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Inference for stationary random fields given Poisson samples

Published online by Cambridge University Press:  01 July 2016

Alan F. Karr
Alan F. Karr*
Affiliation:
The Johns Hopkins University
*
∗Postal address: Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA.
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Abstract

Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

Keywords

NON-PARAMETRIC COVARIANCE APPROXIMATIONSTATE ESTIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 406 - 422
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research support by the Air Force Office of Scientific Research, grant number AFOSR 82-0029B. The United States Government is authorized to reproduce and distribute reprints for governmental purposes.

References

Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.Google Scholar
Bloomfield, P. (1970) Spectral analysis with randomly missing observations. J. R. Statist. Soc. B 32, 369–380.Google Scholar
Brillinger, D. R. (1981) Time Series: Data Analysis and Theory. Holden-Day, San Francisco.Google Scholar
Guarascio, M., David, M., and Huijbregts, C., (Eds.) (1976) Advanced Geostatistics in the Mining Industry. Reidel, Boston.Google Scholar
Jolivet, E. (1981) Central limit theorem and convergence of empirical processes for stationary point processes. In Point Processes and Queueing Problems, ed Bártfai, P. and Tomkó, J., North-Holland, Amsterdam, 117–162.Google Scholar
Kallenberg, O. (1983) Random Measures, 3rd edn. Akademic-Verlag, Berlin; Academic Press, New York.Google Scholar
Karr, A. F. (1982) A partially observed Poisson process. Stoch. Proc. Appl. 12, 249–269.Google Scholar
Karr, A. F. (1984) Estimation and reconstruction for zero-one Markov processes. Stoch. Proc. Appl. 16, 219–255.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and their Statistical Inference. Dekker, New York.Google Scholar
Kingman, J. F. C. (1963) Poisson counts for random sequences. Ann. Math. Statist. 34, 1217–1232.Google Scholar
Krickeberg, K. (1982) Processus ponctuels en statistique. In Lecture Notes in Mathematics 929, Springer-Verlag, Berlin, 205–313.Google Scholar
Leonov, V. P. and Shiryayev, A. N. (1959) On the technique of calculating semi-invariants. Teor. Veroyatnost i Primenen 4, 342–355.Google Scholar
Masry, E. (1978) Poisson sampling and spectral estimation of continuous-time processes. IEEE Trans. Inf. Theory 24, 173–183.Google Scholar
Masry, E. (1983) Nonparametric covariance estimation from irregularly spaced data. Adv. Appl. Prob. 15, 113–132.CrossRefGoogle Scholar
Masry, E. (1984) The estimation of frequency-wavenumber spectra using acoustic arrays, Parts I and II. J. Acoust. Soc. Amer. 76, 139–149; 1123-1131.Google Scholar
Matheron, A. (1976) A simple substitute for conditional expectations: the disjunctive kriging. In Guarascio, et al. (1976), 221–236.Google Scholar
Nguyen, X. X. and Zessin, H. (1979) Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.Google Scholar
Shapiro, H. S. and Silverman, R. A. (1960) Alias-free sampling of random noise. J. SIAM 8, 225–248.Google Scholar
Yadrenko, M. I. (1983) Spectral Theory of Random Fields. Optimization Software, Inc., New York.Google Scholar