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Estimation of the offspring mean in a controlled branching process with a random control function. (English) Zbl 1114.62082
Summary: Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by work of C. Z. Wei and J. Winnicki [Estimation of the mean in the branching process with immigration. Ann. Stat. 18, No. 4, 1757–1773 (1990; Zbl 0736.62071)], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki for branching processes with immigration and provide a unified limit theory of estimation.

MSC:
62M05 Markov processes: estimation; hidden Markov models
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
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[1] Bhat, B.R.; Adke, S.R., Maximum likelihood estimation for branching processes with immigration, Adv. appl. probab., 13, 498-509, (1981) · Zbl 0481.62067
[2] Chow, Y.S.; Teicher, H., Probability theory. independence, interchangeability, martingales, (1997), Springer-Verlag New York · Zbl 0891.60002
[3] Dion, J.P.; Essebbar, B., On the statistics of controlled branching processes, Lecture notes in statist., 99, 14-21, (1995) · Zbl 0821.62046
[4] Ethier, S.N.; Kurtz, T.G., Markov processes: characterization and convergence, (1986), Wiley New York · Zbl 0592.60049
[5] González, M.; Martínez, R.; Mota, M., On the geometric growth in a class of homogeneous multitype Markov chain, J. appl. probab., 42, 1015-1030, (2005) · Zbl 1098.60072
[6] González, M.; Martínez, R.; Mota, M., On the unlimited growth of a class of homogeneous multitype Markov chains, Bernoulli, 11, 559-570, (2005) · Zbl 1073.60069
[7] González, M.; Martínez, R.; del Puerto, I., Nonparametric estimation of the offspring distribution and the Mean for a controlled branching process, Test, 13, 465-479, (2004) · Zbl 1069.62064
[8] González, M.; Martínez, R.; del Puerto, I., Estimation of the variance for a controlled branching process, Test, 14, 199-213, (2005) · Zbl 1069.62063
[9] González, M.; Molina, M.; del Puerto, I., On the class of controlled branching processes with random control functions, J. appl. probab., 39, 804-815, (2002) · Zbl 1032.60077
[10] González, M.; Molina, M.; del Puerto, I., On the geometric growth in controlled branching processes with random control function, J. appl. probab., 40, 995-1006, (2003) · Zbl 1054.60087
[11] González, M.; Molina, M.; del Puerto, I., Limiting distribution for subcritical controlled branching processes with random control function, Statist. probab. lett., 67, 277-284, (2004) · Zbl 1063.60121
[12] González, M.; Molina, M.; del Puerto, I., Asymptotic behavior of critical controlled branching processes with random control functions, J. appl. probab., 42, 463-477, (2005) · Zbl 1079.60073
[13] Hall, P.; Heyde, C.C., Martingale limit theory and its application, (1980), Academic Press San Diego · Zbl 0462.60045
[14] Heyde, C.C., Extension of a result of seneta for the supercritical galton – watson process, Ann. math. statist., 41, 739-742, (1970) · Zbl 0195.19201
[15] Heyde, C.C.; Seneta, E., Analogous of classical limit theorems for the supercritical galton – watson process with immigration, Math. biosci., 11, 249-259, (1971) · Zbl 0224.60042
[16] Heyde, C.C.; Seneta, E., Estimation theory for growth immigration rates in a multiplicative process, J. appl. probab., 9, 235-258, (1972) · Zbl 0243.60047
[17] Heyde, C.C.; Seneta, E., Notes on ‘estimation theory for growth immigration rates in a multiplicative process’, J. appl. probab., 11, 572-577, (1974) · Zbl 0294.60063
[18] Heyde, C.C.; Brown, B.M., An invariance principle and some convergence rate results for branching processes, Z. wahrsch. verw. geb., 20, 271-278, (1971) · Zbl 0212.49505
[19] Jagers, P., Branching processes with biological applications, (1975), Wiley London · Zbl 0356.60039
[20] Kallenberg, O., Foundations of modern probability, (1997), Springer-Verlag New York · Zbl 0892.60001
[21] Klimko, L.A.; Nelson, P.I., On conditional least squares estimation for stochastic processes, Ann. statist., 6, 629-642, (1978) · Zbl 0383.62055
[22] Qi, Y.; Reeves, J., On sequential estimation for branching processes with immigration, Stochastic process appl., 100, 41-51, (2002) · Zbl 1057.62065
[23] Sevast’yanov, B.A.; Zukov, A., Controlled branching processes, Theory probab. appl., 19, 1, 14-21, (1974)
[24] Shete, S; Sriram, T.N., A note on estimation in multitype supercritical branching processes with immigration, Sankhyā, 65 (part 1), 107-121, (2003) · Zbl 1192.60097
[25] Sriram, T.N.; Basawa, I.V.; Huggins, R., Sequential estimation for branching processes with immigration, Ann. statist., 19, 2232-2243, (1991) · Zbl 0850.62636
[26] Smoluchowski, M., Drei vortrage uber diffusion brownsche bewegung und wagulation von kolloidtelchen, Physik. zeits., 17, 557-585, (1916)
[27] Venkataraman, K.N., A time series approach to the study of the simple subcritical galton – watson process with immigration, Adv. appl. probab., 14, 1-20, (1982) · Zbl 0486.62088
[28] Venkataraman, K.N.; Nanthi, K., A limit theorem on a subcritical galton – watson process with immigration, Ann. probab., 10, 1069-1074, (1982) · Zbl 0498.62073
[29] Wei, C.Z.; Winnicki, J., Some asymptotic results for the branching process with immigration, Stochastic process appl., 31, 261-282, (1989) · Zbl 0673.60092
[30] Wei, C.Z.; Winnicki, J., Estimation of the Mean in the branching process with immigration, Ann. statist., 18, 1757-1773, (1990) · Zbl 0736.62071
[31] Yanev, N.M., Conditions for degeneracy of \(\phi\)-branching processes with random \(\phi\), Theory probab. appl., 28, 481-491, (1975)
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