The Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate

References

• Areia , A. , Oliveira , M. M. , Mexia , J. T. ( 2008 ). Models for a series of studies based on geometrical representation . Statist. Methodol. 5 ( 3 ): 277 – 288 .
   Google Scholar
• Barbe , P. , McCormick , W. P. ( 2005 ). Asymptotic expansions of convolutions of regularly varying distributions . J. Aust. Math. Soc. 78 ( 3 ): 339 – 371 .
   Web of Science ®Google Scholar
• Bhattacharya , R. , Ranga Rao , R. ( 1976 ). Normal Approximation and Asymptotic Expansions . New York : John Wiley & Sons .
   Google Scholar
• Bingham , N. H. , Goldie , C. M. , Teugels , J. L. ( 1989 ). Regular Variation . Encyclopedia of Mathematics and its Applications . Vol. 27 . Cambridge : Cambridge University Press .
   Google Scholar
• Blinnikov , S. , Moessner , R. ( 1998 ), Expansions for nearly gaussians distributions . Astron. Astrophys. Suppl. Ser. 130 : 193 – 205 .
   Google Scholar
• Cramer , H. ( 1999 ). Mathematical Methods of Statisitics . Princeton , NJ : Princeton University Press .
   Google Scholar
• Christoph , G. ( 1991 ). Non uniform estimates in asympttotic expansions with a stable limit distribution . Math. Nachr. 153 : 257 – 272 .
   Web of Science ®Google Scholar
• De Bruijn , N. G. ( 1981 ). Asymptotic Methods in Analysis . New York : Dover Publications Inc .
   Google Scholar
• Edgeworth , F. Y. ( 1905 ). The law of error . Trans. Cambridge Philos. Soc. 20 ( 35–65 ): 113 – 141 .
   Google Scholar
• Esquível , M. L. , Mexia , J. T. , da Silva , J. L. , Ramos , L. P. C. ( 2009 ). Some asymptotic expansions and distribution approximations outside a clt context . Proc. 6th St. Petersburg Workshop Simul. I : 444 – 448 .
   Google Scholar
• Esseen , C.-G. ( 1945 ). Fourier analysis of distribution functions. a mathematical study of the laplace-gaussian law . Acta Math. 77 : 1 – 125 .
   Web of Science ®Google Scholar
• Evans , L. C. , Gariepy , R. F. ( 1992 ). Measure Theory and Fine Properties of Functions . Boca Raton , FL : CRC Press .
   Google Scholar
• Federer , H. ( 1969 ). Geometric Measure Theory . New York : Springer-Verlag .
   Google Scholar
• Fonseca , M. , Mexia , J. T. ( 2010 ). Delta method and moment convergence, ICNAAM 2010: International Conference of numerical analysis and applied mathematics 2010 . AIP Conf. Proc. 1281 : 1252 – 1255 .
   Google Scholar
• Feller , W. ( 1971 ). An Introduction to Probability Theory and its Applications . Vol. 2 . 2nd ed. New York : John Wiley & Sons .
   Google Scholar
• Fujikoshi , Y. , Shimizu , R. ( 1989a ). Asymptotic expansions of some mixtures of the multivariate normal distribution and their error bounds . Ann. Statist. 17 ( 3 ): 1124 – 1132 .
   Web of Science ®Google Scholar
• Fujikoshi , Y. , Shimizu , R. (1989b). Error bounds for asymptotic expansions of scale mixtures of univariate and multivariate distributions. J. Multivariate Anal. 30(2):279–291.
   Web of Science ®Google Scholar
• Fujikoshi , Y. , Shimizu , R. ( 1990 ). Asymptotic expansions and error bounds for certain types of probability distributions: the distribution of a sum of independent random variables and the distribution of scale mixtures of random variables . Sugaku Expositions 3 ( 1 ): 75 – 96 .
   Google Scholar
• Fujikoshi , Y. , Ulyanov , V. V. , Shimizu , R. ( 2005 ). L 1-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to hotelling's generalized . J. Multivariate Anal. 96 ( 1 ): 1 – 19 .
   Web of Science ®Google Scholar
• Galambos , J. , Seneta , E. ( 1973 ). Regularly varying sequences . Proc. Amer. Mathemat. Soc. 41 ( 1 ): 110 – 116 .
   Web of Science ®Google Scholar
• Götze , F. , Tikhomirov , A. N. ( 2005 ). Asymptotic expansions in non-central limit theorems for quadratic forms . J. Theoret. Prob. 18 ( 4 ): 757 – 811 .
   Web of Science ®Google Scholar
• Hoffmann-Jørgensen , J. ( 1992 ). Asymptotic Likelihood Theory. Functional Analysis III, Proc. Dubrovnik 1989, Matematisk Institut, Aarhus Universitet, Var. Publ. Ser. no 40, pp. 5–192 .
   Google Scholar
• Hoffmann-Jørgensen , J. ( 1994 ). Probability with a View Towards Statistics . Vol. II . London : Chapman & Hall .
   Google Scholar
• Inzhevitov , P. G. ( 1983 ). Asymptotic expansions in local theorems for sums of independent random variables in the case of a stable limit law . Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 2 : 88 – 91 .
   Google Scholar
• Itô , K. , Ed. ( 1993 ). Encyclopedic Dictionary of Mathematics. , 2nd ed. Cambridge : MIT Press .
   Google Scholar
• Kristof , G. ( 1982 ). Asymptotic expansions in the case of a stable limit law. II . Litovsk. Mat. Sb. 22 ( 2 ): 137 – 146 .
   Google Scholar
• Khaniyev , T. , Kucuk , Z. ( 2004 ). Asymptotic expansions for the moments of the gaussian random walk with two barriers . Statist. Probab. Lett. 69 ( 1 ): 91 – 103 .
   Web of Science ®Google Scholar
• Le Cam , L. ( 1986 ). Asymptotic Methods in Statistical Decision Theory . New York : Springer-Verlag .
   Google Scholar
• Mattila , P. ( 1995 ). Geometry of Sets and Measures in Euclidean Spaces . Cambridge : Cambridge University Press .
   Google Scholar
• Mexia , J. T. , Oliveira , M. M. ( 2010 ). Asymptotic linearity and limit distributions approximations . J. Statist. Plann. Infer. 140 ( 2 ): 353 – 357 .
   Web of Science ®Google Scholar
• Oehlert , G. W. ( 1992 ). A note on the delta method . Amer. Statistician 46 ( 1 ): 27 – 29 .
   Web of Science ®Google Scholar
• Pestana , D. ( 2007 ). Personal communication .
   Google Scholar
• Ramos , L. P. C. ( 2007 ). Quase-normalidade e Inferência para Séries de Estudos Emparelhadas. Ph.D.'s dissertation in Mathematics, speciality Statistics Faculty of Science and Technology, New University of Lisbon .
   Google Scholar
• Ramos , L. P. C. , Ferreira , D. , Ferreira , S. S. , Nunes , C. , Mexia , J. T. ( 2012 ). Approximate normality of low degree polynomials in normal independent variables . Far East J. Math. Sci. 68 ( 2 ): 287 – 296 .
   Google Scholar
• Shiryaev , A. N. ( 1996 ). Probability. , 2nd ed. New York : Springer-Verlag .
   Google Scholar
• Wallace , D. L. ( 1958 ). Asymptotic approximations to distributions . Ann. Math. Statist. 29 ( 3 ): 635 – 654 .
   Google Scholar
• Zorich , V. A. ( 2004 ). Mathematical Analysis II . Berlin : Springer-Verlag .
   Google Scholar