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Estimation of the criticality parameter of a superitical branching process with random environments

Published online by Cambridge University Press:  14 July 2016

K. Nanthi
K. Nanthi*
Affiliation:
University of Madras
*
∗Postal address: Department of Statistics, University of Madras, Madras 600005, India.
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Abstract

For a supercritical branching process x = {xn; n ≧ 0, x0 = 1} with random environments, define when xn > 0; and = 1 when xn = 0. When x is assumed to satisfy the standard regularity assumptions, under the non-extinction hypothesis, is a strongly consistent and asymptotically unbiased estimator for the criticality parameter π and is asymptotically normal. A strongly consistent estimator, is also proposed for the associated variance, σ2.

Keywords

ESTIMATION OF MEANESTIMATION OF VARIANCECONFIDENCE BOUNDSASYMPTOTIC PROPERTIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 890 - 896
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research supported by the University Grants Commission, India.

References

[1] Athreya, K. B. and Karlin, S. (1971) On branching processes with random environments: I and II. Ann. Math. Statist. 42, 1499–1520 and 1843–1858.CrossRefGoogle Scholar
[2] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[3] Becker, N. (1977) Estimation for discrete time branching processes with application to epidemics. Biometrics 33, 515–522.Google Scholar
[4] Dion, J.-P. and Esty, W. W. (1979) Estimation problems in branching processes with random environments. Ann. Statist. To appear.CrossRefGoogle Scholar
[5] Dion, J.-P. and Keiding, N. (1978) In Advances in Probability , ed. Ney, P. Dekker, New York.Google Scholar
[6] Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494.Google Scholar
[7] Heyde, C. C. (1975) Remarks on efficiency in estimation for branching processes. Biometrika 62, 49–55.CrossRefGoogle Scholar
[8] Kesten, H. and Stigum, B. P. (1966) A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37, 1211–1223.Google Scholar
[9] Loève, M. (1968) Probability Theory. Affiliated East-West Press Pvt. Ltd., New Delhi.Google Scholar
[10] Nagaev, A. V. (1967) On estimating the expected number of direct descendants of a particle in a branching process. Theory Prob. Appl. 12, 314–320.CrossRefGoogle Scholar
[11] Rényi, A. (1958) On mixing sequences of sets. Acta Math. Acad. Sci. Hung. 9, 215–228.Google Scholar