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Self-exciting threshold binomial autoregressive processes

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Abstract

We introduce a new class of integer-valued self-exciting threshold models, which is based on the binomial autoregressive model of order one as introduced by McKenzie (Water Resour Bull 21:645–650, 1985. doi:10.1111/j.1752-1688.1985.tb05379.x). Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation and forecasting are addressed. Finally, the performance of these models is illustrated through a simulation study and an empirical application to a set of measle cases in Germany.

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Notes

  1. The density-dependent models by Weiß and Pollett (2014) might also be understood as special SET models with \(N+1\) regimes.

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Acknowledgments

The authors thank the referees for carefully reading the article and for their comments, which greatly improved the article. This work was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.

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Correspondence to Christian H. Weiß.

Appendices

Appendix 1: Proofs

1.1 Unconditional mean and variance

The unconditional mean (6) is a direct consequence of (4):

$$\begin{aligned} \mu _X= & {} E\left[ E[X_{t} | X_{t-1}]\right] = r_{1}\,\mu _{IX} + (1-r_{1})\pi _{1}N\,p\ +\ r_{2}\,(\mu _{X}-\mu _{IX}) \\&+ (1-r_{2})\pi _{2}N\,(1-p)\\= & {} \ r_{2}\,\mu _{X}\ +\ (r_{1}-r_{2})\,\mu _{IX} + Np\,\pi _{1}(1-r_{1}) + N(1-p)\,\pi _{2}(1-r_{2}). \end{aligned}$$

For the unconditional variance (7), consider first

$$\begin{aligned}&E\left[ V[X_{t} | X_{t-1}] \right] \nonumber \\&\quad \overset{(5)}{=} E\left[ I_{t-1}\left( r_{1}(1-r_{1})(1-2\pi _{1})X_{t-1}+N(1-r_{1})\pi _{1}\left( 1-(1-r_{1})\pi _{1}\right) \right) \right] \nonumber \\&\qquad +\,E\left[ (1-I_{t-1})\left( r_{2}(1-r_{2})(1-2\pi _{2})X_{t-1}+N(1-r_{2})\pi _{2}\left( 1-(1-r_{2})\pi _{2}\right) \right) \right] \nonumber \\&\quad = r_{1}(1-r_{1})(1-2\pi _{1})\mu _{IX} + pN(1-r_{1})\pi _{1}\left( 1-(1-r_{1})\pi _{1}\right) \nonumber \\&\qquad +\, r_{2}(1-r_{2})(1-2\pi _{2})(\mu _{X}-\mu _{IX}) + (1-p)N(1-r_{2})\pi _{2}\left( 1-(1-r_{2})\pi _{2}\right) ,\nonumber \\ \end{aligned}$$
(30)

as well as (note that \(E[I_{t-1}(1-I_{t-1})\cdot Y]=0\))

$$\begin{aligned}&V\left[ E[X_{t} | X_{t-1} ] \right] \overset{(4)}{=}\ V\left[ I_{t-1} \left( r_{1}X_{t-1} +(1-r_{1})\pi _{1} N \right) \right. \nonumber \\&\quad \left. + (1-I_{t-1})\left( r_{2}X_{t-1} + (1-r_{2})\pi _{2}N\right) \right] \nonumber \\&\quad =\ V\left[ I_{t-1} \left( r_{1}X_{t-1} +(1-r_{1})\pi _{1} N \right) \right] + V\left[ (1-I_{t-1})\left( r_{2}X_{t-1} + (1-r_{2})\pi _{2}N\right) \right] \nonumber \\&\qquad +\,2\,\hbox {Cov}\left[ I_{t-1} \left( r_{1}X_{t-1} +(1-r_{1})\pi _{1} N \right) , (1-I_{t-1})\left( r_{2}X_{t-1} + (1-r_{2})\pi _{2}N\right) \right] \nonumber \\&\quad =\ V\left[ I_{t-1} r_{1}X_{t-1}\right] + V\left[ I_{t-1}(1-r_{1})\pi _{1} N\right] \nonumber \\&\qquad + 2\, \hbox {Cov}\left[ I_{t-1} r_{1}X_{t-1}, I_{t-1}(1-r_{1})\pi _{1} N\right] \nonumber \\&\qquad +\, V\left[ (1-I_{t-1})r_{2}X_{t-1}\right] + V\left[ (1-I_{t-1})(1-r_{2})\pi _{2}N\right] \nonumber \\&\qquad +\, 2\,\hbox {Cov}\left[ (1-I_{t-1})r_{2}X_{t-1}, (1-I_{t-1})(1-r_{2})\pi _{2}N\right] \nonumber \\&\qquad +\, 0 - 2\, E\left[ I_{t-1} \left( r_{1}X_{t-1} +(1-r_{1})\pi _{1} N \right) \right] {\cdot } E\left[ (1-I_{t-1})\left( r_{2}X_{t-1} + (1-r_{2})\pi _{2}N\right) \right] \nonumber \\&\quad =\ r_{1}^{2}\, V[I_{t-1}X_{t-1}] + (1-r_{1})^{2}\pi _{1}^{2}N^{2}\,p(1-p) \nonumber \\&\qquad + 2r_{1}(1-r_{1})\pi _{1}N\, \hbox {Cov}[I_{t-1}X_{t-1}, I_{t-1}]\nonumber \\&\qquad +\, r_{2}^{2}\,V\left[ (1-I_{t-1})X_{t-1}\right] +(1-r_{2})^{2}\pi _{2}^{2}N^{2}\,p(1-p)\nonumber \\&\qquad +\,2r_{2}(1-r_{2})\pi _{2}N\,\hbox {Cov}\left[ (1-I_{t-1})X_{t-1}, (1-I_{t-1})\right] \nonumber \\&\qquad -\, 2r_{1}r_{2}\mu _{IX} (\mu _{X}-\mu _{IX}) - 2r_{1}(1-r_{2})\pi _{2}(1-p)N\mu _{IX}\nonumber \\&\qquad -\,2 r_{2}(1-r_{1})\pi _{1}pN(\mu _{X}-\mu _{IX}) - 2p(1-p)(1-r_{1})\pi _{1}(1-r_{2})\pi _{2}N^{2}\nonumber \\&\quad =\ r_{1}^{2} (\mu _{IX,2}-\mu _{IX}^2) + (1-r_{1})^{2}\pi _{1}^{2}N^{2}p(1-p) + 2r_{1}(1-r_{1})\pi _{1}N(1-p)\mu _{IX}\nonumber \\&\qquad +\, r_{2}^{2}\,(\sigma _X^2+2\mu _X\mu _{IX}-\mu _{IX,2}-\mu _{IX}^2)+(1-r_{2})^{2}\pi _{2}^{2}N^{2}p(1-p)\nonumber \\&\qquad +\,2r_{2}(1-r_{2})\pi _{2}Np(\mu _{X}-\mu _{IX})\nonumber \\&\qquad -\, 2r_{1}r_{2}\mu _{IX} (\mu _{X}-\mu _{IX}) - 2r_{1}(1-r_{2})\pi _{2}(1-p)N\mu _{IX}\nonumber \\&\qquad -\,2 r_{2}(1-r_{1})\pi _{1}pN(\mu _{X}-\mu _{IX}) - 2p(1-p)(1-r_{1})\pi _{1}(1-r_{2})\pi _{2}N^{2}. \end{aligned}$$
(31)

Insertion of (30) and (31) into \(\sigma _X^2 = E\left[ V[X_{t} | X_{t-1}] \right] + V\left[ E[X_{t} | X_{t-1} ] \right] \) and reordering gives

$$\begin{aligned}&(1-r_{2}^{2})\sigma _{X}^{2}\\&\quad = r_{1}(1-r_{1})(1-2\pi _{1})\mu _{IX} + r_{2}(1-r_{2})(1-2\pi _{2})(\mu _{X}-\mu _{IX})\\&\quad \quad + Np(1-r_{1})\pi _{1}\left( 1-(1-r_{1})\pi _{1}\right) + N(1-p)(1-r_{2})\pi _{2}\left( 1-(1-r_{2})\pi _{2}\right) \\&\quad \quad +\ r_{1}^{2} (\mu _{IX,2}-\mu _{IX}^2) + 2r_{2}^{2}\,\mu _X\mu _{IX} - r_{2}^{2}\,(\mu _{IX,2}+\mu _{IX}^2) - 2r_{1}r_{2}\mu _{IX} (\mu _{X}-\mu _{IX})\\&\quad \quad + (1-r_{1})^{2}\pi _{1}^{2}N^{2}p(1-p) + (1-r_{2})^{2}\pi _{2}^{2}N^{2}p(1-p)\\&\quad \quad - 2(1-r_{1})\pi _{1}(1-r_{2})\pi _{2}N^{2}p(1-p)\\&\quad \quad + 2r_{1}(1-r_{1})\pi _{1}N(1-p)\mu _{IX} - 2r_{1}(1-r_{2})\pi _{2}N(1-p)\mu _{IX}\\&\quad \quad +2r_{2}(1-r_{2})\pi _{2}Np(\mu _{X}-\mu _{IX}) -2 r_{2}(1-r_{1})\pi _{1}Np(\mu _{X}-\mu _{IX})\\&\quad = \left( r_{1}(1-r_{1})(1-2\pi _{1}) - r_{2}(1-r_{2})(1-2\pi _{2})\right) \mu _{IX} + r_{2}(1-r_{2})(1-2\pi _{2})\,\mu _{X}\\&\quad \quad + Np(1-r_{1})\pi _{1}\left( 1-(1-r_{1})\pi _{1}\right) + N(1-p)(1-r_{2})\pi _{2}\left( 1-(1-r_{2})\pi _{2}\right) \\&\quad \quad + (r_{1}^{2}-r_{2}^{2})\,\mu _{IX,2} -(r_{1} - r_{2})^{2}\,\mu _{IX}^2 - 2r_{2}(r_{1}-r_{2})\,\mu _X\mu _{IX}\\&\quad \quad + N^{2}p(1-p)\,\left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) ^{2}\\&\quad \quad + 2N(1-p)r_{1}\left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) \,\mu _{IX}\\&\quad \quad - 2 Npr_{2} \left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) \,(\mu _{X}-\mu _{IX})\\&\quad = r_{2}(1-r_{2})(1-2\pi _{2})\,\mu _{X} - 2 Npr_{2} \left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) \,\mu _{X}\\&\quad \quad - 2r_{2}(r_{1}-r_{2})\,\mu _X\mu _{IX}\ +\ (r_{1}^{2}-r_{2}^{2})\,\mu _{IX,2} -(r_{1} - r_{2})^{2}\,\mu _{IX}^2\\&\quad \quad + 2N\left( r_{1}-p(r_{1} - r_{2})\right) \left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) \,\mu _{IX}\\&\quad \quad + \left( r_{1}(1-r_{1})(1-2\pi _{1}) - r_{2}(1-r_{2})(1-2\pi _{2})\right) \mu _{IX}\\&\quad \quad + Np(1-r_{1})\pi _{1}\left( 1-(1-r_{1})\pi _{1}\right) + N(1-p)(1-r_{2})\pi _{2}\left( 1-(1-r_{2})\pi _{2}\right) \\&\quad \quad + N^{2}p(1-p)\,\left( (1-r_{1})\pi _{1} - (1-r_{2})\pi _{2}\right) ^{2}. \end{aligned}$$

This completes the proof of the variance formula (7).

To get the properties of the LSET-BAR(1) model, we have to insert \(r := r_{1} =r_{2}\) into the Eqs. (6) and (7). We start with the mean (10):

$$\begin{aligned} (1-r)\,\mu _X = 0 + Np\,\pi _{1}(1-r) + N(1-p)\,\pi _{2}(1-r). \end{aligned}$$

The derivation of the variance (11) is more tedious:

$$\begin{aligned}&(1-r^{2})\sigma _{X}^{2} \\&\quad = r(1-r)(1-2\pi _{2})\,\mu _{X} \ +\ 2 Np\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{X}\\&\qquad -\ 2N\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{IX} \ +\ r(1-r)\,\left( (1-2\pi _{1}) - (1-2\pi _{2})\right) \,\mu _{IX}\\&\qquad +\ Np(1-r)\pi _{1}\left( 1-(1-r)\pi _{1}\right) \ +\ N(1-p)(1-r)\pi _{2}\left( 1-(1-r)\pi _{2}\right) \\&\qquad +\ N^{2}p(1-p)\,(1-r)^2\,(\pi _{2} - \pi _{1})^{2}\\&\quad = r(1-r)(1-2\pi _{2})\,N\left( p\,\pi _{1}\ +\ (1-p)\,\pi _{2}\right) \\&\qquad \ +\ 2 Np\,r(1-r)\,(\pi _{2} - \pi _{1})\,N\left( \pi _{2}\ -\ p(\pi _{2}-\pi _{1})\right) \\&\qquad -\ 2(N-1)\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{IX} \ +\ Np\,(1-r^2)\,\pi _{1}(1-\pi _{1})\ \\&\qquad -\ Np\,r(1-r)\,\pi _{1}(1-2\pi _{1})\\&\qquad +\ N(1-p)\,(1-r^2)\,\pi _{2}(1-\pi _{2})\ -\ N(1-p)\,r(1-r)\,\pi _{2}(1-2\pi _{2}) \\&\qquad +\ N^{2}p(1-p)\,(1-r^2)\,(\pi _{2} - \pi _{1})^{2}\ -\ 2\,N^{2}p(1-p)\,r(1-r)\,(\pi _{2} - \pi _{1})^{2}\\&\quad = Np\,r(1-r)\,\pi _{1}\left( (1-2\pi _{2})\ -\ (1-2\pi _{1})\right) \\&\qquad +\ 2\, N^2p\,r(1-r)\,\pi _{2}(\pi _{2} - \pi _{1}) \ -\ 2\, N^2p^2\,r(1-r)\,(\pi _{2} - \pi _{1})^2 \\&\qquad -\ 2\,N^{2}p(1-p)\,r(1-r)\,(\pi _{2} - \pi _{1})^{2}\ -\ 2(N-1)\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{IX} \\&\qquad +\ Np\,(1-r^2)\,\pi _{1}(1-\pi _{1})\ +\ N(1-p)\,(1-r^2)\,\pi _{2}(1-\pi _{2})\\&\qquad \ +\ N^{2}p(1-p)\,(1-r^2)\,(\pi _{2} - \pi _{1})^{2}\\&\quad = -\ 2\,Np\,r(1-r)\,\pi _{1}(\pi _{2} - \pi _{1}) \ +\ 2\, N^2p\,r(1-r)\,\pi _{2}(\pi _{2} - \pi _{1})\\&\qquad -\ 2\, N^2p\,r(1-r)\,(\pi _{2} - \pi _{1})^2\ -\ 2(N-1)\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{IX} \\&\qquad +\ Np\,(1-r^2)\,\pi _{1}(1-\pi _{1})\ +\ N(1-p)\,(1-r^2)\,\pi _{2}(1-\pi _{2})\\&\qquad \ +\ N^{2}p(1-p)\,(1-r^2)\,(\pi _{2} - \pi _{1})^{2}\\&\quad = 2\,(N-1)\,r(1-r)(\pi _{2} - \pi _{1})\, Np\,\pi _{1} \ -\ 2(N-1)\,r(1-r)\,(\pi _{2} - \pi _{1})\,\mu _{IX} \\&\qquad +\ Np\,(1-r^2)\,\pi _{1}(1-\pi _{1})\ +\ N(1-p)\,(1-r^2)\,\pi _{2}(1-\pi _{2})\\&\qquad +\ N^{2}p(1-p)\,(1-r^2)\,(\pi _{2} - \pi _{1})^{2}. \end{aligned}$$

This completes the proof.

1.2 Binomial index of dispersion

First, we consider the denominator of the \(\hbox {BID}\) (15) for the case of the LSET model (\(r_{1}=r_{2}=r\)). Using (10), we obtain

$$\begin{aligned} \frac{\mu _X}{N}\left( 1-\frac{\mu _X}{N}\right)&= \left( p\,\pi _{1} + (1-p)\,\pi _{2}\right) \left( 1-\left( p\,\pi _{1} + (1-p)\,\pi _{2}\right) \right) \nonumber \\&= p\pi _{1}+(1-p)\pi _{2}\ -p^{2}\pi _{1}^{2}\ -2p(1-p)\pi _{1}\pi _{2}\ -(1-p)^{2}\pi _{2}^{2}\nonumber \\&= p\pi _{1}+(1-p)\pi _{2}\ -p\pi _{1}^{2}\nonumber \\&\qquad +p(1-p)\pi _{1}^{2} -(1-p)\pi _{2}^{2}+p(1-p)\pi _{2}^{2}\ -2p(1-p)\pi _{1}\pi _{2}\nonumber \\&= p\,\pi _{1}(1-\pi _{1})\ +\ (1-p)\,\pi _{2}(1-\pi _{2})\ +\ p(1-p)\,(\pi _{2}-\pi _{1})^{2}. \end{aligned}$$
(32)

Then we look for matching terms in the numerator. First note that

$$\begin{aligned} N^{2}p(1-p)\,(\pi _{2}-\pi _{1})^{2} = Np(1-p)\,(\pi _{2}-\pi _{1})^{2} + N(N-1)\,p(1-p)\,(\pi _{2}-\pi _{1})^{2}. \end{aligned}$$

Using this together with (11) and (32), we find

$$\begin{aligned} \sigma _X^{2}= & {} \ \mu _X\,(1-\mu _X/N)\ +\ N(N-1)\,p(1-p)\,(\pi _{2}-\pi _{1})^{2}\\&+\ \frac{2r}{1+r}\,(N-1)\,(\pi _{2}-\pi _{1})\,\left( Np\,\pi _{1} - \mu _{IX}\right) . \end{aligned}$$

Bringing the results together leads to (16) for the \(\hbox {BID}\). Note that only the last term, \(\frac{2r}{1+r}\ldots \), might become negative.

If \(r=0\), (16) reduces to

$$\begin{aligned} \hbox {BID} = 1+ \frac{p(1-p)N(N-1)(\pi _{2}-\pi _{1})^{2}}{p\pi _{1}(1-\pi _{1})+(1-p)\pi _{2}(1-\pi _{2})+p(1-p)(\pi _{2}-\pi _{1})^{2}}\ \ge 1. \end{aligned}$$

1.3 Autocovariance function

By the law of total covariance, we obtain

$$\begin{aligned} \gamma (k)&\ :=\ \hbox {Cov}[X_{t} , X_{t-k}] = \hbox {Cov}\left[ E[ X_{t} | X_{t-1}, \ldots ], E[ X_{t-k} | X_{t-1}, \ldots ]\right] \ +\ 0\\&\ \overset{(8)}{=}\ \hbox {Cov}\left[ rX_{t-1} + N(1-r)\left( \pi _{2}\ +\ I_{t-1}\,(\pi _{1}-\pi _{2})\right) , X_{t-k}\right] \\&= r \cdot \hbox {Cov}[X_{t-1} , X_{t-k}]\ +\ N(1-r)\cdot (\pi _{1}-\pi _{2})\cdot \hbox {Cov}[I_{t-1}, X_{t-k}]\\&= \cdots = r^{k}\, V[X_{t-k}]\ +\ N(1-r) \cdot (\pi _{1}-\pi _{2}) \cdot \sum \limits _{s=1}^{k} r^{s-1}\,\hbox {Cov}[I_{t-s},X_{t-k}], \end{aligned}$$

which proves (18).

Appendix 2: Tables

See Tables 5, 6, 7 and 8.

Table 5 Conditional least squares and conditional maximum likelihood estimates for \((r,\pi _{1},\pi _{2})\) and R in the LSET-BAR(1) model. Model M1: \((N,R; r,\pi _{1},\pi _{2}) = (40,10; 0.3, 0.15, 0.4)\)
Table 6 Conditional least squares and conditional maximum likelihood estimates for \((r,\pi _{1},\pi _{2})\) and R in the LSET-BAR(1) model. Model M2: \((N,R; r,\pi _{1},\pi _{2}) = (20,4; 0.3, 0.15, 0.4)\)
Table 7 Conditional least squares and conditional maximum likelihood estimates for \((r,\pi _{1},\pi _{2})\) and R in the LSET-BAR(1) model. Model M3: \((N,R; r,\pi _{1},\pi _{2}) = (40,10; 0.7, 0.15, 0.4)\)
Table 8 Conditional least squares and conditional maximum likelihood estimates for \((r,\pi _{1},\pi _{2})\) and R in the LSET-BAR(1) model. Model M4: \((N,R; r,\pi _{1},\pi _{2}) = (20,5; 0.7, 0.15, 0.4)\)

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Möller, T.A., Silva, M.E., Weiß, C.H. et al. Self-exciting threshold binomial autoregressive processes. AStA Adv Stat Anal 100, 369–400 (2016). https://doi.org/10.1007/s10182-015-0264-6

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