Recursive calculation model for a special multivariate normal probability of first-order stationary sequence. (English) Zbl 07284460
Summary: The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved.
MSC:
| 60-XX | Probability theory and stochastic processes |
Keywords:
multivariate normal probability; recursive calculation; first-order; antiaircraft; hit probabilityReferences:
| [1] | Ambartzumian R, Der Kiureghian A, Ohaniana V, Sukiasiana H (1998) Multinormal probability by sequential conditioned importance sampling: Theory and application. Probab. Engrg. Mech. 13(4):299-308.Crossref, Google Scholar · doi:10.1016/S0266-8920(98)00003-4 |
| [2] | Caflisch RE (1998) Monte Carlo and quasi-Monte Carlo methods. Acta Numerica 7:1-49.Crossref, Google Scholar · Zbl 0949.65003 · doi:10.1017/S0962492900002804 |
| [3] | Craig P (2008) A new reconstruction of multivariate normal orthant probabilities. J. Royal Statist. Soc. Ser. B (Statist. Methodology) 70(1):227-243.Google Scholar · Zbl 05563352 |
| [4] | Deak I (1980) Three digit accurate multiple normal probabilities. Numerica Math. 35(4):369-380.Crossref, Google Scholar · Zbl 0453.65007 · doi:10.1007/BF01399006 |
| [5] | Deak I (2003) Probabilities of simple n-dimensional sets for the normal distribution. IIE Trans. 35(3):285-293.Crossref, Google Scholar · doi:10.1080/07408170304362 |
| [6] | Drezner Z (1990) Approximations to the multivariate normal integral. Commun. Statist. Simulation Comput. 19(2):527-534.Crossref, Google Scholar · Zbl 0718.62033 · doi:10.1080/03610919008812873 |
| [7] | Drezner Z (1992) Computation of the multivariate normal integral. ACM Trans. Math. Software 18(4):470-480.Crossref, Google Scholar · Zbl 0894.65077 · doi:10.1145/138351.138375 |
| [8] | Gassmann HI, Deák I, Szántai T (2002) Computing multivariate normal probabilities: A new look. J. Comput. Graphical Statist. 11(4):920-949.Crossref, Google Scholar · doi:10.1198/106186002385 |
| [9] | Genz A (1992) Numerical computation of multivariate normal probabilities. J. Comput. Graphical Statist. 1(2):141-149.Google Scholar |
| [10] | Genz A (2004) Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Statist. Comput. 14(3):251-260.Crossref, Google Scholar · doi:10.1023/B:STCO.0000035304.20635.31 |
| [11] | Genz A, Bretz F (2002) Comparison of methods for the computation of multivariate t probabilities. J. Comput. Graphical Statist. 11(4):950-971.Crossref, Google Scholar · doi:10.1198/106186002394 |
| [12] | Genz A, Bretz F (2009) Computation of Multivariate Normal and t Probabilities, vol 195 (Springer Science & Business Media, Berlin Heidelberg).Crossref, Google Scholar · Zbl 1204.62088 · doi:10.1007/978-3-642-01689-9 |
| [13] | Genz A, Kwong KS (2000) Numerical evaluation of singular multivariate normal distributions. J. Statist. Comput. Simulation 68(1):1-21.Crossref, Google Scholar · Zbl 0963.62047 · doi:10.1080/00949650008812053 |
| [14] | Geweke J (1991) Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. Comput. Sci. Statist. Proc. 23rd Sympos. Interface (Citeseer, Seattle), 571-578.Google Scholar |
| [15] | Gupta SS (1963) Probability integrals of multivariate normal and multivariate t. Ann. Math. Statist. 34(3):792-828.Crossref, Google Scholar · Zbl 0124.35505 · doi:10.1214/aoms/1177704004 |
| [16] | Hajivassiliou V, McFadden D, Ruud P (1996) Simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results. J. Econom. 72(1):85-134.Crossref, Google Scholar · Zbl 0849.62064 · doi:10.1016/0304-4076(94)01716-6 |
| [17] | Henery R (1981) An approximation to certain multivariate normal probabilities. J. Royal Statist. Soc. Ser. B (Statist. Methodology) 43(1):81-85.Google Scholar · Zbl 0451.62016 |
| [18] | Joe H (1995) Approximations to multivariate normal rectangle probabilities based on conditional expectations. J. Amer. Statist. Assoc. 90(431):957-964.Crossref, Google Scholar · Zbl 0843.62016 · doi:10.1080/01621459.1995.10476596 |
| [19] | Lerman S, Manski C (1981) On the use of simulated frequencies to approximate choice probabilities. Structural Analysis of Discrete Data with Econometric Applications, vol 10 (MIT Press, Cambridge, MA), 305-319.Google Scholar |
| [20] | McFadden D (1989) A method of simulated moments for estimation of discrete response models without numerical integration. Econom. J. Econom. Soc. 57(5):995-1026.Google Scholar · Zbl 0679.62101 |
| [21] | Milton RC (1972) Computer evaluation of the multivariate normal integral. Technometrics 14(4):881-889.Crossref, Google Scholar · Zbl 0243.62015 · doi:10.1080/00401706.1972.10488983 |
| [22] | Miwa T, Hayter A, Kuriki S (2003) The evaluation of general non-centred orthant probabilities. J. Royal Statist. Soc. Ser. B (Statist. Methodology) 65(1):223-234.Crossref, Google Scholar · Zbl 1063.62082 · doi:10.1111/1467-9868.00382 |
| [23] | Morrison DF (1998) Multivariate Analysis, Overview (Wiley Online Library, Hoboken, NJ).Google Scholar |
| [24] | Nash JC (1990) Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation (CRC Press, Boca Raton, FL).Google Scholar · Zbl 0697.68004 |
| [25] | Nikoloulopoulos AK (2013) On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood. J. Statist. Planning Inference 143(11):1923-1937.Crossref, Google Scholar · Zbl 1279.62116 · doi:10.1016/j.jspi.2013.06.015 |
| [26] | Nikoloulopoulos AK (2016) Efficient estimation of high-dimensional multivariate normal copula models with discrete spatial responses. Stochastic Environ. Res. Risk Assessment 30(2):493-505.Crossref, Google Scholar · doi:10.1007/s00477-015-1060-2 |
| [27] | Robert CP (2004) Monte Carlo Methods (Wiley Online Library, Hoboken, NJ).Crossref, Google Scholar · doi:10.1007/978-1-4757-4145-2 |
| [28] | Schervish MJ (1984) Algorithm as 195: Multivariate normal probabilities with error bound. J. Royal Statist. Soc. Ser. C (Appl. Statist.) 33(1):81-94.Google Scholar · Zbl 0547.65097 |
| [29] | Šidák Z (1967) Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62(318):626-633.Google Scholar · Zbl 0158.17705 |
| [30] | Sidák Z (1968) On multivariate normal probabilities of rectangles: Their dependence on correlations. Ann. Math. Statist. 39(5):1425-1434.Crossref, Google Scholar · Zbl 0169.50102 · doi:10.1214/aoms/1177698122 |
| [31] | Slepian D (1962) The one-sided barrier problem for Gaussian noise. Bell Labs Tech. J. 41(2):463-501.Crossref, Google Scholar · doi:10.1002/j.1538-7305.1962.tb02419.x |
| [32] | Somerville PN (1998) Numerical computation of multivariate normal and multivariate-t probabilities over convex regions. J. Comput. Graphical Statist. 7(4):529-544.Google Scholar |
| [33] | Szántai T (2000) Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function. Ann. Oper. Res. 100(1-4):85-101.Crossref, Google Scholar · Zbl 0973.65006 · doi:10.1023/A:1019211000153 |
| [34] | Tong YL (2012) The Multivariate Normal Distribution (Springer Science & Business Media, New York).Google Scholar |
| [35] | Vijverberg WP (1997) Monte Carlo evaluation of multivariate normal probabilities. J. Econom. 76(1):281-307.Crossref, · Zbl 0880.62016 · doi:10.1016/0304-4076(95)01792-5 |
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