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Suppose $\vec y \sim N(\vec 0,\Sigma)$ with $\Sigma$ singular. Is the distribution of $y^TAy$ known in the case that $A$ is indefinite?

kjetil b halvorsen
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Mikkel Rev
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  • what does indefinite mean? – Taylor Sep 05 '17 at 23:20
  • @Taylor http://mathworld.wolfram.com/IndefiniteQuadraticForm.html – Mikkel Rev Sep 05 '17 at 23:58
  • @Taylor See https://stats.stackexchange.com/a/301714/919 on this site. Marius: $y^\prime A y$ is a linear combination of $\chi^2(1)$ variates. I believe you can unearth this answer, with details, by [searching our site](https://stats.stackexchange.com/search?q=normal+quadratic+chi). The singularity of $\Sigma$ is irrelevant. – whuber Sep 06 '17 at 14:26
  • @whuber Thanks. But I already had to look. I just searched your suggest phrase, and I didn't find an answer. – Mikkel Rev Sep 06 '17 at 18:25
  • Here's one at https://stats.stackexchange.com/questions/72479. It shows how to find a sum of Gammas. (A $\chi^2(1)$ is a multiple of a Gamma$(1/2)$ distribution.) Somewhere we have a thread about general linear combinations of Gammas, but it only cites references. Linear algebra shows that $y^\prime A y$ is distributed like a linear combination of sums of squared Normal variates and therefore is a linear combination of Gammas--those linear algebraic arguments are quoted in many threads concerning quadratic forms in Normal variables (and they're simple). – whuber Sep 06 '17 at 19:15
  • @whuber Thanks, I need the details, is it possible that you suggest a paper or textbook which derives the distribution of $y'Ay$ when $A$ is indefinite? I looked at Mathai, Provost: Quadratic Forms in Random Variables. From what I can tell, it does not contain the answer. – Mikkel Rev Sep 07 '17 at 00:09
  • This is called the [generalized chisquare distribution](https://en.wikipedia.org/wiki/Generalized_chi-squared_distribution) or https://stats.stackexchange.com/questions/338989/generalized-chi-squared-distribution-pdf/339984#339984. Look at [this](https://math.stackexchange.com/questions/442472/sum-of-squares-of-dependent-gaussian-random-variables/442916#442916) and as an exercise, extend the arguments to singular covariance matrix using generalized inverses – kjetil b halvorsen Feb 05 '21 at 12:01

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