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What is known about the distribution of the random variable $X$ that is a cubic form of independent central and noncentral normal variables $$X\sim y_1 y_2 y_3- y_1 x_1 x_2 - y_2 x_3 x_4- y_3 x_5 x_6+x_1 x_4 x_5 + x_2 x_3 x_6$$ with

$y_i\sim\mathcal{N}\left(\mu_i,1\right)\\x_i\sim\mathcal{N}\left(0,1\right)$

Maybe someone knows a paper that deals with such patterns of cubic forms.

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    $\begingroup$ Given the vague similarity to your previous question at stats.stackexchange.com/questions/591696/…, could you please explain what you mean by "such patterns"? Arbitrary polynomial functions of independent Normal random variables, perhaps? $\endgroup$
    – whuber
    Oct 12, 2022 at 16:33
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    $\begingroup$ By pattern I mean cubic forms of normal variables, i.e. sums of products of 3 factors of normals, mixed central and noncentral. I am interested in the special given case but maybe there is a theory about a similar or more general case. Therefore I used "pattern". $\endgroup$ Oct 13, 2022 at 13:24
  • $\begingroup$ That helps us understand the direction of your question. Could you explain the statistical setting in which this arises? $\endgroup$
    – whuber
    Oct 13, 2022 at 16:36
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    $\begingroup$ The direction to look, then, would be into the distribution of determinants of random matrices. $\endgroup$
    – whuber
    Oct 13, 2022 at 21:02
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    $\begingroup$ Can you please add new information in comments as an edit to the post! We prefer posts to be self-contained, few people read comments, and comments can be deleted. $\endgroup$ Oct 13, 2022 at 22:45

1 Answer 1

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Because your profile says you have access to Mathematica, you can easily find moments of $Z$:

z = y[1] y[2] y[3] - y[1] x[1] x[2] - y[2] x[3] x[4] - y[3] x[5] x[6] + x[1] x[4] x[5] +
    x[2] x[3] x[6];

dist = TransformedDistribution[z, Flatten[{Table[x[i] \[Distributed] NormalDistribution[0, 1], {i, 6}],
  Table[y[i] \[Distributed] NormalDistribution[μ[i], 1], {i, 3}]}]];

Mean[dist]
(* μ[1] μ[2] μ[3] *)
Variance[dist]
(* 6+2 μ[1]^2+2 μ[2]^2+μ[1]^2 μ[2]^2+2 μ[3]^2 + μ[1]^2 μ[3]^2 + μ[2]^2 μ[3]^2 *)
Skewness[dist]

Skewness

Kurtosis[dist]

Kurtosis

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