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A new Monte Carlo technique: antithetic variates

Published online by Cambridge University Press:  24 October 2008

J. M. Hammersley  and
K. W. Morton
J. M. Hammersley
Affiliation:
Atomic Energy Research EstablishmentHarwell, Didcot, Berks
K. W. Morton
Affiliation:
Atomic Energy Research EstablishmentHarwell, Didcot, Berks
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Extract

As we have stressed in a previous paper (9), the main concern in Monte Carlo work is to achieve without inordinate labour a respectably small standard error in the final result. Mere replication of the Monte Carlo results is unrewarding; for, to reduce a standard error by a factor k, the labour must be increased k2-fold, and this will be beyond the resources of even electronic computers when k = 1000, say. The remedy lies in a skilful choice of sampling technique and the substitution of analytical methods for random processes wherever possible. The efficiency of a Monte Carlo process may be taken as inversely proportional to the product of the sampling variance of the final estimate and the amount of labour expended in obtaining this estimate; and it is profitable to allow some increase in the labour if that produces an overwhelming decrease in the variance. For instance, in the last example quoted below (Table 2), we reduce the variance by a factor of four million at the expense of only multiplying the labour sixteenfold, thereby attaining a 250,000-fold gain of efficiency.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 3 , July 1956 , pp. 449 - 475
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

(1)Fieller, E. C. and Hartley, H. O.Sampling with control variables. Biometrika, 41 (1954), 494–501.CrossRefGoogle Scholar
(2)Fort, T.Finite differences (Oxford, 1948).Google Scholar
(3)Glaisher, J. W. L.Remarks on the calculation of π. Messeng. Math. 2 (1873), 119–28.Google Scholar
(4)Hall, A.On an experimental determination of π. Messeng. Math. 2 (1873), 113–14.Google Scholar
(5)Hammersley, J. M.On a certain type of integral associated with circular cylinders. Proc. roy. Soc. A, 210 (1951), 98–110.Google Scholar
(6)Hammersley, J. M.Lagrangian integration coefficients for distance functions taken over right circular cylinders. J. Math. Phys. 31 (1952), 139–50.CrossRefGoogle Scholar
(7)Hammersley, J. M. The absorption of radioactive radiation in rods. (National Bureau of Standards' unpublished working paper no. 1929, 1952.)Google Scholar
(8)Hammersley, J. M. and Mauldon, J. G.General principles of antithetic variates. Proc. Camb. phil. Soc. 52 (1956), 476–81.CrossRefGoogle Scholar
(9)Hammersley, J. M. and Morton, K. W.Poor man's Monte Carlo. J. R. statist. Soc. B, 16 (1954), 23–38.Google Scholar