Celebratio Mathematica — Rosenblatt

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F. J. Breidt, R. A. Davis, N.-J. Hsu, and M. Rosenblatt: “Pile-up probabilities for the Laplace likelihood estimator of a non-invertible first order moving average,” pp. 1–19 in Time series and related topics: In memory of Ching-Zong Wei (Taipei, Taiwan, 12–14 December 2005). Edited by H.-C. Ho, C.-K. Ing, and T. L. Lai. IMS Lecture Notes–Monograph Series 52. Institute of Mathematical Statistics (Beachwood, OH), 2006. MR 2427836 Zbl 1268.62107 incollection

The first-order moving average model or \( \mathrm{MA}(1) \) is given by \[ X_t = Z_t - \theta_0 Z_{t-1} ,\] with independent and identically distributed \( \{Z_t\} \). This is arguably the simplest time series model that one can write down. The \( \mathrm{MA}(1) \) with unit root (\( \theta_0 = 1 \)) 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 \( \mathrm{MA}(1) \) 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 \( \mathrm{MA}(1) \) 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 (\( \theta = 1 + \beta/n \), \( \beta\leq 0 \)). Unlike standard \( 1/\sqrt{n} \)-asymptotics, these estimation procedures have \( 1/n \)-asymptotics and a so-called pile-up effect, in which \( \mathrm{P}(\hat{\theta} = 1) \) converges to a positive value. One explanation for this pile-up phenomenon is the lack of identifiability of \( \theta \) in the Gaussian case. That is, the Gaussian likelihood has the same value for the two sets of parameter values \( (\theta,\sigma^2) \) and \( (1/\theta,\theta^2\sigma^2) \). It follows that \( \theta = 1 \) is always a critical point of the likelihood function. In contrast, for non-Gaussian noise, \( \theta \) 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 \( \theta_0 \) 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.

Nan-Jung Hsu	Related
Ching-Kang Ing	Related
F. Jay Breidt	Related
Tze Leung Lai	Related
Richard Alan Davis	Related
Ching Zong Wei	Related
Hwai-Chung Ho	Related