Parameter Change Test for Poisson Autoregressive Models

Abstract

In this paper, we consider the problem of testing for a parameter change in Poisson autoregressive models. We suggest two types of cumulative sum (CUSUM) tests, namely, those based on estimates and residuals. We first demonstrate that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal and then construct the CMLE-based CUSUM test. It is shown that under regularity conditions, its limiting null distribution is a function of independent Brownian bridges. Next, we construct the residual-based CUSUM test and derive its limiting null distribution. Simulation results are provided for illustration. A real-data analysis is performed on data for polio incidence and campylobacteriosis infections.

Robust CUSUM test for time series of counts and its application to analyzing the polio incidence data

“…Since the data include some strongly deviating observations, we employ the robust cumulative sum (CUSUM) test proposed by Kang and Song (2015) to perform the test for parameter change. Contrary to the result of Kang and Lee (2014), our data analysis indicates that there is no significant change in the case of the CUSUM test with strong robustness and the same result is obtained after ridding the polio data of outliers. We additionally consider the comparison of the forecasting performance.…”

“…To perform the above test, Kang and Lee (2014) proposed the CUSUM test based on maximum likelihood estimator (MLE):…”

“…For historical background and general review, Csörgő and Horváth (1997) and Chan and Gupta (2000), and the articles cited therein. As Kang and Lee (2014) addressed, integer-valued time series (particularly in the field of epidemiology) can also undergo a significant change in parameters due to, for example, changes in the quality of health care and the state of patients' health. However, only recently have researchers begun to study on the detection of parameter changes for integer-valued time series, Kang and Lee (2014) and Doukhan and Kengne (2015).…”

“…This data set has been previously studied by many researchers, including Zeger (1988), Davis et al (2000) and Jung and Tremayne (2011). Recently, Kang and Lee (2014) performed the CUSUM test to detect change points in this data based on Poisson autoregressive models. They showed that there exist a significant change in parameter, which may be because the outliers affect the test procedure.…”

“…They showed that there exist a significant change in parameter, which may be because the outliers affect the test procedure. In this study, we employ the robust CUSUM test of Kang and Song (2015) to analyze the polio data, and compare with the CUSUM test of Kang and Lee (2014). This paper is organized as follows.…”