Multivariate Normal Probability Research Articles

Fast computation of high-dimensional multivariate normal probabilities

A new efficient method is proposed to compute multivariate normal probabilities over rectangles in high dimensions. The method exploits four variance reduction techniques: conditional Monte Carlo, importance sampling, splitting and control variates. Simulation results are presented that evaluate the performance of the new proposed method. The new method is designed for computing small exceedance probabilities.

Likelihood Analysis of Multivariate Probit Models Using a Parameter Expanded MCEM Algorithm

Multivariate binary data arise in a variety of settings. In this article we propose a practical and efficient computational framework for maximum likelihood estimation of multivariate probit regression models. This approach uses the Monte Carlo expectation maximization (MCEM) algorithm, with parameter expansion to complete the M-step, to avoid the direct evaluation of the intractable multivariate normal orthant probabilities. The parameter expansion not only enables a closed-form solution in the M-step, but also improves efficiency. Using the simulation studies, we compare the performance of our approach with the MCEM algorithms developed by Chib and Greenberg (1998) and Song and Lee (2005), as well as the iterative approach proposed by Li and Schafer (2008). Our approach is further illustrated using a real-world example.

A Chance Constrained Approach to Fractional Programming with Random Numerator

This paper presents a chance constrained programming approach to the problem of maximizing the ratio of two linear functions of decision variables which are subject to linear inequality constraints. The coefficient parameters of the numerator of the objective function are assumed to be random variables with a known multivariate normal probability distribution. A deterministic equivalent of the stochastic linear fractional programming formulation has been obtained and a subsidiary convex program is given to solve the deterministic problem.

New Numerical Algorithm for Multivariate Normal Probabilities in Package mvtnorm

New Numerical Algorithm for Multivariate Normal Probabilities in Package mvtnorm

So Many Correlated Tests, So Little Time! Rapid Adjustment of P Values for Multiple Correlated Tests

Contemporary genetic association studies may test hundreds of thousands of genetic variants for association, often with multiple binary and continuous traits or under more than one model of inheritance. Many of these association tests may be correlated with one another because of linkage disequilibrium between nearby markers and correlation between traits and models. Permutation tests and simulation-based methods are often employed to adjust groups of correlated tests for multiple testing, since conventional methods such as Bonferroni correction are overly conservative when tests are correlated. We present here a method of computing P values adjusted for correlated tests (P(ACT)) that attains the accuracy of permutation or simulation-based tests in much less computation time, and we show that our method applies to many common association tests that are based on multiple traits, markers, and genetic models. Simulation demonstrates that P(ACT) attains the power of permutation testing and provides a valid adjustment for hundreds of correlated association tests. In data analyzed as part of the Finland-United States Investigation of NIDDM Genetics (FUSION) study, we observe a near one-to-one relationship (r(2)>.999) between P(ACT) and the corresponding permutation-based P values, achieving the same precision as permutation testing but thousands of times faster.

A New Reconstruction of Multivariate Normal Orthant Probabilities

SummaryA new method is introduced for geometrically reconstructing orthant probabilities for non-singular multivariate normal distributions. Orthant probabilities are expressed in terms of those for auto-regressive sequences and an efficient method is developed for numerical approximation of the latter. The approach allows more efficient accurate evaluation of the multivariate normal cumulative distribution function than previously, for many situations where the original distribution arises from a graphical model. An implementation is available as a package for the statistical software R and an application is given to multivariate probit models.

Searching for extensible Korobov rules

Extensible lattice sequences have been proposed and studied in [F.J. Hickernell, H.S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, in: G.H. Golub, S.H. Lui, F.T. Luk, R.J. Plemmons (Eds.), Proceedings of the Workshop on Scientific Computing (Hong Kong), Singapore, Springer, Berlin, 1997, pp. 209–215; F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138; F.J. Hickernell, H.Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286–300]. For the special case of extensible Korobov sequences, parameters can be found in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C.Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138]. The searches made to obtain these parameters were based on quality measures that look at several projections of the lattice. Because it is often the case in practice that low-dimensional projections are very important, it is of interest to find parameters for these sequences based on measures that look more closely at these projections. In this paper, we prove the existence of “good” extensible Korobov rules with respect to a quality measure that considers two-dimensional projections. We also report results of experiments made on different problems where the newly obtained parameters compare favorably with those given in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138].

Calculation of Multivariate Normal Probabilities by Simulation, with Applications to Maximum Simulated Likelihood Estimation

We discuss methods for calculating multivariate normal probabilities by simulation and two new Stata programs for this purpose: mdraws for deriving draws from the standard uniform density using either Halton or pseudorandom sequences, and an egen function, mvnp(), for calculating the probabilities themselves. Several illustrations show how the programs may be used for maximum simulated likelihood estimation.

Calculation of Multivariate Normal Probabilities By Simulation, With Applications to Maximum Simulated Likelihood Estimation

We discuss methods for calculating multivariate normal probabilities by simulation and two new Stata programs for this purpose: - mdraws - for deriving draws from the standard uniform density using either Halton or pseudo-random sequences, and an egen function - mvnp() - for calculating the probabilities themselves. Several illustrations show how the programs may be used for maximum simulated likelihood estimation.

Pricing of multi-period rate of return guarantees

The basis for this paper is the pricing of multi-period rate of return guarantees. These guarantees can typically be found in life insurance and pension contracts. We derive closed form solutions, expressed by the cumulative multivariate normal probability distribution, for multi-period rate of return guarantees on both a money market account and a stock. The guarantees of Hipp (1996), Persson and Aase (1997), and Miltersen and Persson (1999) [Options for guaranteed index-linked life insurance, in: Proceedings of the AFIR 1996, 1996, pp. 1463–1483; J. Risk Insur. 64 (4) (1997) 599; Insur.: Math. Econ. 25 (3) (1999) 307] are special cases of our results.

Alternative sampling methods for estimating multivariate normal probabilities

We study the performance of alternative sampling methods for estimating multivariate normal probabilities through the GHK simulator. The sampling methods are randomized versions of some quasi-Monte Carlo samples (Halton, Niederreiter, Niederreiter-Xing sequences and lattice points) and some samples based on orthogonal arrays (Latin hypercube, orthogonal array and orthogonal array based Latin hypercube samples). In general, these samples turn out to have a better performance than Monte Carlo and antithetic Monte Carlo samples. Improvements over these are large for low-dimensional (4 and 10) cases and still significant for dimensions as large as 50.

Multivariate Normal Probabilities: Implementing an Old Idea of Plackett's

This article describes a reduction formula for the computation of multivariate normal probabilities first developed by Plackett. The method is recursive and, despite very promising results, seems not to have been implemented before. Extensive numerical results are given to compare the effectiveness of the approach to other methods from the literature.

A probabilistic one-centre location problem on a network

In this paper we consider the one-centre problem on a network when the speeds on links are stochastic rather than deterministic. Given a desirable time to reach customers residing at the nodes, the objective is to find the location for a facility such that the probability that all nodes are reached within this time threshold is maximized. The problem is formulated, analyzed and solved by using multivariate normal probabilities. The procedure is demonstrated on an example problem.

The Evaluation of General Non-Centred Orthant Probabilities

Summary The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non-centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non-centred orthant probability can be expressed as differences between at most (m−1)! non-centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice.

Onsager Machlup Functional for Stochastic Evolution Equations in a Class of Norms

In this note, we show that the Onsager Machlup functional for stochastic evolution equations on a given separable Hilbert space H, computed in Bardina, Rovira, and Tindel (Onsager Machlup functional for stochastic evolution equations. In Annales Institut Henri Poincaré, 2003, 39, 69–93.), does not depend on the measurable norm considered on the functions from [0, 1] to H, dominating the norm on L 2([0, 1]; H), and satisfying some lower bound conditions for the stochastic convolution associated to our evolution system. The main ingredient of the proof is the extension of some Gaussian correlation inequalities of Hargé (Une inégalité de décorrélation pour la mesure gaussienne, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 11, (1998), 1325–1328) and S˘idak (On multivariate normal probabilities on rectangles (their dependence on correlation), Ann. Math. Statist. 39, (1968), 1425–1434) to the infinite dimensional case.

Computing Multivariate Normal Probabilities: A New Look

This article describes and compares several numerical methods for finding multivariate probabilities over a rectangle. A large computational study shows how the computation times depend on the problem dimensions, the correlation structure, the magnitude of the sought probability, and the required accuracy. No method is uniformly best for all problems and—unlike previous work—this article gives some guidelines to help establish the most promising method a priori. Numerical tests were conducted on approximately 3,000 problems generated randomly in up to 20 dimensions. Our findings indicate that direct integration methods give acceptable results for up to 12-dimensional problems, provided that the probability mass of the rectangle is not too large (less than about 0.9). For problems with small probabilities (less than 0.3) a crude Monte Carlo method gives reasonable results quickly, while bounding procedures perform best on problems with large probabilities (> 0.9). For larger problems numerical integration ...

Comparison of Methods for the Computation of Multivariate t Probabilities

This article compares methods for the numerical computation of multivariate t probabilities for hyper-rectangular integration regions. Methods based on acceptance-rejection, spherical-radial transformations, and separation-of-variables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz for multivariate normal probabilities. These methods allow moderately accurate multivariate t probabilities to be quickly computed for problems with as many as 20 variables. Methods for the noncentral multivariate t distribution are also described.

Computing Multivariate Normal Probabilities: A New Look

This article describes and compares several numerical methods for finding multivariate probabilities over a rectangle. A large computational study shows how the computation times depend on the problem dimensions, the correlation structure, the magnitude of the sought probability, and the required accuracy. No method is uniformly best for all problems and—unlike previous work—this article gives some guidelines to help establish the most promising method a priori. Numerical tests were conducted on approximately 3,000 problems generated randomly in up to 20 dimensions. Our findings indicate that direct integration methods give acceptable results for up to 12-dimensional problems, provided that the probability mass of the rectangle is not too large (less than about 0.9). For problems with small probabilities (less than 0.3) a crude Monte Carlo method gives reasonable results quickly, while bounding procedures perform best on problems with large probabilities (> 0.9). For larger problems numerical integration with quasirandom Korobov points may be considered, as may a decomposition method due to Deák. The best method found four-digit accurate probabilities for every 20-dimensional problem in less than six minutes on a 533MHz Pentium III computer.

Detecting dose-response using contrasts: asymptotic power and sample size determination for binomial data.

Recently, Stewart and Ruberg proposed the use of contrast tests for detecting dose-response relationships. They considered in particular bivariate contrasts for healing rates and gave several possibilities of defining adequate sets of coefficients. This paper extends their work in several directions. First, asymptotic power expressions for both single and multiple contrast tests are derived. Secondly, well known trend tests are rewritten as multiple contrast tests, thus alleviating the inherent problem of choosing adequate contrast coefficients. Thirdly, recent results on the efficient calculation of multivariate normal probabilities overcome the traditional simulation-based methods for the numerical computations. Modifications of the power formulae allow the calculation of sample sizes for given type I and II errors, the spontaneous rate, and the dose-response shape. Some numerical results of a power study for small to moderate sample sizes show that the nominal power is a reasonably good approximation to the actual power. An example from a clinical trial illustrates the practical use of the results.

Probability bounds given by hypercherry trees

In this article new lower and upper bounds are given for the probability of the union of events. For this purpose the new concept of hypercherry trees has been introduced. Earlier the concept of cherry tree and its application for bounding the probability of union of events was introduced by Bukszár and Prékopa. This, based on the cherry tree bound, is always an upper bound, and it can be regarded as a generalisation of the upper bound introduced by Hunter by means of maximum weight spanning trees. Later the Hunter bound was generalised by Tomescu. He used the concept of hypertrees in the framework of uniform hypergraphs and on the basis of these new hypergraph structures it became possible to define not only upper but also lower bounds on the probability of union of events. The new bounds of the paper are generalisations of Tomescu's bounds in the same sense as the upper bound by Bukszár and Prékopa was a generalisation of the Hunter bound. The efficiency of the new bounds is illustrated on some test problems according to multivariate normal probability distribution function calculations.