# Computation of Multivariate Normal and t Probabilities by Alan Genz

By Alan Genz

This ebook describes lately built equipment for exact and effective computation of the mandatory likelihood values for issues of or extra variables. It comprises examples that illustrate the chance computations for numerous applications.

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Extra info for Computation of Multivariate Normal and t Probabilities (Lecture Notes in Statistics)

Example text

K For importance sampling techniques, let hk (x) = j=1 h(xj ) denote the continuous density of k independent and identically distributed random variables, which is decomposable into a product of univariate densities h. 13) where y1 , . . , yN is a sample from the importance density hk . The basic idea of importance sampling is to choose hk so that the generated random numbers are concentrated in the region where φk,Ik is large. 13) simpliﬁes to N k 1 Φ(aj ≤ ltj yi ≤ bj ). 2). 5), then the resulting computations are equivalent to GHK importance sampling.

Q = ⎢ . . . ⎥ , ⎣? ··· ? ⎦ ⎣ 0 ? ··· ··· ? 0⎦ ? ··· ··· ? 0 0 ··· 0 ? ’s are either zero or nonzero entries. Now, −∞ < Ly ≤ b becomes −∞ < LQQt y = LQz ≤ b, with z = Qt y, and Φk (−∞ ≤ x ≤ b; Σ) = Φk (−∞ ≤ LQz ≤ b; Ik ). t t t t because e−z z/2 = e−y QQ y/2 = e−y y/2 . Then, Fourier-Motzkin elimination (Schechter, 1998) can be used to replace −∞ < LQz ≤ b by k − 1 inequalities (1) of the form −∞ < Mj z ≤ bj , for j = 1, 2, . . , k − 1, where ⎡ (1) Mj ? ⎢ .. ⎢. ⎢ = ⎢? ⎢ ⎣? 0 ⎤ 0 .. ⎥ ⎥ .

0 .. (1) Now, with signs sj = ±1, depending on the Mj diagonal entries, k−1 Φk (−∞ ≤ x ≤ b; Σ) = e− sj j=1 (1) zt z 2 (2π)m dz. −∞