Abstract
Known methods are described for evaluating the multivariate normal distribution function in general and in a number of special cases. Methods are presented for evaluation of the one-dimensional normal distribution function and its inverse.
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Literature cited
V. I. Avramenko, “On the computation of the probability that a normal vector lie in an n-dimensional parallelepiped,” in: Electronics and Modeling [in Russian], No. 12, Naukova Dumka, Kiev (1976), pp. 65–69.
A. I. Avramenko and V. V. Poznyakov, “Approximate evaluation of n-dimensional normal distribution functions by means of Edgeworth series,” Kibern. Vychisl. Tekh., Resp. Mezhved. Sb., No. 18, 31–36 (1972).
V. I. Avramenko and V. V. Poznyakov, “Evaluation of the distribution function of a 4-dimensional normal vector by means of two-dimensional Edgeworth series,” Tochnost' Nadezhnost' Kibern. Sist., Resp. Mezhved. Sb., No. 1, 36–41 (1973).
V. S. Vedrov and N. V. Derevnina, “Approximate formulas for computing the Mills ratio and the La-place distribution function,” Uch. Zap. Tsentr. Aerogidrodin. Inst.,9, No. 2, 122–126 (1978).
V. S. Godlevskii and A. N. Zavarin, “A method of approximate computation of multidimensional normal integrals,” Tochnost' Nadezhnost' Kibern. Sist., Resp. Mezhved. Sb., No. 4, 30–35 (1976).
P. N. Dubner, Computation of Direct and Inverse Distribution Functions, Mosk. Univ., Moscow (1971); Knizh. Leotpis', No. 1, 22 (1972).
B. B. Dunaev, “A method of computing the two-dimensional normal distribution,” Prikl. Mat., No. 3, 82–94 (1971).
M. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 1, Distribution Theory, Hafner (1969).
P. I. Kuznetsov and A. S. Yudina, “Some asymptotic expansions of an incomplete probability Integral,” Teor. Veroyatn. Ee Primen.,18, No. 2, 367–371 (1973).
V. A. Kutin, “A formula for inverting the probability integral,” Uch. Zap. Perm. Univ., No. 309, 99–101 (1974).
B. R. Levin and Ya. A. Fomin, “On a method of approximate evaluation of multidimensional integral distribution functions of stochastic processes,” Probl. Peredachi Inf.,6, No. 4, 102–108 (1970).
G. V. Martynov, “Iterative formulas for normal quantiles,” in: Numerical Methods of Mathematical Statistics. Algorithms and Programs [in Russian], Moscow State Univ., Moscow (1976), pp. 22–29.
A. K. Mitropol'skii, The Probability Integral [in Russian], 2nd ed., Leningr. Univ., Leningrad (1972).
V. S. Mukha, “Evaluation of integrals connected with a multidimensional Gaussian distribution,” Izv. Leningr. Elektrotekh. Inst., No. 160, 27–30 (1974).
E. S. Pereverzev, “On an approximate method of evaluating multidimensional normal distribution functions,” Avtom. Vychisl. Tekh., No. 5, 89–91 (1971).
E. S. Pereverzev, “On an algorithm for the approximate evaluation of a multidimensional normal distribution,” Gidroaeromekh. Teor. Uprugosti, Mezhvuz. Nauch. Sb., No. 14, 188–191 (1972).
V. V. Poznyakov, “Approximate evaluation of the multidimensional normal distribution,” Kibern. Vychisl. Tekh. Resp. Mezhved. Sb., No. 10, 16–24 (1971).
V. V. Poznyakov, “On a representation of a multidimensional normal distribution,” Ukr. Mat. Zh.,23, No. 4, 562–566 (1971).
V. V. Poznyakov, “A method of constructing approximate estimates for multidimensional normal integrals,” Tochnost' Nadezhnost' Kibern. Sist., Resp. Mezhved. Sb., No. 4, 27–30 (1976).
E. V. Rakova, “On ‘inverse probability functions,’” Tr. Uchebn. Inst. Svyazi, Minstvo. Svyazi SSSR, No. 52, 166–172 (1970).
N. V. Smirnov and L. N. Bol'shev, Tables for Evaluating Two-Dimensional Normal Distribution Functions [in Russian], Akad. Nauk SSSR, Moscow (1962).
R. S. Sudakov and A. N. Chekanov, “An approximate method for evaluating multidimensional normal integrals in reliability problems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 1, 69–75 (1972).
Tables of Probability Functions [in Russian], Vols. I, II, Computer Center, Akad. Nauk SSSR, Moscow (1970).
D.I. Tveritinov, “On an approach to the evaluation of the multidimensional normal distribution,” in: Probabilistic and Statistical Methods of Investigating Complex Systems [in Russian], Naukova Dumka, Kiev (1976), pp. 88–93.
D.I. Tveritinov, “On evaluating multidimensional integrals of the normal function of probability density,” Tr. Mosk. Vyssh. Tekh. Uchil. im. N. E. Bauman, No. 218, 21–23 (1976).
I. G. Abrahamson, “Orthant probabilities for the quadrivariate normal distribution,” Ann. Math. Stat.,35, No. 4, 1685–1703 (1964).
A. G. Adams, “Algorithm 39. Areas under normal curve,” Comput. J.,12, 197–198 (1969).
Jiří Anděl, “On multiple normal probabilities of rectangles,” Appl. Math.,16, No. 3, 172–181 (1971).
A. A. Anis and E. H. Lloid, “On the range of partial sums of a finite number of independent normal variates,” Biometrika,40, Nos. 1–2, 35–42 (1953).
R. H. Bacon, “Approximations to multivariate normal orthant probabilities,” Ann. Math. Stat.,34, No. 1, 191–198 (1963).
J. D. Beasley and S. G. Springer, “Algorithm AS 111. The percentage points of the normal distribution,” Appl. Stat.,26, No. 1, 118–121 (1977).
R. E. Beckhofer and A. C. Tamhane, “An iterated integral representation for a multivariate normal integral having block covariance structure,” Biometrika,61, No. 3, 615–619 (1974).
Z. W. Birnhaum, “An inequality for Mills' ratio,” Ann. Math. Stat.,13, No. 2, 245–246 (1942).
D. M. Borth, “A modification of Owen's method for computing the bivariate normal integral,” Appl. Stat.,22, No. 1, 82–85 (1973).
A. V. Boyd, “Inequalities for Mills' ratio,” Repts. Stat. Appl. Res., Union Jpn. Sci. Eng.,6, No. 2, 44–46 (1959).
I. W. Burr, “A useful approximation to the normal distribution function, with application to simulation,” Technometrics,9, No. 4, 647–651 (1967).
T. Cacoullos and M. Sobel, “An inverse-sampling procedure for selecting the most probable event in a multinomial distribution,” in: Multivariate Analysis, Academic Press, New York-London (1966), pp. 423–455.
J. H. Cadwell, “The bivariate normal integral,” Biometrika,38, Nos. 3–4, 475–479 (1951).
M. C. Cheng, “The orthant probabilities of four Gaussian variates,” Ann. Math. Stat.,40, No. 1, 152–161 (1969).
D. R. Childs, “Reduction of the multivariate normal integral to characteristic form,” Biometrika,54, Nos. 1–2, 293–300 (1967).
Jae-Rong Choi, “An equality involving orthant probabilities,” Commun. Stat.,4, No. 12, 1167–1175 (1975).
Jae-Rong Choi, “On method of computing singular orthant probability and its application,” Mem. Fac. Sci. Kyushu Univ.,A30, No. 1, 7–13 (1976).
J. T. Chu, “On bounds for the normal integral,” Biometrika,42, Nos. 1–2, 263–265 (1955).
B. E. Cooper, “Algorithm AS 2. The normal integral,” Appl. Stat.,17, No. 2 186–188 (1968).
B. E. Cooper, “Algorithm AS 4. An auxiliary function for distribution integrals,” Appl. Stat.,17, No. 2, 190–192 (1968).
B. E. Cooper, “Algorithm AS 4. An auxiliary function for distribution integrals. (Corrigendum),” Appl. Stat.,19, No. 2, 204 (1970).
S. W. Cunningham, “From normal integral to deviate. Algorithm AS 24,” Appl. Stat.,18, No. 3, 290–293 (1969).
R. N. Curnow and C. W. Dunnett, “The numerical evaluation of certain multivariate normal integrals,” Ann. Math. Stat.,33, No. 2, 571–579 (1962).
M. G. Dagenais and G. Lonergan, “Accurate evaluation of bivariate normal integrals,” Cah. Cent. Etud. Rech. Oper.,16, No. 2, 153–160 (1974).
D. J. Daley, “Computation of bi- and tri-variate normal integrals,” Appl. Stat.,23, No. 3, 435–438 (1974).
Sisir Chandra Das, “The numerical evaluation of a class of integrals,” Proc. Cambridge Philos. Soc.,52, No. 3, 442–448 (1956).
F. N. David, “A note of the evaluation of the multivariate normal integral,” Biometrika,40, Nos. 3–4, 458–459 (1953).
F. N. David and C. L. Mallows, “The variance of Spearman's rho in normal samples,” Biometrika,48, Nos. 1–2, 19–28 (1961).
H. A. David and F. B. Six, “Sign distribution of standard multinormal variables with equal positive correlation,” Rev. Inst. Int. Stat.,39, No. 1, 1–3 (1971).
Deák István, A többdimenziós normális eloszlásfüggvény Monte Carlo integrálással történő kiszámit-ásának számitógépes tapasztalatai. Magy. tud. akad. Számítástechn. és automatiz. kut. intéz. közl., No. 19, 47–59 (1978).
T. G. Donnelly, “Algorithm 462. Bivariate normal distribution,” Commun. ACM,16, No. 10, 638 (1973).
O. J. Dunn, “Estimation of the means of dependent variables,” Ann. Math. Stat.,29, No. 4, 1095–1111 (1958).
C. W. Dunnett and M. Sobel, “Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distributions,” Biometrika,42, Nos. 1–2, 258–260 (1955).
J. E. Dutt, “A representation of multivariate normal probability integrals by integral transforms,” Biometrika,60, No. 3, 637–645 (1973).
J. E. Dutt, “Numerical aspects of multivariate normal probabilities in econometric models,” Ann. Econ. Social Meas.,5, No. 4, 547–561 (1976).
J. E. Dutt and T. K. Lin, “A short table for computing normal orthant probabilities dimensions 4 and 5,” Stat. Comput. Simul.,4, No. 2, 95–120 (1975).
A. W. F. Edwards, “A linkage for drawing the normal distribution,” Appl. Stat.,12, No. 1, 44–45 (1963).
Y. Escoufier, “Calculs de probabilités par une méthode de Monte-Carlo pour une variable p-normale,” Rev. Stat. Appl.,15, No. 4, 5–15 (1967).
R. D. Gordon, “Values of Mills' ratio of area to bounding ordinate of normal robability integral for large values of the argument,” Ann. Math. Stat.,12, No. 3, 364–366 (1941).
H. L. Gray and W. R. Schucany, “On the evaluation of distribution functions,” J. Am. Stat. Assoc.,63, No. 322, 715–720 (1968).
S. S. Gupta, “Probability integrals of multivariate normal and multivariate t,” Ann. Math. Stat.,34, No. 3, 792–828 (1963).
S. S. Gupta, “Bibliography on the multivariate normal integrals and related topics,” Ann. Math. Stat.,34, No. 3, 829–838 (1963).
S. S. Gupta, “A note on some inequalities for multivariate normal distribution,” Bull. Calcutta Stat. Assoc.,18, Nos. 71–72, 179–180 (1969).
S. S. Gupta and M. N. Waknis, “A system of inequalities for the incomplete gamma function and the normal integral,” Ann. Math. Stat.,36, No. 1, 139–149 (1965).
J. B. S. Haldane, “Simple approximations to the probability integral and P(χ1, 1) when both are small,” Sankhya,A23, No. 1, 9–10 (1961).
Hugo C. Hamaker, “Approximating the cumulative normal distribution and its inverse,” Appl. Stat.,27, No. 1, 76–77 (1978).
C. Hastings, “Approximations for Digital Computers, Univ. Press, Princeton, New York, Oxford Univ. Press, London (1955).
F. Hebrant, “Numerical techniques and algorithms for the normal and chi-square probability functions and their inverses,” Biometrie-Praximetrie,17, Nos. 3–4 (1977).
F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York-London (1956).
I. D. Hill, “A remark on algorithm AS 2 ‘The normal integral,’” Appl. Stat.,18, No. 3, 299–300 (1969).
I. D. Hill, “Algorithm AS 66. The normal integral,” Appl. Stat.,22, No. 3, 424–427 (1973).
I. D. Hill and S. A. Jouse, “Algorithm 304. Normal curve integral,” Commun. ACM,10, 374–375 (1967).
G. Hornecker, “Évaluation approchée de la meileure approximation polinomiale d'ordre n de f(x) sur un segment fini (a, b),” Chiffres,1, 157–169 (1958).
D. Ibbetson, “Algorithm 209,” Commun. ACM,6, 616 (1963).
P. Ihm, “Numerical evaluation of certain multivariate normal integrals,” Sankhya,21, Nos. 3–4, 363–366 (1959).
S. John, “On the evaluation of the probability integral of a multivariate normal distribution,” Sankhya,21, Nos. 3–4, 367–370 (1959).
N. L. Johnson and S. Kotz, “Continuous multivariate distributions,” Publishers' Weekly,202, No. 17, 61 (1972).
M. G. Kendall, “Proof of relations connected with the tetrachoric series and its generalization,” Biometrika,32, Nos. 1–2, 196–198 (1941).
D. F. Kerridge and C. W. Cook, “Yet another series for the normal integral,” Biometrika,63, No. 2, 401–403 (1976).
C. G. Khatri, “On certain inequalities for normal distributions and their applications to simultaneous confidence bounds,” Ann. Math. Stat.,38, No. 6, 1853–1867 (1967).
W. F. Kibble, “An extension of a theorem of Mehler's on Hermite polynomials,” Proc. Cambridge Philos. Soc.,41, 12–15 (1945).
P. S. Laplace, Théorie Analitique de Probabilités. Part 2, Courcier, Paris (1812).
G. Marsaglia, “Expressing the normal distribution with covariance matrix A + B in terms of one with covariance matrix A,” Biometrika,50, Nos. 3–4, 535–538 (1963).
J. A. McFadden, “Urn models of correlation and a comparison with the multivariate normal integral,” Ann. Math. Stat.,26, No. 3, 478–489 (1955).
J. A. McFadden, “An approximation for the symmetric quadrivariate normal integral,” Biometrika,43, No. 1, 206–207 (1956).
J. A. McFadden, “Two expansions for the quadrivariate normal integral,” Biometrika,47, Nos. 3–4, 325–333 (1960).
J. A. McFadden and J. L. Lewis, “Multivariate normal integrals for highly correlated samples from a Wiener process,” J. Appl. Probab.,4, No. 2, 303–312 (1967).
J. P. Mills, “Tables of ratio: area to bounding ordinate, for any portion of the normal cure,” Biometrika,18, 395–400 (1926).
R. C. Milton, “Computer evaluation of the multivariate normal integral,” Technometrics,14, No. 4, 881–889 (1972).
R. C. Milton and R. Hotchkiss, “Computer evaluation of the normal and inverse normal distribution functions,” Technometrics,11, No. 4, 817–822 (1969).
P. A. P. Moran, “Rank correlation and product-moment correlation,” Biometrika,35, Nos. 1–2, 203–206 (1948).
P. A. P. Moran, “The numerical evaluation of a class of integrals,” Proc. Cambridge Philos. Soc.,52, No. 2, 230–233 (1956).
Yoshisada Murotsu, Masaaki Yonezawa, Fuminori Oba, and Kazukuni Niwa, “A method for calculating multidimensional Gaussian distribution,” Bull. Univ. Osaka Prefect.,A24, No. 2, 193–204 (1975).
V. N. Murti, “On a result of Birnbaum regarding the skewness of X in a bivariate normal population,” J. Indian Soc. Agric. Stat.,4, 85–87 (1962).
C. Nicholson, “The probability integral for two variables,” Biometrika,33, No. 1, 59–72 (1943).
R. E. Odeh and J. O. Evans, “Algorithm AS 70. The percentage points of the normal distribution,” Appl. Stat.,23, No. 1, 96–97 (1974).
D. B. Owen, “Tables for computing bivariate normal probabilities,” Ann. Math. Stat.,27, No. 4, 1075–1090 (1956).
D. B. Owen and J. M. Wiesen, “A method of computing bivariate normal probabilities with an application to handling errors in testing and measuring,” Bell Syst. Tech. J.,38, No. 2, 553–572 (1959).
E. Page, “Approximations to the cumulative normal function and its inverse for use on a pocket calculator,” Appl. Stat.,26, No. 1, 75–76 (1977).
K. Pearson, “Mathematical contributions to the theory of evolution. VII. On the correlation of characters not quantitatively measurable,” Philos. Trans. R. Soc. London,A195, 1–47 (1901).
R. L. Plackett, “A reduction formula for normal multivariate integrals,” Biometrika,41, Nos. 3–4, 351–360 (1954).
G. Polyà, “Remarks on computing the probability integral in one and two dimensions,” Proc. Berkeley Symp., 63–78 (1949).
W. D. Ray and A. E. N. T. Pitman, “Chebyshev polynomial and other new approximations to Mills's ratio,” Ann. Math. Stat.,34, No. 3, 892–902 (1963).
H. Ruben, “On the moments of order statistics in samples from normal populations,” Biometrika,41, Nos. 1–2, 200–227 (1954).
H. Ruben, “Probability content of regions under spherical normal distributions. III. The bivariate normal integral,” Ann. Math. Stat.,32, No. 1, 171–186 (1961).
H. Ruben, “On the numerical evaluation of a class of multivariate normal integrals,” Proc. Soc. Edinburgh,A65, No. 3, 272–281 (1959–1960).
H. Ruben, “A new asymptotic expansion for the normal probability integral and Mills' ratio,” J. &.R. Stat. Soc.,B24, No. 1, 177–179 (1962).
H. Ruben, “An asymptotic expansion for a class of multivariate normal integrals,” J. Austral. Math. Soc.,2, No. 3, 253–264 (1962).
H. Ruben, “A convergent asymptotic expansion for Mills' ratio and the normal probability integral in terms of rational functions,” Math. Ann.,151, No. 4, 355–364 (1963).
H. Ruben, “An asymptotic expansion for the multivariate normal distribution and Mills' ratio,” J. Res. Nat. Bur. Stand.,B68 No. 1, 3–11 (1964).
H. Ruben, “Irrational fraction approximations to Mills' ratio,” Biometrika,51, Nos. 3–4, 339–345 (1964).
M. R. Sampford, “Some inequalities on Mills' ratio and related functions,” Ann. Math. Stat.,24, No. 1, 130–132 (1953).
I. R. Savage, “Mills' ratio for multivariate normal distributions,” J. Res. Nat. Bur. Stand.,B36, No. 3, 93–96 (1962).
A. Scott, “A note on conservative confidence regions for the mean of a multivariate normal,” Ann. Math. Stat.,38, No. 1, 278–280 (1967).
L. Schläfli, “On the multiple integral ∫ndxdy...dz whose limits are p1=a1x+b1y+...+h1z>0, p2>0,..., pn>0 and x2+...+z2<1,” Q. J. Pure Appl. Math.,2, 261–301 (1858).
L. Schläfli, “On the multiple integral ∫ndxdy...dz whose limits are p1=a1x+b1y+...+h1z>0, p2>0,..., pn>0 and x2+z2<1,” Q. J. Pure Appl. Math.,3, 54–68, 97–107 (1860).
O. Schlömilch, “Compedium der Höheren analysis,”2, 265–270 (1895).
L. R. Shenton, “Inequalities for the normal integral including a new continued fraction,” Biometrika,41, Nos. 1–2, 177–189 (1954).
W. F. Sheppard, “On the calculation of the double-integral expressing normal correlation,” Trans, Cambridge Philos. Soc.,19, 23–26 (1900).
W. F. Sheppard, “The probability integral,” British Assoc. Math. Tables,7 (1939).
Z. Sidác, “Remarks on Anděl's paper ‘On multiple normal probabilities of rectangles,’” Appl. Math.,16, No. 3, 182–187 (1971).
E. A. Silver, “A safety factor approximation based on Tukey's lambda distribution,” Oper. Res. Q.,28, No. 3/ii, 743–746 (1977).
P. N. Somerville, “Some problems on optimum sampling,” Biometrika,41, Nos. 3–4, 420–429 (1954).
Man Mohan Sondhi, “A note on the quadrivariate normal integral,” Biometrika,48, Nos. 1–2, 201–203 (1961).
R. R. Sowden and J. R. Ashford, “Computation of the bivariate normal integral,” Appl. Stat.,18, No. 2, 169–180 (1969).
G. P. Steck, “A table for computing trivariate normal probabilities,” Ann. Math. Stat.,29, No. 3, 780–800 (1958).
G. P. Steck, “Orthant probabilities for the equicorrelated multivariate normal distribution,” Biometrika,49, Nos. 3–4, 433–445 (1962).
G. P. Steck and D. B. Owen, “A note on the equicorrelated multivariate normal distribution,” Biometrika,49, Nos. 1–2, 269–271 (1962).
A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, New Jersey (1966).
A. Stuart, “Equally correlated variates and the multinormal integral,” J. R. Stat. Soc.,B20, No. 2, 373–378 (1958).
Tamás Szántai, “Egy eljárás a többdimenziós normális eloszlásfüggvénty és gradiense értékeinek meghatározására,” Alkalm. Mat. Lapok.,2, Nos. 1–2, 27–39 (1976).
R. F. Tate, “On a double inequality of the normal distribution,” Ann. Math. Stat.,24, No. 1, 132–134 (1953).
Yung Liang Tong, “Some probability inequalities of multivariate normal and multivariate t.,” J. Am. Stat. Assoc.,65, No. 331, 1243–1247 (1970).
H. R. van der Vaart, “The content of certain spherical polyhedra for any number of dimensions,” Experientia,9, No. 3, 88–90 (1953).
H. R. van der Vaart, “The content of some classes of non-Euclidean polyhedra for any number of dimensions, with several applications. I. II,” Indag. Math.,A58, No. 2, 199–221 (1955).
J. T. Webster, “On the application of the method of Das in evaluating a multivariate normal integral,” Biometrika,57, No. 3, 657–660 (1970).
A. van Wijngaarden, “A transformation of formal series. I. II,” Indag. Math.,A56, No. 5, 522–543 (1953).
J. D. Williams, “An approximation to the probability integral,” Ann. Math. Stat.,17, No. 3, 363–365 (1946).
J. C. Young and Ch. E. Minder, “Algorithm AS 76. An integral useful in calculating noncentral t and bivariate normal probabilities,” Appl. Stat.,23, No. 3, 455–457 (1974).
M. Zelen and N. C. Severo, “Graphs for bivariate normal probabilities,” Ann. Math. Stat.,31, No. 3, 619–624 (1960).
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Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 17, pp. 57–84, 1979.
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Martynov, G.V. Evaluation of the normal distribution function. J Math Sci 17, 1857–1875 (1981). https://doi.org/10.1007/BF01085187
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DOI: https://doi.org/10.1007/BF01085187