Duchesne, Pierre and Francq, Christian (2010): On testing for the mean vector of a multivariate distribution with generalized and {2}inverses.

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Abstract
Generalized Wald's method constructs testing procedures having chisquared limiting distributions from test statistics having singular normal limiting distributions by use of generalized inverses. In this article, the use of twoinverses for that problem is investigated, in order to propose new test statistics with convenient asymptotic chisquare distributions. Alternatively, Imhofbased test statistics can also be defined, which converge in distribution to weighted sum of chisquare variables; The critical values of such procedures can be found using Imhof's (1961) algorithm. The asymptotic distributions of the test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.
Item Type:  MPRA Paper 

Original Title:  On testing for the mean vector of a multivariate distribution with generalized and {2}inverses 
Language:  English 
Keywords:  twoinverses; generalized Wald's method; generalized inverses; multivariate analysis; singular normal distribution 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C12  Hypothesis Testing: General 
Item ID:  19740 
Depositing User:  Christian Francq 
Date Deposited:  07 Jan 2010 08:34 
Last Modified:  01 Oct 2019 04:42 
References:  Andrews, D.W.K. (1987), `Asymptotic results for generalized Wald tests', Econometric Theory 3, 348358. Anderson, T.W. (1984), An Introduction to Multivariate Statistical Analysis, Wiley, New York. Bahadur, R.R. (1960), `Comparisons of tests', The Annals of Mathematical Statistics 31, 276295. Bhimasankaram, P. and Sengupta, D. (1991), `Testing for the mean vector of a multivariate normal distribution with a possibly singular dispersion matrix and related results', Statistics \& Probability Letters 11, 473478. Bilodeau, M. and Brenner, D. (1999), Theory of Multivariate Statistics, Springer, New York. Box, G.E.P. and Pierce, D.A. (1970), `Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models', Journal of the American Statistical Association 65, 15091526. Casty, C., Wanner, H., Luterbacher, J., Esper, J. and Böhm,R. (2005), `Temperature and precipitation variability in the European Alps since 1500', International Journal of Climatology 25, 18551880. Casty, C., Raible, C.C., Stocker, T.F., Luterbacher, J. and H. Wanner. (2007), `A European pattern climatology 17662000', Climate Dynamics, DOI: 10.1007/s0038200702576. Duchesne, P. and Francq, C. (2008), `On diagnostic checking time series models with portmanteau test statistics based on generalized inverses and \{ 2 \}inverses', COMPSTAT 2008, Proceedings in Computational Statistics, 143154. Eaton, M. L. (2007), Multivariate Statistics. A Vector Space Approach, Institute of Mathematical Statistics, Lecture Notes  Monograph Series, vol. 53, Ohio. Francq, C., Roy, R. and Zako\"ian, J.M. (2005), `Diagnostic checking in ARMA models with uncorrelated errors', Journal of the American Statistical Association 100, 532544. Getson, A. J. and Hsuan, F. C. (1988), $\{ 2 \}$Inverses and Their Statistical Application, Lecture Notes in Statistics 47, SpringerVerlag, New York. Ghosh, B. K. (1973), `Some monotonicity theorems for x2*F and t distributions with applications', Journal of the Royal Statistical Society, Series B 35, 480492. Hadi, A.S. and Wells, M.T. (1990), `A note on generalized Wald's method', Metrika 37, 309315. Imhof, J. P. (1961), `Computing the distribution of quadratic forms in normal variables' Biometrika 48, 419426. Li, W. K. (2004), Diagnostic Checks in Time Series. Chapman \& Hall/CRC, New York. Li, W.K. (1992), `On the asymptotic standard errors of residual autocorrelations in nonlinear time series modelling', Biometrika 79, 435437. Li,W.K. and McLeod,A.I. (1981), `Distribution of the residual autocorrelations in multivariate ARMA time series models', Journal of the Royal Statistical Society, Series B 43, 231239. Ljung, G.M. (1986), `Diagnostic testing of univariate time series models', Biometrika 73, 725730. Moore, D.S. (1978), `Chisquare tests', in Studies in Statistics, vol. 19, Hogg, R. V. (Editor), 66106, The Mathematical Association of America, Washington DC. Moore, D.S. (1977), `Generalized inverses, Wald's method, and the construction of chisquared tests of fit', Journal of the American Statistical Association 72, 131137. Muirhead, R.J. (1982), Aspects of Multivariate Statistical Theory, Wiley, New York. Rao, C.R. and Mitra, S.K.M. (1971), Generalized Inverse of Matrices and its Applications, Wiley, New York. Rao, C.R. (1973), Linear Statistical Inference and Its Applications, Wiley, New York. Robinson, P.M. (1972), `Nonlinear regression for multiple timeseries', Journal of Applied Probability 9, 758768. Srivastava, M.S. and Khatri, C.G. (1979), An Introduction to Multivariate Statistics, NorthHolland, New York. Tyler, D.E. (1981), `Asymptotic inference for eigenvectors', The Annals of Statistics 4, 725736. Wald, A. (1943), `Tests of statistical hypothesis concerning several parameters when the number of observations is large', Transaction of the American Mathematical Society 54, 426482. Zolotarev, V.M. (1961), `Concerning a certain probability problem', Theory of Probability and its Applications 6, 201204. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/19740 