对左截断数据估计平稳序列

设平稳信号过程$\{X_t\}$被白噪声序列$\{Y_t\}$干扰. 只有$X_t>Y_t$时可以观测到信号过程$X_t$, 否则只能观测到白噪声$Y_t$. 这种数据模型被称为左截断数据模型. 本文在左截断数据模型下估计平稳信号过程的$\{X_t\}$均值, 自协方差函数, 和自相关系数. 证明所给的估计量是强相合估计. 当$X_t$是自回归序列时, 本文给出自回归模型的强相合的参数估计.

Estimate a Stationary Process under Left Censoring

Let $\{X_t\}$ be a stationary signal process interfered by an white noise $\{Y_t\}$. The signal $X_t$ is detected and observed only when $X_t>Y_t$, otherwise only the white noise $Y_t$ is observed. This model is called the left censored model and the observation is called the left censored observation. In this paper we use the nonparametric MLE of the marginal distributions of $X_t$ and $Y_t$ to construct estimates of the mean, autocovariance and autocorrelation functions of the original signal process $\{X_t\}$. The strong consistency of the estimates is derived. When $\{X_t\}$ is a causal autoregression process, consistent estimates of the autoregression parameters are provided.