6. Conclusion

change in KS counts due to the coding practice, and the φ<sup>i</sup> denote the coefficients for the

Model AIC ω β<sup>0</sup> β<sup>1</sup> φ<sup>1</sup> φ<sup>2</sup> σ PDZIB(1) 922.98 0.248 �3.349 �0.249 �0.223 0.430

PDZIP(1) 923.31 0.248 �3.389 �0.242 �0.241 0.410

(0.061) (0.024) (0.086) (0.002)

(0.061) (0.046) (0.084) (0.004) PDZIB(2) 922.98 0.248 �3.359 �0.237 �0.120 0.264 0.426

PDZIP(2) 924.09 0.248 �3.395 �0.230 �0.119 0.263 0.402

ODZIB(2) 1038.11 0.341 �3.250 �0.275 �0.008 0.007

ODZIP(2) 1028.49 0.341 �3.288 �0.266 �0.007 0.007

ODZIB(1) 1039.80 0.341 �3.184 �0.319 �0.007

ODZIP(1) 1030.04 0.341 �3.224 �0.309 �0.007

(0.034) (0.051) (0.120) (0.160) (0.044)

(0.034) (0.051) (0.116) (0.166) (0.043)

(0.034) (0.054) (0.126) (0.166) (0.153) (0.046)

(0.034) (0.052) (0.118) (0.178) (0.158) (0.045)

p

(0.061) (0.033) (0.088) (0.002) (0.002)

(0.061) (0.058) (0.087) (0.004) (0.004)

, (50)

: (53)

i¼1 φi yt�i

� � <sup>¼</sup> log ð Þþ nt <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>β</sup>1xt <sup>þ</sup> zt, (51)

p

i¼1 φi yt�i

zt�<sup>i</sup> þ εt: (52)

logitð Þ¼ <sup>π</sup><sup>t</sup> <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>β</sup>1xt <sup>þ</sup><sup>X</sup>

where β<sup>1</sup> and φ<sup>i</sup> reflect parameters analogous to those defined for the parameter-driven setting. In addition, we consider four comparable Poisson-type models based on the work by Yang

For the two ODZIB(p) models, we employ the following linear predictor:

Table 11. Model fitting results for eight different zero-inflated models.

et al. [5, 7]. For the two PDZIP(p) models, we employ the linear predictor

zt <sup>¼</sup> <sup>X</sup> p

i¼1 φi

� � <sup>¼</sup> log ð Þþ nt <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>β</sup>1xt <sup>þ</sup><sup>X</sup>

log μ<sup>t</sup>

For the two ODZIP(p) models, we employ the linear predictor

log μ<sup>t</sup>

autoregressive process.

148 Time Series Analysis and Applications

Count time series featuring a preponderance of zeros are commonly encountered in a variety of scientific applications. In characterizing such series, modeling frameworks that assume a Poisson mixture distribution have been extensively studied. However, minimal work has been focused on modeling frameworks that assume a binomial mixture distribution. When data are more naturally assumed to arise from the latter, a Poisson-type model with an offset is often employed; however, the propriety of such an approximation is unclear.

We propose two general classes of models to effectively characterize a count time series that arises from a zero-inflated binomial mixture distribution. The observation-driven ZIB model, formulated in the partial likelihood framework, is fit using the Newton–Raphson algorithm. The parameter-driven ZIB model, formulated in the state-space framework, is fit using the MCEM algorithm. When data are generated from a binomial mixture, our proposed ZIB models outperform their Poisson-type counterparts. We illustrate our methodology with an application that assesses a particular level change for a diagnosis code.

Future work involves extending the current frameworks to the zero-inflated beta-binomial (ZIBB) model. Both observation-driven and parameter-driven ZIBB models can be formulated and fit based on methodological developments similar to those presented in this work. However, weak identifiability could arise as a potentially problematic issue in fitting the parameter-driven ZIBB model, as not only the overdispersion explicitly induced by the beta distribution but also the correlated random effects account for any excess variability in the data [5]. In addition, we could consider more complicated correlation structures by incorporating moving average components in the linear predictors for parameter-driven models. Such an extension necessitates non-trivial revisions to the state-space model formulation and the complete-data likelihood, which warrant further investigation.
