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On the Spectrum of Asymptotic Expansions for an Asymptotic Normal Sequence
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Open Journal of Statistics·Volume 02 (2012)·Pages 98–105·Published 6 January 2012·DOI10.4236/ojs.2012.21010
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Abstract
We present a family of formal expansions for the density function of a general one-dimensional asymptotic normal sequence X n . Members of the family are indexed by a parameter τ with an interval domain which we refer to as the spectrum of the family. The spectrum provides a unified view of known expansions for the density of X n . It also provides a means to explore for new expansions. We discuss such applications of the spectrum through that of a sample mean and a standardized mean. We also discuss a related expansion for the cumulative distribution function of X n .
KeywordsAsymptotic ExpansionAsymptotic Normal SequenceEdgeworth ExpansionSaddlepoint ExpansionSaddlepoints ExpansionHermite Polynomials
- F. Y. Edgeworth, “The Law of Error,” Cambridge Philosophical Society Transactions, Vol. 20, 1905, pp. 33- 66.
- D. L. Wallace, “Asymptotic Approximations to Distributions,” Annals of Mathematical Statistics, Vol. 29, No. 3, 1958, pp. 635-654. doi:10.1214/aoms/1177706528
- H. Cramér, “On the Composition of Elementary Errors,” Skand Aktuarietidskr, Vol. 11, 1928, pp. 13-74.
- H. Cramér, “Random Variables and Probability Distributions,” Cambridge University Press, Cambridge, 1937.
- W. Feller, “An Introduction to Probability Theory and Its Applications,” Wiley, New York, 1966.
- R. N. Bhattacharya and J. K. Ghosh, “On the Validity of the Formal Edgeworth Expansion,” Annals of Statistics, Vol. 6, No. 2, 1978, pp. 434-451. doi:10.1214/aos/1176344134
- R. Strawderman, G. Casella and M. Wells, “Practical Small- Sample Asymptotics for Regression Problems,” Journal of the American Statistical Association, Vol. 91, No. 434, 1996, pp. 643-654. doi:10.2307/2291660
- C. A. Field, “Tail Areas of Linear Combinations of Chi- Squares and Non-Central Chi-Squares,” Journal of Statistical Computation and Simulation, Vol. 45, No. 3-4, 1996, pp. 243-248. doi:10.1080/00949659308811484
- H. E. Daniels, “Saddlepoint Approximations in Statistics,” Annals of Mathematical Statistics, Vol. 25, No. 4, 1954, pp. 631-649. doi:10.1214/aoms/1177728652
- M. S. Srivastava and W. K. Yau, “Tail Probability Approximations of General Statistics,” Technical Report No. 88-38, Center for Multivariate Analysis, University of Pittsburgh, Pittsburgh, 1988.
- G. S. Easton and E. Ronchetti, “General Saddlepoint Approximation with Application to Statistics,” Journal of the American Statistical Association, Vol. 81, No. 394, 1986, pp. 420-429. doi:10.2307/2289231
- R. D. Routledge and M. Tsao, “Uniform Validity of Saddlepoint Expansion on Compact Set,” Canadian Journal of Statistics, Vol. 23, No. 4, 1995, pp. 425-431. doi:10.2307/3315386
- H. Callaert, P. Janssen and N. Veraverbeke, “An Edgeworth Expansion for U-Statistics,” Annals of Statistics, Vol. 8, No. 2, 1980, pp. 299-312. doi:10.1214/aos/1176344955
- P. J. Bickel, F. G?tze and W. R. van Zwet, “The Edgeworth Expansion for U-Statistic of Degree Two,” Annals of Statistics, Vol. 14, No. 4, 1986, pp. 1463-1484. doi:10.1214/aos/1176350170