Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

Area of Science:

• Statistics
• Probability Theory
• Stochastic Processes

Background:

• Markov branching processes are fundamental in modeling population dynamics.
• Estimating parameters in continuous-time processes observed discretely presents challenges.
• Likelihood-based methods are standard but can be computationally intensive or ill-defined.

Purpose of the Study:

• To investigate quasi-likelihood and pseudo-likelihood estimation for continuous-time multi-type Markov branching processes.
• To compare the properties of conventional and conditional estimation approaches.
• To identify conditions under which these estimators are asymptotically equivalent and robust.

Main Methods:

• Analysis of quasi-likelihood and pseudo-likelihood estimation techniques.
• Comparison of "conventional" and conditional estimation strategies.
• Application to pure birth, linear birth-and-death, and two-type branching processes.
• Monte Carlo simulation studies for finite sample performance evaluation.

Main Results:

• Quasi- and pseudo-likelihood methods demonstrate robustness in parameter estimation.
• Asymptotic equivalence between estimators is identified under specific conditions.
• Limitations of linear quasi-likelihood functions are addressed through non-linear functions or moment conditioning.
• Maximum likelihood estimation is sometimes achieved by these approaches.

Conclusions:

• Quasi- and pseudo-likelihood provide viable and robust alternatives for parameter estimation in discrete-time observations of continuous-time Markov branching processes.
• Careful selection of functions and conditioning is crucial for effective parameter estimation.
• The investigated methods offer practical solutions for complex branching process models.