不可逆一阶移动平均Laplace似然估计的堆积概率
Pile-up probabilities for the Laplace likelihood estimator of a non-invertible first order moving average
摘要：
一阶移动平均模型或MA（1）由$X_t=Z_t-θ_0Z_{t-1}$给出，其中独立且同分布$\{Z_t\}$。这可以说是人们可以写下的最简单的时间序列模型。具有单位根的MA（1）（$θ_0=1$）在各种时间序列应用中自然出现。例如，如果基础时间序列由线性趋势加上白噪声误差组成，则差分序列是具有单位根的MA（1）。在这种情况下，对差分序列的单位根的测试相当于对趋势加噪声模型的充分性的测试。单位根问题也自然出现在信号加噪声模型中，其中信号被建模为随机游走。差分序列遵循MA（1）模型，当且仅当随机游走信号实际上是常数时，差分序列具有单位根。针对单位根情况和近单位根情况（$θ=1+β/n,β\le0$），已经建立了基于高斯似然的各种估计量的渐进理论。与标准$1/\sqrt{n}$ -渐近性不同，这些估计过程具有$1/n$ -渐近性和所谓的堆积效应，其中P $(\hatθ=1)$收敛于正值。对这种堆积现象的一种解释是在高斯情况下$θ$缺乏可识别性。也就是说，高斯似然对于两组参数值$(θ,σ^2)$和$(1/θ,θ^2σ^2$具有相同的值。因此，$θ=1$总是似然函数的临界点。相比之下，对于非高斯噪声，$θ$对于所有实值都是可识别的。因此，在非高斯情况下是否会持续同样的堆积现象不再清楚。在本文中，我们专注于限制堆积概率估计$θ_0$拉普拉斯似然的基础上。在某些情况下，这些估计值可以被视为最小绝对偏差（LAD）估计值。仿真结果验证了极限理论。
英文摘要：
The first-order moving average model or MA(1) is given by $X_t=Z_t-θ_0Z_{t-1}$, with independent and identically distributed $\{Z_t\}$. This is arguably the simplest time series model that one can write down. The MA(1) with unit root ($θ_0=1$) arises naturally in a variety of time series applications. For example, if an underlying time series consists of a linear trend plus white noise errors, then the differenced series is an MA(1) with unit root. In such cases, testing for a unit root of the differenced series is equivalent to testing the adequacy of the trend plus noise model. The unit root problem also arises naturally in a signal plus noise model in which the signal is modeled as a random walk. The differenced series follows a MA(1) model and has a unit root if and only if the random walk signal is in fact a constant. The asymptotic theory of various estimators based on Gaussian likelihood has been developed for the unit root case and nearly unit root case ($θ=1+β/n,β\le0$). Unlike standard $1/\sqrt{n}$-asymptotics, these estimation procedures have $1/n$-asymptotics and a so-called pile-up effect, in which P$(\hatθ=1)$ converges to a positive value. One explanation for this pile-up phenomenon is the lack of identifiability of $θ$ in the Gaussian case. That is, the Gaussian likelihood has the same value for the two sets of parameter values $(θ,σ^2)$ and $(1/θ,θ^2σ^2$). It follows that $θ=1$ is always a critical point of the likelihood function. In contrast, for non-Gaussian noise, $θ$ is identifiable for all real values. Hence it is no longer clear whether or not the same pile-up phenomenon will persist in the non-Gaussian case. In this paper, we focus on limiting pile-up probabilities for estimates of $θ_0$ based on a Laplace likelihood. In some cases, these estimates can be viewed as Least Absolute Deviation (LAD) estimates. Simulation results illustrate the limit theory.
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