Computation of Multivariate Normal and t Probabilities

By Alan Genz, Frank Bretz. 2009

977 citations recorded by the source corpus.

Multivariate normal and t probabilities are needed for statistical inference in many applications. Modern statistical computation packages provide functions for the computation of these probabilities for problems with one or two variables. This book describes recently developed methods for accurate and efficient computation of the required probability values for problems with two or more variables. The book discusses methods for specialized problems as well as methods for general problems. The book includes examples that illustrate the probability computations for a variety of applications.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Nonlinear Statistical Learning with Truncated Gaussian Graphical Models (2016, methodology): “…not alter the quadratic energy function or the conditional dependency structure of the GGM, it is possible for a TGGM inference algorithm to utilize many well-studied properties of multi-variate Gaussian distributions and the associated numerical methods (Johnson et al., 1994; Genz & Bretz, 2009).”
The normal law under linear restrictions: simulation and estimation via minimax tilting (2016, methodology): “Genz and Bretz (2009) suggest the following improvement of the SOV algorithm.”
Flexible dependence modeling of operational risk losses and its impact on total capital requirements (2014, methodology): “Gaussian or Student’s t copulas may, on the other hand, be used here, since fast algorithms for the evaluation of CW are available [Genz and Bretz, 2009].”
Simultaneous Statistical Inference (2014, methodology): “This has been demonstrated, for instance, by Hothorn et al. (2008) and Bretz et al. (2010) in general parametric models. For particular applications in genetic association studies (cf. Chap. 9), central limit theorems for multinomial distributions, together with positive dependency properties of mul”
Upper tail dependence in rainfall extremes: would we know it if we saw it? (2015, methodology): “The analyses were performed in R (R Core Team 2013) by using the contributed packages POT (Ribatet 2006), mvtnorm (Genz and Bretz 2009; Genz et al. 2011), and copula (Yan 2007; Kojadinovic and Yan 2010).”
Efficient inference and simulation for elliptical Pareto processes (2013, methodology): “ψˆ. A practical inconvenience for maximum likelihood inference based on L1 or L2 is the need to calculate the tprobabilities in (11) and (14). They can be calculated using Monte Carlo approximations (Genz and Bretz, 2009), but the use of full likelihood inference might be too slow if D>50. In lar”
Rank-based multiple test procedures and simultaneous confidence intervals (2012, methodology): “(2001) [3] and Genz and Bretz (2009) [19].”
Modeling asymptotically independent spatial extremes based on Laplace random fields (2015, methodology): “To avoid plain Monte-Carlo approaches for calculating probabilities pr(X 6≤ x) = pr(Xj > xj for at least one j) which can be too inaccurate or too slow in certain cases, we can instead exploit tailor-made algorithms for calculating multivariate normal probabilities ΦΣ(x) (Genz and Bretz, 2009).”
nparcomp: An R Software Package for Nonparametric Multiple Comparisons and Simultaneous Confidence Intervals (2015, methodology): “Multiplicity adjusted p-values for the individual hypotheses are computed by using the cumulative multivariate t distribution function (Genz and Bretz 2009) instead of the univariate t distribution function.”
Asymptotic Analysis of Multivariate Tail Conditional Expectations (2012, background): “…example, if loss variables follow a multivariate t distribution as is the case in the numerical examples discussed in Section 3, then the proportionality constants can be evaluated through numerical integration or simulation on multivariate normally distributed variables (see Genz and Bretz 2009).”
Ecologists overestimate the importance of predictor variables in model averaging: a plea for cautious interpretations (2014, methodology): “We controlled for the correlation structure both between the response variables and predictor variables and among predictor variables using a Cholesky decomposition (Genz & Bretz 2009).”
Approximate Gaussian Integration using Expectation Propagation (2011, methodology): “The integrand of Equation 1 is transformed to the unit cube, and heuristic choices on integration order are made before quasi-random integration or lattice-point rules are used (Genz and Bretz, 2009).”
Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures (2016, background): “The computation of G in (4) and G I in (6) relies on the Gaussian distribution function in dimension D and | I c | respectively, which can be estimated without bias (Genz and Bretz, 2009).”
Excursion and contour uncertainty regions for latent Gaussian models (2012, methodology): “…can then eﬃciently be evaluated using a quasi MC (QMC) method where the uniform random numbers in the ordinary MC integrator are replaced by some deterministic sequence of points chosen to reduce the probabilistic error bound of the crude MC integrator, see Genz and Bretz (2009) for details.”
The Bivariate Normal Copula (2009, methodology): “For background on multivariate methods we refer to the recent book by Bretz & Genz (2009).”