Section 2d

2.4 Distribution of Quadratic Forms

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Problem 2d.1

Show that the m.g.f. for (2.10) is ∏1n(1−2tdi)−1/2\prod_{1}^{n}\left(1-2 t d_{i}\right)^{-1 / 2}.

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Problem 2d.2

Let Y∼Nn(0,In)\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathrm{I}_{n}\right) and let A\mathbf{A} be symmetric.

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(a)

Show that the m.g.f. of YTAY\mathbf{Y}^{T} \mathbf{A Y} is [det⁡(In−2tA)]−1/2\left[\operatorname {det}\left(\mathbf{I}_{n}-2 t \mathbf{A}\right)\right]^{-1 / 2}.

(b)

If A\mathbf{A} is idempotent of rank⁡r\operatorname {rank} r, show that the m.g.f. is (1−2t)−r/2(1-2 t)^{-r / 2}.

(c)

Find the m.g.f. if Y∼Nn(0,Σ)\mathbf{Y} \sim N_{n}(\mathbf{0}, \mathbf{\Sigma }).

Problem 2d.3

If Y∼N2(0,I2)\mathbf{Y} \sim N_{2}\left(0, \mathbf{I}_{2}\right), find values of aa and bb such that

a(Y1−Y2)2+b(Y1+Y2)2∼χ22 a\left(Y_{1}-Y_{2}\right)^{2}+b\left(Y_{1}+Y_{2}\right)^{2} \sim \chi _{2}^{2}
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Problem 2d.4

Suppose that Y∼N3(0,In)\mathbf{Y} \sim N_{3}\left(\mathbf{0}, \mathbf{I}_{n}\right). Show that

13[(Y1−Y2)2+(Y2−Y3)2+(Y3−Y1)2] \frac{1}{3}\left[\left(Y_{1}-Y_{2}\right)^{2}+\left(Y_{2}-Y_{3}\right)^{2}+\left(Y_{3}-Y_{1}\right)^{2}\right]

has a χ22\chi_{2}^{2} distribution. Does some multiple of

(Y1−Y2)2+(Y2−Y3)2+⋯+(Yn−1−Yn)2+(Yn−Y1)2 \left(Y_{1}-Y_{2}\right)^{2}+\left(Y_{2}-Y_{3}\right)^{2}+\cdots +\left(Y_{n-1}-Y_{n}\right)^{2}+\left(Y_{n}-Y_{1}\right)^{2}

have a chi-squared distribution for general nn?

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Problem 2d.5

Let Y∼Nn(0,In)\mathbf{Y} \sim N_{n}\left(0, \mathrm{I}_{n}\right) and let A\mathbf{A} and B\mathbf{B} be symmetric. Show that the joint m.g.f. of YTAY\mathbf{Y}^{T} \mathbf{A Y} and YTBY\mathbf{Y}^{T} \mathbf{B Y} is [det⁡(In−2sA−2tB)]−1/2\left[\operatorname {det}\left(\mathbf{I}_{n}-2 s \mathbf{A}-2 t B\right)\right]^{-1 / 2}. Hence show that the two quadratic forms are independent if AB=0\mathbf{A B} = 0.

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