Censored Regression Models with Autoregressive Errors: A Likelihood-Based Perspective

Fernanda L. Schumacher
Víctor H. Lachos
Dipak K. Dey

In many studies that involve time series variables, limited or censored data are naturallycollected. This occurs, in several practical situations, for reasons such as limitations of mea-suring equipment or from experimental design. Hence, the exact true value is recorded only ifit falls within an interval range, so the responses can be either left, interval or right censored.Practitioners commonly disregard censored data cases or replace these observations with somefunction of the limit of detection, which often results in biased estimates. In this paper, wepropose an analytically tractable and efficient stochastic approximation of the EM (SAEM)algorithm to obtain the maximum likelihood estimates of the parameter of censored regressionmodels with autoregressive errors of order p. This approach permits easy and fast estimationof the parameters of autoregressive models when censoring is present and as a byproduct, en-ables predictions of unobservable values of the response variable. The observed informationmatrix is derived analytically to account for standard errors. We use simulations to investigatethe asymptotic properties of the SAEM estimates and prediction accuracy. In this simulationstudy comparisons are also made between inferences based on the censored data and thosebased on complete data obtained by crude/ad-hoc imputation methods. Finally, the method isillustrated using a meteorological time series dataset on cloud ceiling height, where the mea-surements are subject to the detection limit of the recording device. The proposed algorithmand methods are implemented in the new R package ARCensReg.

Autoregressive AR(p) models
Censored data
Limit of detection
SAEM algorithm