Smoothing Estimation for Discrete-Valued Time Series

We deal with the smoothed estimators for conditional probability functions of discrete-valued time series fY t g under two diierent settings. When the conditional distribution of Y t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya-Watson estimator stands out as it shares the advantages of both Nadaraya-Watson and local linear regression estimators. The asymp-totic normality of the proposed estimators is established in the manner of sparse asymptotics, which shows that the proposed smoothed methods outperform their conventional (unsmoothed) parametric counterparts under very mild conditions. Simulation results lend further support to the above assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994 { 1995. An ad hoc model selection procedure based on local AIC is also proposed to select the signiicant pollutant indices.