Dimension Reduction Methods with Application to ... - Rice University
Dimension Reduction Methods with Application to ... - Rice University
Dimension Reduction Methods with Application to ... - Rice University
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OUTLINE OF PROOF<br />
◮ Gamma-Poisson Relationship: Suppose X ∼ Gamma(d, 1), for<br />
d = 1, 2, 3, . . . , and Y ∼ Poisson(x), then<br />
P(X ≥ x) =<br />
∫ ∞<br />
◮ ||y|| 2 = ∑ k<br />
j=1 y2 j ∼ χ 2 k<br />
x<br />
1<br />
∑d−1<br />
Γ(d) zd−1 e −z x y e −x<br />
dz =<br />
y!<br />
y=0<br />
◮ Let d = k/2, α 1 = k(1 + ɛ), and α 2 = k(1 − ɛ), then,<br />
◮ Right tail prob.:<br />
◮ Left tail prob.:<br />
= P(Y ≤ d − 1)<br />
∫ ∞<br />
P[||y|| 2 1<br />
∑d−1<br />
≥ α 1 ] =<br />
α 1 /2 Γ(a) zd−1 e −z (α 1 /2) y e −α 1/2<br />
dz =<br />
y!<br />
y=0<br />
∫ α2<br />
P[||y|| 2 /2 1<br />
≤ α 2 ] =<br />
0 Γ(a) zd−1 e −z dz =<br />
∞∑ (α 2 /2) y e −α 2/2<br />
y=d<br />
y!<br />
(10)<br />
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