Abstract
The three asymptotic tests, Neyman and Pearson Likelihood Ratio (LR), Wald’s statistic (W) and Rao’s score (RS)are referred to in statistical literature on testing of hypotheses as the Holy Trinity. All these tests are equivalent to the first-order of asymptotics, but differ to some extent in the second-order properties. Some of the merits and defects of these tests are presented.
Some applications of the score test, recent developments on refining the score test and problems for further investigation are presented.
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Aitchison, J. and Silvey, S. D. (1958). Maximum likelihood estimation of parameters subject to restraints, Annals of Mathematical Statistics, 29, 813–828.
Amemiya, T. (1985). Advanced Econometrics, Harvard University Press, Boston, Massachusetts.
Bartlett, M. S. (1937). Properties of sufficiency and statistical tests, Proceedings of the. Royal Society, Series A, 160, 268–282.
Bera, A. K. and Bilias, Y. (2001). Rao’s score, Neyman’s C(α)) and Silvey’s LM tests: An essay on historical developments and new results, Journal of Statistical Planning and Inference, 97, 9–44.
Bera, A. K. and Jarque. C. M. (1981). An efficient large sample test for normality of observations and regression residuals, Working Papers in Economics and Econometrics. 40. The Australian National University, Canberra, Australia.
Bera, A. K. and Ullah. A. (1991). Rao’s score test in Econometrics, Journal of Quantitative Economics, 7, 189–220.
Bickel, P. J. and Doksurn. K. A. (2001). Mathematical Statistics, Vol. 1, Second Edition, Prentice-Hall, Inglewood Cliffs, New Jersey.
Box, G. E. P. and Raminez, J. (1992). Cumulative score charts, Quality and Reliability Engineering. 8, 17–27.
Bradley. R. A. (1953). Some statistical methods in taste testing and quality evaluation, Biometrics, 9, 22–38.
Breusch, T.S. (1978). Testing for autocorrelation in dynamic linear models, Australian Economic Papers. 17, 334–355.
Breusch, T. S. and Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation, Econometrics, 47, 1287–1294.
Breusch. T. S. and Pagan, A. R. (1980). The Lagrange Multiplier test and its application to model specification in econometrics, Review of Economic Studies, 47, 239–253.
Byron, R. P. (1968). Methods for estimating demand equations using prior information, A series of experiments with Australian data, Australian Economic Papers. 7, 227–248.
Chandra. T. K. and Joshi, S. N. (1983). Comparison of the likelihood ratio, Wald’s and Rao’s tests, Sankhyā, Series A, 45, 226–246.
Chandra, T. K. and Mukherjee, R. (1984). On the optimality of Rao’s statistic, Communications in Statistics—Theory and Methods, 13, 1507–1515.
Chandra, T. K. and Mukherjee, R. (1985). Comparison of the likelihood ratio, Wald’s and Rao’s tests, Sankhyā, Series A, 47, 271–284.
Chandra, T. K. and Samanta, T. (1988). On second order local comparisons between perturbed maximum likelihood estimators and Rao’s statistic as test statistics, Journal of Multivariate Analysis, 25, 201–222.
Conniffe, D. (1990). Testing hypotheses with estimated scores, Biometrika, 77, 97–106.
Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika. 64, 247–254.
Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74, 33–43.
Dean, C. and Lawless, J. F. (1989). Tests for detecting over dispersion in Poisson regression models, Journal of the. American Statistical Association, 84, 467–472.
Engle, R. F. (1984). Wald, likelihood ratio and Lagrangian Multiplier test in econometrics, In Handbook of Econometrics, Vol. 2 (Eds., Z. Griliches and M. Intriligator), North-Holland Science Publishers, Amsterdam.
Fears, T. R.. Benichow, J. and Gail, M. H. (1996). A remainder of the fallibility of the Wald statistic, The American Statistician, 50, 226–227.
Ghosh, J. K. (1991). Higher order asymptotics for the likelihood ratio, Rao’s and Wald’s test, Statistics & Probability Letters, 12, 505–509.
Ghosh, J. K. and Mukherjee, R. (2001). Test statistic arising from quasilikelihood: Bartlett adjustment and higher-order power, Journal of Statistical Planning and Inference, 97, 45–55.
Godfrey, L. G. (1978a). Testing against general autoregressive and moving average models when the regression include lagged dependent variables, Econometrica, 46, 227–236.
Godfrey, L. G. (1978b). Testing for higher order serial correlation in regression equations when the regression includes lagged dependent variables, Econometrica, 46, 1303–1310.
Godfrey, L. G. (1988). Mis specification Tests in Econometrics, Cambridge University Press, London, England.
Godfrey, L. G. and Orme, C. D. (2001). On improving the robustness and reliability of Rao’s score test, Journal of Statistical Planning and Inference, 97, 153–176.
Godfrey, L. G. and Wickens, M. R. (1981). Testing linear and log-linear regression for functional form, Review of Economic Studies, 48, 487–496.
Green, W. H. (1990). Econometric Analysis, Macmillan Publishers Limited. Hampshire, England.
Gregory, A. W. and Veal, M. R. (1985). Formulating Wald tests of nonlinear restrictions, Econometrica, 53, 1465–1468.
Hall, W. J. and Mathiason, D. J. (1990). On large sample estimation and testing in parameter models, International Statistical Review, 58, 77–97.
Harris, P. (1985). An asymptotic expansion for the null distribution of the efficient score statistics, Biometrika, 72, 653–659.
Harvey, A. (1990). The Econometric Analysis of Time Series, Philip Allan, Oxford, England.
Hauck, W. W. and Donner, A. (1977). Wald test as applied to hypotheses in logit analysis, Journal of the American Statistical Association, 72, 851–853.
Journal of Statistical Planning and Inference, 2001, Vol. 97, No. 1, Special Issue on Rao’s Score Test (12 papers, pp. 1–200).
Judge, G. G.. Griffiths, W. E., Hill, R. C., Lütkepohl, H. and Lee, T. C. (1985). The Theory and Practice of Econometrics, Second edition, John Wiley & Sons, New York.
Kmenta, J. (1986). Elements of Econometrics, Second edition, Macmillan Publishers Limited, Hampshire, England.
Kramer. W. and Sonnberger, H. (1986). The Linear Regression Model Test, Physica-Verlag, Heidelberg, Germany.
Le Cam, L. (1990). On the standard asymptotic confidence ellipsoids of Wald, International Statistical Review, 58, 129–152.
Lehmann, E. (1999). Theory of Asymptotic Inference, Springer-Verlag, New York.
Li, Bing (2001). Sensitivity of Rao’s score test, the Wald test and the likelihood ratio test to nuisance parameters, Journal of Statistical Planning and Inference, 97, 57–66.
Maddala, G. S. (1988). Introduction to Econometrics, Macmillan Publishers Limited, Hampshire, England.
Mantel, N. (1987). Understanding Wald’s test for exponential families, The American Statistician, 41, 147–149.
Mukherjee, R. (1990). Comparison of tests in the multiparameter case I: Second order power, Journal of Multivariate Analysis, 33, 17–30.
Mukherjee, R. (1993). Rao’s score test: Recent asymptotic results, In Handbook of Statistics 11 (Eds.. G. S. Maddala, C. R. Rao and H. Vinod), pp. 363–379, North-Holland Science Publishers, Amsterdam.
Neyman, J. (1959). Optimal asymptotic test of composite statistical hypothesis, In Probability and Statistics (Ed., U. Grenander), John Wiley & Sons, New York.
Neyman, J, (1979). C(α) tests and their uses, Sankhyā, Series A, 41, 1–21.
Neyman, J. and Pearson. E. S. (1928). On the use and interpretation of certain test criteria, Biometrika, 20A, 175–240, 263–294.
Pawitan, Y. (2000). A remainder of the fallibility of the Wald statistic: Likelihood explanation, The American Statistician, 54, 54–56.
Peers, H. W. (1971). Likelihood ratio and associated test criteria, Biometrika, 58, 577–587.
Rao, C. R. (1948). Large sample tests of statistical hypotheses concerning several parameters with application to problems of estimation, Proceedings of the Cambridge Philosophical Society, 44, 50–57.
Rao, C. R. (1950). Methods of scoring linkage data giving the simultaneous segregation of three factors, Heredity, 4, 37–59.
Rao, C. R. (1951). Sequential tests of null hypotheses, Sankhyā, 10, 361–370.
Rao, C. R. (1961). A study of large sample test criteria through properties of efficient estimates. Sankhyā, Series A, 23, 25–40.
Rao, C. R. (1973). Linear Statistical Inference and its Applications, Second edition, John Wiley & Sons, New York.
Rao, C. R. and Poti, S. J. (1946). On locally most powerful tests when alternatives are one-sided, Sankhyā, 7, 439–440.
Sen, P. K. (1997). Introduction to Rao (1948). In Breakthroughs in Statistics (Eds., S. Kotz and N. L. Johnson). Vol. III, Springer-Verlag, New York.
Serfling, R.. J. (1980). Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New York.
Silvey, S. D. (1959). The Lagrangian multiplier test, Annals of Mathematical Statistics, 30, 389–407.
Spanos, A. (1986). Statistical Foundations of Econometric Modeling, Cambridge University Press, London, England.
Taniguchi, M. (2001). On large deviation asymptotics of some tests in time series, Journal of Statistical Planning and Inference, 97, 191–200.
Taniguchi, M. (1988). Asymptotic expansions of the distribution of some statistics for Gaussian ARMA process, Journal of Multivariate Analysis, s, 494–511.
Taniguchi, M. (1991). Higher Order Asymptotic Theory for Time Series Analysis, Lecture Notes in Statistics, Vol. 68, Springer-Verlag, New York.
Terril, G. R. (2001). The gradient statistic, Personal communication.
Tu, D.. Chen, J. and Shi, P. (2004). A Bartlett type correction for Rao Score test in Cox regression model, Technical Report.
Vaeth, M. (1985). On the use of Wald’s test in exponential families, International Statistical Review, 53, 199–214.
Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large, Transactions of the Americal Mathematical Soceity, 54, 426–482.
White, H. (1982). Maximum likelihood estimation of misspecified models, Econometrica, 50, 1–25.
White, H. (1984). Asymptotic Theory for Econometricians, Academic Press, New York.
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Rao, C.R. (2005). Score Test: Historical Review and Recent Developments. In: Balakrishnan, N., Nagaraja, H.N., Kannan, N. (eds) Advances in Ranking and Selection, Multiple Comparisons, and Reliability. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4422-9_1
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DOI: https://doi.org/10.1007/0-8176-4422-9_1
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