Estimation of subcritical Galton Watson processes with correlated immigration

TL;DR

The paper advances parameter estimation in stationary subcritical Galton–Watson processes with immigration by treating correlated immigration. It first derives consistent moment estimators and a CLT when immigration becomes ultimately uncorrelated, then extends to general correlated immigration under mixing, showing that the estimator of $\ln\lambda_0$ converges slowly at rate $1/\ln n$ with a two-term stochastic expansion under stronger decay. A Berk-type spectral estimator provides a consistent long-run variance estimate, enabling valid inference for the asymptotic covariance in both independent and correlated cases. Numerical results illustrate the methods, highlighting robustness of spectral-based variance estimation and the unusually slow convergence when immigration exhibits correlation. Overall, the work broadens applicability of GW-based models with dependent immigration and offers practical inference tools for the reproduction and immigration parameters.

Abstract

We consider an observed subcritical Galton Watson process $\{Y_n,\ n\in \mathbb{Z} \}$ with correlated stationary immigration process $\{ε_n,\ n\in \mathbb{Z} \}$. Two situations are presented. The first one is when $\mbox{Cov}(ε_0,ε_k)=0$ for $k$ larger than some $k_0$: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when $\{ε_n,\ n\in \mathbb{Z} \}$ has general correlation structure: under mixing assumptions, we exhibit an estimator for the the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.

Figures (8)

• Figure 1: Infected immigrants $\{\epsilon_n,\ n\in \mathbb{Z}\}$
• Figure 2: The moment estimators and the estimators proposed by KN78 of $N=1,000$ independent simulations of model \ref{['model']} of size $n=5,000$ with unknown parameter $\lambda_0=0.5$ and $m_0=1$, when the immigration is independent (left panels, with $\epsilon_n \sim IID{\cal P}(1)$) and when the immigration is correlated (right panels, with $\epsilon_n=Z_n Z_{n-1}$ where $Z_n \sim IID{\cal P}(1)$). The reproduction sequence $\xi\sim\mathcal{B}(\lambda_0)$. The panels display the distribution of the estimators $\hat{R}_{k_0,n}$ and $\hat{M}_{k_0,n}$. The two top panels (resp. two bottom panels) correspond to our proposed estimators (resp. to the estimators proposed by KN78 and AOA87).
• Figure 3: The panels at the top present the distribution of the estimation error of the estimates $\lambda_0$ and $m_0$ given in Figure \ref{['fig:param']}. The left (resp. right) panel corresponds to the case of independent (correlated) immigration. Points 1 and 2, in the box-plots, display the distribution of the estimation error $\hat{R}_{k_0,n}-\lambda_0$ and $\hat{M}_{k_0,n}-m_0$. The two top panels (resp. two bottom panels) correspond to our proposed estimators (resp. to the estimators proposed by KN78 and AOA87).
• Figure 4: The panels at the top present the Q–Q plot of the estimates $\lambda_0$ given in Figure \ref{['fig:param']}. The left (resp. right) panel corresponds to the case of independent (correlated) immigration. The two top panels (resp. two bottom panels) correspond to our proposed estimators (resp. to the estimators proposed by KN78 and AOA87).
• Figure 5: The panels at the top present the Q–Q plot of the estimates $m_0$ given in Figure \ref{['fig:param']}. The left (resp. right) panel corresponds to the case of independent (correlated) immigration. The two top panels (resp. two bottom panels) correspond to our proposed estimators (resp. to the estimators proposed by KN78 and AOA87).

Theorems & Definitions (38)

• Proposition 3.1: Stationary distribution and existence of moments
• Lemma 3.1
• Corollary 3.2
• Example 4.1
• Example 4.2
• Proposition 4.1: Consistency
• Remark 4.1
• Theorem 4.2
• Theorem 4.3: Asymptotic normality
• Remark 4.2