积分Galton--Watson过程中的加权最小二乘法：截距推断与最优权重

中文摘要 AI 辅助

在带移民的积分Galton--Watson过程中，Wei和Winnicki（1990）使用了权重为$(1+X_{t-1})^{-1}$的加权最小二乘法（WLS），并遗留了在递归情形下所得截距估计量的渐近分布问题。Lu（2026）通过提出权重为$1/t$的时间加权WLS绕过了这一困难。虽然该估计量在所有机制下产生一个有效的高斯过程，但其速率仅为$\sqrt{\log n}$。我们将Wei--Winnicki的状态加权WLS估计量推广到权重为$(a+X_{t-1})^{-1}$的情形，其中$a$为任意固定正数。我们解决了这一分布问题，并证明在严格递归下收敛速率是$n$的多项式，而在边界处为$\log n$，此时极限分布是非正态的。我们还证明了在所有三种机制下一种常见的估计偏移过程的合理性。模拟实验展示了所得推断的有限样本性能，并表明施加单位根可显著提高覆盖率，尤其是在边界附近。对加拿大洪水灾害计数的应用表明，状态加权漂移估计对样本起始年份的敏感性远低于时间加权估计。

英文摘要

We study inference on the drift of a critical Galton--Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights $1/t$, and improved state-weighted confidence-interval coverage when the unit root is imposed.

发表机构

• Concordia University（康考迪亚大学）

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