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Normal approximations for binary lattice systems

  • Richard J. Kryscio (a1), Roy Saunders (a1) and Gerald M. Funk (a2)

Abstract

Consider an array of binary random variables distributed over an m 1(n) by m 2(n) rectangular lattice and let Y 1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y 1(n), · ··, Yr (n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

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Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, U.S.A.
∗∗ Postal address: Department of Mathematical Sciences, Loyola University of Chicago, 6525 North Sheridan Rd., Chicago, IL 60626, U.S.A.

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Research supported in part by NSF Grant No. MCS 77–03582.

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References

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Besag, J. E. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192235.
Bloemena, A. R. (1964) Sampling From a Graph. Mathematical Centre Tracts 2, Amsterdam.
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Feller, W. (1967) An Introduction to Probability Theory and its Applications , Volume I, 3rd ed. Wiley, New York.
Fraser, D. A. S. (1957) Nonparametric Methods in Statistics. Wiley, New York.
Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.
Saunders, R., Kryscio, R. J. and Funk, G. M. (1979) Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions. J. Appl. Prob. 16, 554566.
Strauss, D. J. (1975) Analyzing binary lattice data with the nearest neighbor property. J. Appl. Prob. 12, 702712.
Strauss, D. J. (1977) Clustering on coloured lattices. J. Appl. Prob. 14, 135143.

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Normal approximations for binary lattice systems

  • Richard J. Kryscio (a1), Roy Saunders (a1) and Gerald M. Funk (a2)

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