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# Series Expansions in Statistical Theory of Detection and Estimation

Chapter

## Abstract

Methods of series expansions have been used extensively in many areas of science and engineering. In particular, since the early part of the century, they have been used in the study of approximations to distribution functions and in the study of approximations to quantiles, especially for complicated distributions. These approximations are of great importance in the general theory of statistical inference so that one can evaluate (approximately) the performance and/or robustness of some standard tests of hypotheses and of optimal estimators.

## Keywords

Equivalence Class Asymptotic Expansion Series Expansion Noise Distribution Edgeworth Expansion
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## References

- [1]J.I. Marcum, A statistical theory of target detection by pulsed radar,
*IRE Trans. Inform. Theory*, Vol. IT–6, pp. 59–267, 1960.Google Scholar - [2]P. Swerling, Probability of detection for fluctuating targets,
*IRE Trans. Inform. Theory*, Vol. IT–6, pp. 269–308, 1960.Google Scholar - [3]S.C. Schwartz, A series technique for the optimum detection of stochastic signals in noise,
*IEEE Trans. Inform. Theory*, Vol. IT–15, pp. 362–369, 1969.Google Scholar - [4]A. Bodharamik et al., Optimal detection and signal design for channels with non- but near-Gaussian additive noise,
*IEEE Trans. Comm.*, Vol. COM–20, pp. 1087–1096, 1972.Google Scholar - [5]L.F. Eastwood and R. Lugannani, Approximate likelihood ratio detectors for linear processes,
*IEEE Trans. Inform. Theory*, Vol. IT–23, pp. 482–489, 1977.Google Scholar - [6]Y.F. Huang and J.B. Thomas, Sensitivity of some optimal detectors to noise skewness,
*Proc. of the 1982 Intl. Conf. Comm.*, pp. 2H.1.1–2H. 1. 5, 1982.Google Scholar - [7]Y.F. Huang and J.B. Thomas, Signal detection in nearly-Gaussian skewed noise,
*J. Acoust. Soc. Amer.*, Vol. 74, pp. 1399–1405, 1983.MathSciNetzbMATHCrossRefGoogle Scholar - [8]D. Middleton, Man-made noise in urban environments and transportation systems: Models and measurements,
*IEEE Trans. Comm.*, Vol. COM–21, pp. 1232–1241, 1973.Google Scholar - [9]R.S. Freedman, On Gram-Charlier approximations,
*IEEE Trans. Comm.*, Vol. COM–29, pp. 122–125, 1981.Google Scholar - [10]J.B. Thomas, Nonparametric detection,
*Proc. IEEE*, Vol. 58, pp. 623–631, 1970.CrossRefGoogle Scholar - [11]V.G.Hansen, Detection performance of some noriparametric rank tests and an application to radar, IEEE Trans. Inform. Theory, Vol. IT–16, pp. 309–318, 1970.Google Scholar
- [12]P.J. Huber, Robust estimation of a location parameter, Ann. Math. Statist., Vol. 35, pp. 73–101, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
- [13]P.J. Huber, A robust version of the probability ratio test, Ann. Math. Statist., Vol. 36, pp. 1753–1758, 1965.MathSciNetzbMATHCrossRefGoogle Scholar
- [14]A.H. El-Sawy and V.D. Vandelinde, Robust sequential detection of signals in noise,
*IEEE Trans. Inform. Theory*, Vol. IT–25, pp. 346–353, 1979.Google Scholar - [15]D.L. Wallace, Asymptotic approximation to distributions,
*Ann. Math. Statist.*, Vol. 29, pp. 635–654, 1958.MathSciNetzbMATHCrossRefGoogle Scholar - [16]A. Erdelyi,
*Asymptotic Expansions*, New York, NY: Dover, 1956.zbMATHGoogle Scholar - [17]P.J. Bickel, Edgeworth expansions in nonparametric statistics,
*Ann. Math. Statist.*, Vol. 2, pp. 1–20, 1974.MathSciNetzbMATHGoogle Scholar - [18]E.A. Cornish and R.A. Fisher, Moments and cumulants in the specification of distributions, Paper 30 in R.A. Fisher,
*Contributions to Mathematical Statistics*, New York, NY: Wiley, 1950.Google Scholar - [19]W. Feller,
*An Introduction to Probability Theory and Its Applications*, Vol. II, 2nd ed., New York, NY: Wiley, 1971.zbMATHGoogle Scholar - [20]H.L. Royden,
*Real Analysis*, 2nd ed., New York, NY: The Macmillan Co., 1972.Google Scholar - [21]A.R. Milne and J.H. Ganton, Ambient noise under Arctic sea ice,
*J. Acoust. Soc. Amer*,, Vol. 36, pp. 855–863, 1964.CrossRefGoogle Scholar - [22]R.F. Dwyer, FRAM II single channel ambient noise statistics,
*101st Meeting of Acoust. Soc. Amer.*, May, 1981.Google Scholar - [23]M.E. Frazer, Some statistical properties of lake surface reverberation,
*J. Acoust. Soc. Amer.*, Vol. 64, pp. 858–868, 1978.CrossRefGoogle Scholar - [24]G.R. Wilson and D.R. Powell, Experimental and modeled density estimates of underwater acoustic returns,
*Signal Processing in the Ocean Environment Workshop*, Maryland, May, 1982.Google Scholar - [25]F.W. Machell and C.S. Penrod, Probability density functions of ocean acoustic noise processes,
*Proc. of the Office of Naval Research Workshop on Signal Processing*, 1982.Google Scholar - [26]H. Cramer,
*Mathematical Methods of Statistics*, Princeton, NJ: Princeton University Press, 1946.zbMATHGoogle Scholar - [27]S.A. Kassam, G. Moustakides and J.G. Shin, Robust detection of known signals in asymmetric noise,
*IEEE Trans. Inform. Theory*, Vol. IT–28, pp. 84–91, 1982.Google Scholar - [28]R.F. Dwyer, A technique for improving detection and estimation of signals contaminated by under-ice noise, Naval Underwater Systems Center, Tech. Document 6717, July, 1982.Google Scholar
- [29]N.J. Johnson, Modified T-test and confidence intervals for asymmetrical populations,
*JASA*, Vol. 73, pp. 536–544, 1978.zbMATHGoogle Scholar - [30]R.J. Carroll, On estimating variances of robust estimators when the errors are asymmetric,
*JASA*, Vol. 74, pp. 674–679, 1979.zbMATHGoogle Scholar - [31]S.C Schwartz, Estimation of probability density by an orthogonal series,
*Ann. Math. Statist.*, Vol. 38, pp. 1261–1265, 1967.MathSciNetzbMATHCrossRefGoogle Scholar - [32]M. Kendall and A. Stuart,
*The Advanced Theory of Statistics*, Vol. 1, 4th ed., New York, NY: Macmillan, 1977.zbMATHGoogle Scholar - [33]Y.F. Huang, On series expansions and signal detection,
*Proc. 17th Annual Conf. Inform. Sci. and Sys.*, Johns Hopkins University, pp. 534–538, 1983.Google Scholar - [34]Y. V. Linnik and N.M. Mitrofanova, Some asymptotic expansions for the distribution of the maximum likelihood estimate,
*Sankhya Ser. A*, Vol. 27, pp. 73–82, 1965.MathSciNetzbMATHGoogle Scholar - [35]N.M. Mitrofanova, An asymptotic expansion for the maximum likelihood estimate of vector parameters,
*Theor. Prob. Appl.*, Vol. 12, pp. 364–372, 1967.MathSciNetzbMATHCrossRefGoogle Scholar - [36]J. Pfanzagl, Asymptotic expansions in parametric statistical theory, Chapter 1 in
*Developments in Statistics*, Vol. 3, (P.R. Krishnaiah, Ed.), New York, NY: Academic Press, 1980.Google Scholar - [37]I.M. Skovgaard, Transformations of an Edgeworth expansion by a sequence of smooth functions,
*Scand. J. Statist.*, Vol. 8, pp. 207–217, 1981.MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag New York Inc. 1986