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Linear combination of independent noncentral chi-squared random variables

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Travis AppVeyor

This package estimates cumulative density functions of linear combinations of independent noncentral χ² random variables and a standard Normal distribution. Formally, it estimates P[Q<q], where:

Q = λ₁X₁ + ... + λₙXₙ + σX₀.

Xᵢ (𝚒≠𝟶) is an independent random variable following a noncentral χ² distribution with nᵢ degrees of freedom and noncentrality parameter λᵢ. X₀ is an independent random variable having a standard Normal distribution.


It can be installed using the pip command

pip install chi2comb


Consider the following linear combination of four random variables:

Q = 6⋅X₁ + 3⋅X₂ + 1⋅X₃ + 2⋅X₀,

where X₁, X₂, and X₃ are noncentral χ² random variables having degrees of freedom n₁=n₂=1 and n₃=2 and noncentrality parameters λ₁=0.5 and λ₂=λ₃=0. Let us estimate P[Q<1]:

>>> from chi2comb import chi2comb_cdf, ChiSquared
>>> gcoef = 2
>>> ncents = [0.5, 0, 0]
>>> q = 1
>>> dofs = [1, 1, 2]
>>> coefs = [6, 3, 1]
>>> chi2s = [ChiSquared(coefs[i], ncents[i], dofs[i]) for i in range(3)]
>>> result, errno, info = chi2comb_cdf(q, chi2s, gcoef)
>>> result
>>> errno
>>> info
Info(emag=0.6430413191446991, niterms=43, nints=1, intv=0.03462571527167856, truc=1.4608856930426104, sd=0.0, ncycles=21)

The estimated value is P[Q<1] ≈ 0.0587.


If you encounter any issue, please, submit it.



This project is licensed under the MIT License.

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