Unlike the traditional approach of using generating function methods, the author analyses continuous-time Markov branching processes with time- dependent random immigration rate on the basis of multivariate point process theory [J. Jacod, Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 235-253 (1975; Zbl 0302.60032), M. Jacobsen, Statistical analysis of counting processes, Lect. Notes Stat. 12 (1982; Zbl 0518.60065)].
Two situations are distinguished where randomness comes from an external source or from state-dependence. The asymptotic parametric inference is derived for the subcritical case \(m<1\) [for the supercritical case \(m>1\), see I. L. Hudson, Aust. J. Stat. 25, 47-57 (1983; Zbl 0532.62059)]. Particularly, the limit distributions of various estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.