General Theory of Antithetic Time Series


A generalized antithetic time series theory for exponentially derived antithetic random variables is developed. The correlation function between a generalized gamma distributed random variable and its pth exponent is derived. We prove that the correlation approaches minus one as the exponent approaches zero from the left and the shape parameter approaches infinity.

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Ngnepieba, P. and Ridley, D. (2015) General Theory of Antithetic Time Series. Journal of Applied Mathematics and Physics, 3, 1726-1741. doi: 10.4236/jamp.2015.312197.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Ferrenberg, A.M., Lanau, D.P. and Wong, Y.J. (1992) Monte Carlo Simulations: Hidden Errors from? Good Random Number Generators? Physical Review Letters, 69, 3382-3384.
[2] Griliches, Z. (1961) A Note on Serial Correlation Bias in Estimates of Distributed Lags. Econometrica, 29, 65-73.
[3] Nerlove, M. (1958) Distributed Lags and Demand Analysis for Agricultural and Other Commodities. U.S.D.A Agricultural Handbook No. 141, Washington.
[4] Koyck, L.M. (1954) Distributed Lags and Investment Analysis. North-Holland Publishing Co., Amsterdam.
[5] Klein, L.R. (1958) The Estimation of Distributed Lags. Econometrica, 26, 553-565.
[6] Fuller, W.A. and Hasza, D.P. (1981) Properties of Predictors from Autoregressive Time Series. Journal of the American Statistical Association, 76, 155-161.
[7] Dufour, J. (1985) Unbiasedness of Predictions from Estimated Vector Autoregressions. Econometric Theory, 1, 381-402.
[8] Chandan, S. and Jones, P. (2005) Asymptotic Bias in the Linear Mixed Effects Model under Non-Ignorable Missing Data Mechanisms. Journal of the Royal Statistical Society: Series B, 67, 167-182.
[9] Li, B., Nychka, D.W. and Ammann, C.M. (2010) The Value of Multiproxy Reconstruction of Past Climate. Journal of the American Statistical Association, 105, 883-911.
[10] Bunn, D.W. (1979) The Synthesis of Predictive Models in Marketing Research. Journal of Marketing Research, 16, 280-283.
[11] Diebold, F.X. (1989) Forecast Combination and Encompassing: Reconciling Two Divergent Literatures. International Journal of Forecasting, 5, 589-592.
[12] Clemen, R.T. (1989) Combining Forecasts: A Review and Annotated Bibliography. International Journal of Forecasting, 5, 559-583.
[13] Makridakis, S., Anderson, A., Carbone, R., Fildes, R., Hibon, M., Lewandowski, R., Newton, J., Parzen, E. and Winkler, R. (1982) The Accuracy of Extrapolation (Times Series) Methods: Results of a Forecasting Competition. Journal of Forecasting, 1, 111-153.
[14] Winkler, R.L. (1989) Combining Forecasts: A Philosophical Basis and Some Current Issues. International Journal of Forecasting, 5, 605-609.
[15] Hendry, D.F. and Mizon, G.E. (1978) Serial Correlation as a Convenient Simplification, Not a Nuisance: A Commentary on a Study of the Demand for Money by the Bank of England. The Economic Journal, 88, 549-563.
[16] Hendry, D.F. (1976) The Structure of Simultaneous Equations Estimators. Journal of Econometrics, 4, 551-588.
[17] Mizon, G.E. (1977) Model Selection Procedures. In: Artis, M.J. and Nobay, A.D., Eds., Studies in Modern Economic Analysis, Basil Blackwell, Oxford.
[18] Pindyck, R.S. and Rubinfeld, D.L. (1976) Econometric Models and Economic Forecasts. McGraw-Hill, New York.
[19] Durbin, J. and Watson, G.S. (1950) Testing for Serial Correlation in Least Squares Regression: I. Biometrika, 37, 409-428.
[20] Durbin, J. (1970) Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors Are Lagged Dependent Variables. Econometrica, 38, 410-421.
[21] Osborn, D.R. (1976) Maximum Likelihood Estimation of Moving Average Processes. Journal of Economic and Social Measurement, 5, 75-87.
[22] Espasa, D. (1977) The Spectral Maximum Likelihood Estimation of Econometric Models with Stationary Errors. 3, Applied Statistics and Economics Series. Vanderhoeck and Ruprecht, Gottingen.
[23] Hammersley, J.M. and Morton, K.W. (1956) A New Monte Carlo Technique: Antithetic Variates. Mathematical Proceedings of the Cambridge Philosophical Society, 52, 449-475.
[24] Kleijnen, J.P.C. (1975) Antithetic Variates, Common Random Numbers and Optimal Computer Time Allocation in Simulations. Management Science, 21, 1176-1185.
[25] Ridley, A.D. (1999) Optimal Antithetic Weights for Lognormal Time Series Forecasting. Computers & Operations Research, 26, 189-209.
[26] Ridley, A.D. (1995) Combining Global Antithetic Forecasts. International Transactions in Operational Research, 4,387-398.
[27] Ridley, A.D. (1997) Optimal Weights for Combining Antithetic Forecasts. Computers & Industrial Engineering, 2, 371-381.
[28] Ridley, A.D. and Ngnepieba, P. (2014) Antithetic Time Series Analysis and the CompanyX Data. Journal of the Royal Statistical Society: Series A, 177, 83-94.
[29] Ridley, A.D., Ngnepieba, P. and Duke, D. (2013) Parameter Optimization for Combining Lognormal Antithetic Time Series. European Journal of Mathematical Sciences, 2, 235-245.
[30] MATLAB (2008) Application Program Interface Reference, Version 8. The Math Works, Inc.
[31] Hogg, R.V. and Ledolter, J. (2010) Applied Statistics for Engineers and Physical Scientists. 3rd Edition, Prentice Hall, Upper Saddle River, 174.
[32] Box, G.E.P. and Cox, D.R. (1964) An Analysis of Transformations. Journal of the Royal Statistical Society: Series B, 26, 211-252.
[33] Abramowitz, M. and Stegun, I.A. (1964) Handbook of Mathematical Functions. Dover Publications, New York, 260 p.
[34] Bernado, J.M. (1976) Algorithm AS 103: Psi (Digamma) Function. Journal of the Royal Statistical Society: Series C (Applied Statistics), 25, 315-317.
[35] Fuller, W.A. (1996) Introduction to Statistical Times Series. Wiley, New York.

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