Quasi-Likelihood Estimation of a Censored Autoregressive Model With Exogenous Variables

ABSTRACT Maximum likelihood estimation of a censored autoregressive model with exogenous variables (CARX) requires computing the conditional likelihood of blocks of data of variable dimensions. As the random block dimension generally increases with the censoring rate, maximum likelihood estimation becomes quickly numerically intractable with increasing censoring. We introduce a new estimation approach using the complete-incomplete data framework with the complete data comprising the observations were there no censoring. We introduce a system of unbiased estimating equations motivated by the complete-data score vector, for estimating a CARX model. The proposed quasi-likelihood method reduces to maximum likelihood estimation when there is no censoring, and it is computationally efficient. We derive the consistency and asymptotic normality of the quasi-likelihood estimator, under mild regularity conditions. We illustrate the efficacy of the proposed method by simulations and a real application on phosphorus concentration in river water.

[1]  D. Cox,et al.  A General Definition of Residuals , 1968 .

[2]  A. Genz,et al.  Computation of Multivariate Normal and t Probabilities , 2009 .

[3]  J. Tobin.  Estimation of Relationships for Limited Dependent Variables , 1958 .

[4]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[5]  Scott L. Zeger,et al.  Regression analysis with censored autocorrelated data , 1986 .

[6]  Tuan Pham,et al.  Some mixing properties of time series models , 1985 .

[7]  B. Levit,et al.  Asymptotically efficient estimation of analytic functions in Gaussian noise , 1996 .

[8]  B. Ripley,et al.  Editorial Time Series in R 1.5.0 , .

[9]  J. Jawitz.  Moments of truncated continuous univariate distributions , 2004 .

[10]  T. Hothorn,et al.  Multivariate Normal and t Distributions , 2016 .

[11]  Donald W. K. Andrews,et al.  An empirical process central limit theorem for dependent non-identically distributed random variables , 1989 .

[12]  R. Libra,et al.  Nitrogen and phosphorus budgets for Iowa and Iowa watersheds , 2004 .

[13]  G. M. Tallis.  The Moment Generating Function of the Truncated Multi‐Normal Distribution , 1961 .

[14]  Andreas N. Philippou,et al.  A generalized geometric distribution and some of its properties , 1983 .

[15]  Takeshi Amemiya,et al.  Regression Analysis when the Dependent Variable is Truncated Normal , 1973 .

[16]  M. A. Arcones,et al.  Central limit theorems for empirical andU-processes of stationary mixing sequences , 1994 .

[17]  Kung-Sik Chan,et al.  Limiting properties of the least squares estimator of a continuous threshold autoregressive model , 1998 .

[18]  I. James,et al.  Linear regression with censored data , 1979 .

[19]  Kung-Sik Chan,et al.  Time Series Analysis: With Applications in R , 2010 .

[20]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[21]  Kung-Sik Chan,et al.  Quantifying the effect of land use land cover change on increasing discharge in the Upper Mississippi River , 2010 .

[22]  Marc G. Genton,et al.  Censored time series analysis with autoregressive moving average models , 2007 .

[23]  P. Robinson,et al.  ON THE ASYMPTOTIC PROPERTIES OF ESTIMATORS OF MODELS CONTAINING LIMITED DEPENDENT VARIABLES , 1982 .

[24]  S L Hillis.  Residual plots for the censored data linear regression model. , 1995, Statistics in medicine.

[25]  Harold R. Parks,et al.  A Primer of Real Analytic Functions , 1992 .

[26]  P. Robinson.  Estimation and forecasting for time series containing censored or missing observations , 1980 .

[27]  P. Robinson.  Analysis of Time Series from Mixed Distributions , 1982 .