The DOI matches its Crossref record; 9 of the 9 references it cites were found in a registry or an index.
Numerical Computation of Multivariate Normal Probabilities
Abstract
The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation with currently available software.
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Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“As a result, unlike many existing estimators (Vijverberg, 1997;Genz, 1992), we can construct an exact (albeit conservative) confidence interval for ℓ as follows.”
From The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting (2017) · cites this paper as Genz, 1992
In the citing paper
“Because closed-form evaluation of the MVN CDF is generally unavailable in moderate and high dimensions, efficient numerical methods remain essential across these application areas (Genz 1992;Genz and Bretz 2002).”
From Doubly blocked variant of Genz’s method for multivariate normal integration (2026) · cites this paper as Genz 1992
In the citing paper
“Genz's algorithm (Genz 1992;Genz and Bretz 2002) computes this probability by applying a series of transformations that map the integration region to the unit hypercube [0, 1] n-1 .”
From Doubly blocked variant of Genz’s method for multivariate normal integration (2026) · cites this paper as Genz 1992
In the citing paper
“However, MVN probabilities are analytically intractable, and direct application of numerical integration rules often leads to highly inaccurate estimates (Genz 1992).”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Genz 1992
In the citing paper
“However, MVN probabilities are analytically intractable, and direct application of numerical integration rules often leads to highly inaccurate estimates (Genz 1992).”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Genz 1992
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Citations
Cites
The Multivariate Normal Distribution
Izenman, A. J., & Tong, Y. L. (1991) · Technometrics · doi:10.1080/00401706.1991.10484897 · Cited by 662
The Multivariate Normal Distribution
Izenman, A. J., & Tong, Y. L. (1991) · Technometrics · doi:10.2307/1269441 · Cited by 647
The Multivariate Normal Distribution
Tong, Y. L. (1990) · Springer series in statistics · doi:10.1007/978-1-4613-9655-0 · Cited by 488
Algorithm AS 195: Multivariate Normal Probabilities with Error Bound
Schervish, M. J. (1984) · Applied Statistics · doi:10.2307/2347670 · Cited by 265
On the computation of the bivariate normal integral
Drezner, Z., & Wesolowsky, G. O. (1990) · Journal of Statistical Computation and Simulation · doi:10.1080/00949659008811236 · Cited by 201
Berntsen, J., Espelid, T. O., & Genz, A. C. (1991) · ACM Transactions on Mathematical Software · doi:10.1145/210232.210234 · Cited by 176
Self-citationGenerating Correlation Matrices
Marsaglia, G., & Olkin, I. (1984) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0905034 · Cited by 108
Algorithm 462: bivariate normal distribution
Donnelly, T. G. (1973) · Communications of the ACM · doi:10.1145/362375.362414 · Cited by 89
A Simple Approximation for Bivariate and Trivariate Normal Integrals
Cox, D. R., & Wermuth, N. (1991) · International Statistical Review · doi:10.2307/1403446 · Cited by 62
Cited by
Simultaneous Inference in General Parametric Models
Hothorn, T., Bretz, F., & Westfall, P. (2008) · Biometrical Journal · doi:10.1002/bimj.200810425 · Cited by 13,847
Unbiased Recursive Partitioning: A Conditional Inference Framework
Hothorn, T., Hornik, K., & Zeileis, A. (2006) · Journal of Computational and Graphical Statistics · doi:10.1198/106186006x133933 · Cited by 4,021
Lichtenstein, P., Yip, B. H., Björk, C., Pawitan, Y., Cannon, T. D., Sullivan, P. F., & Hultman, C. M. (2009) · The Lancet · doi:10.1016/s0140-6736(09)60072-6 · Cited by 2,125
Implementing a Class of Permutation Tests: The coin Package
Hothorn, T., Hornik, K., Wiel, M. A. V. D., & Zeileis, A. (2008) · Journal of Statistical Software · doi:10.18637/jss.v028.i08 · Cited by 1,492
Fitting Fully Observed Recursive Mixed-process Models with cmp
Roodman, D. (2011) · The Stata Journal: Promoting communications on statistics and Stata · doi:10.1177/1536867x1101100202 · Cited by 1,485
Metadata
- Authors
- Genz, A.
- Affiliations
- Washington State University (US)
- Corresponding author
- Genz, Alan
- Venue
- Journal of Computational and Graphical Statistics
- Publisher
- Informa UK Limited
- Published
- 1992-06
- Volume
- 1
- Issue
- 2
- Pages
- 141-149
- ISSN
- 1061-8600 (ISSN-L)
- 1537-2715
- OpenAlex
- W2061750459
- Semantic Scholar
- 860c2384fcd7aeea66c7421b80863527cb1e24d4
- Cited by
- 1,020
- References
- 8
- Citations by year
- 2026: 22 · 2025: 15 · 2024: 26 · 2023: 25 · 2022: 66 · 2021: 46 · 2020: 49 · 2019: 33 · 2018: 39 · 2017: 35 · 2016: 51 · 2015: 51 · 2014: 39 · 2013: 61 · 2012: 57
- FWCI
- 2.78
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Computational and Graphical Statistics (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Keywords
- multivariate normal distribution, multivariate normal probabilities, numerical integration, coordinate transformation
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- CoChat. (2026). Citation check: Numerical Computation of Multivariate Normal Probabilities. https://app.cochat.ai/papers/10.1080/10618600.1992.10477010
- The paper
- Genz, A. (1992). Numerical Computation of Multivariate Normal Probabilities. Journal of Computational and Graphical Statistics. https://doi.org/10.1080/10618600.1992.10477010
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
