On the Spectrum of Asymptotic Expansions for an Asymptotic Normal Sequence

Author(s)
Min Tsao

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}.

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

Cite this paper

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.

M. Tsao, "On the Spectrum of Asymptotic Expansions for an Asymptotic Normal Sequence,"

References

[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.

[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.