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Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis. (English) Zbl 1221.62126

The first-order autoregressive model with integer-valued data is defined by the ‘thinning’ operator designated by F. W. Steutel and K. van Harn [Ann. Probab. 7, 893–899 (1979; Zbl 0418.60020)]. Let \(X\) be an integer-valued random variable and let \(\psi\in [0,1]\). The ‘thinning’ operator “\(\circ\)” is defined as \(\psi\circ X=\sum_{i=1}^{X}B_i\), where \(\{B_i\}\) is an independent of \(X\) i.i.d. Bernoulli random sequence with \(P(B_i=1)=\psi\). With this operator, the INAR(l) model [M.A. Al-Osh and A.A. Alzaid, J. Time Ser. Anal. 8, 261–275 (1987; Zbl 0617.62096)] is defined as \(X_t=\psi\circ X_{t-1}+Z_t,\;t\geq1,\) where \(\{Z_t\}\) is a sequence of independent of \(X_0\) i.i.d. non-negative integer-valued random variables with mean \(\lambda\) and variance \(\sigma^2_z\). The RCINAR(l) model introduced by H. Zheng, I.V. Basawa and S. Datta [J. Stat. Plann. Inference 137, No. 1, 212–229 (2007; Zbl 1098.62117)] is defined by the recursive equation \(X_t=\psi_t\circ X_{t-1}+Z_t,\;\geq1,\) where \(\{\psi_t\}\) is an i.i.d. sequence with cumulative distribution function defined on \([0,1)\), and \(\{Z_t\}\) is a sequence of non-degenerate and non-negative integer-valued random variables with \(E(Z_t^4)<\infty\). Let \(\psi=E(\psi_t)\), \(\sigma_{\psi}^2=\text{Var}(\psi_t)\), \(\lambda=E(Z_t)\), \(\tau^2=\psi^2=\sigma^2_{\psi}\), \(\sigma^2_z=\text{Var}(Z_t)\). The process \(\{X_t\}\) is an ergodic Markov chain and has a stationary distribution.
The authors deal with the problem of testing for a parameter change in a first-order random coefficient integer-valued autoregressive RCINAR(l) model. They employ the cumulative sum (CUSUM) test for detecting changes of the mean autoregressive rate \(\psi=E(\psi_t)\) and the mean innovation \(\lambda=E(Z_t)\) which is based on the conditional least-squares and modified quasi-likelihood estimators. It is shown that, under regularity conditions, the CUSUM test has the same limiting distribution as the supremum of the squares of independent Brownian bridges. The CUSUM test is then applied to the analysis of a monthly polio counts data set and demonstrates that there is a parameter change in the RCINAR(l) model.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M07 Non-Markovian processes: hypothesis testing
62F03 Parametric hypothesis testing
62P10 Applications of statistics to biology and medical sciences; meta analysis

Software:

Fahrmeir
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References:

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