x = 0 is a singular point when degrees of freedom ν = 1 (approaching infinity), which is labeled as Unavailable.

References

Primary references mapping noncentral forms and computational strategies:

SciPy Statistical Functions: noncentral chi-square (ncx2)

Noncentral Chi-Square Poisson Mixture