On the Spectrum of Asymptotic Expansions for an Asymptotic Normal Sequence
Min Tsao
DOI: 10.4236/ojs.2012.21010   PDF   HTML   XML   3,473 Downloads   5,750 Views  


We present a family of formal expansions for the density function of a general one-dimensional asymptotic normal sequence Xn. 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 Xn. 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 Xn.

Share and Cite:

M. Tsao, "On the Spectrum of Asymptotic Expansions for an Asymptotic Normal Sequence," Open Journal of Statistics, Vol. 2 No. 1, 2012, pp. 98-105. doi: 10.4236/ojs.2012.21010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. Y. Edgeworth, “The Law of Error,” Cambridge Philosophical Society Transactions, Vol. 20, 1905, pp. 33- 66.
[2] D. L. Wallace, “Asymptotic Approximations to Distributions,” Annals of Mathematical Statistics, Vol. 29, No. 3, 1958, pp. 635-654. doi:10.1214/aoms/1177706528
[3] H. Cramér, “On the Composition of Elementary Errors,” Skand Aktuarietidskr, Vol. 11, 1928, pp. 13-74.
[4] H. Cramér, “Random Variables and Probability Distributions,” Cambridge University Press, Cambridge, 1937.
[5] W. Feller, “An Introduction to Probability Theory and Its Applications,” Wiley, New York, 1966.
[6] 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
[7] 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
[8] 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
[9] H. E. Daniels, “Saddlepoint Approximations in Statistics,” Annals of Mathematical Statistics, Vol. 25, No. 4, 1954, pp. 631-649. doi:10.1214/aoms/1177728652
[10] 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.
[11] 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
[12] 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
[13] 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
[14] 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
[15] I. M. Skovgaard, “On Multivariate Edgeworth Expansions,” International Statistical Review, Vol. 54, No. 2, 1986, pp. 169-186. doi:10.2307/1403142
[16] J. L. Jensen, “Saddlepoint Approximations,” Clarendon Press, Oxford, 1995.
[17] M. G. Kendall and A. Stuart, “Advanced Theory of Sta- tistics,” 3rd Edition, High Wycombe, London, 1969.

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.