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Testing for an excessive number of zeros in time series of bounded counts

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Abstract

For the modeling of bounded counts, the binomial distribution is a common choice. In applications, however, one often observes an excessive number of zeros and extra-binomial variation, which cannot be explained by a binomial distribution. We propose statistics to evaluate the number of zeros and the dispersion with respect to a binomial model, which is based on the sample binomial index of dispersion and the sample binomial zero index. We apply this index to autocorrelated counts generated by a binomial autoregressive process of order one, which also includes the special case of independent and identically (i. i. d.) bounded counts. The limiting null distributions of the proposed test statistics are derived. A Monte-Carlo study evaluates their size and power under various alternatives. Finally, we present two real-data applications as well as the derivation of effective sample sizes to illustrate the proposed methodology.

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Notes

  1. www.forecastingprinciples.com/index.php/crimedata, file PghCarBeat.csv.

  2. In the Bavarian school system, Hauptschule constitutes the lowest level of secondary school, Gymnasium the highest level, and Realschule is ranked between them.

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Acknowledgements

The authors thank the editor and the referees for carefully reading the article and for their comments, which greatly improved the article. Main parts of this research were completed while the first author stayed as a guest professor at the Helmut Schmidt University in Hamburg. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07045707).

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Correspondence to Christian H. Weiß.

Appendices

A Proofs

In the sequel, the following properties of a BAR(1) process are used: \((X_t)_{\mathbb {N}}\) is a stationary, ergodic and \(\phi \)-mixing finite Markov chain with marginal distribution \(\text {Bin}(n,\pi )\) (McKenzie 1985; Weiß and Kim 2013). Denoting \(\beta _h:=\pi \, (1-\rho ^h)\) and \(\alpha _h:=\beta _h +\rho ^h\), the (truly positive) h-step-ahead transition probabilities \(p_{k|l}^{(h)}=P(X_t=k\ |\ X_{t-h}=l)\) are given by

$$\begin{aligned} p_{k|l}^{(h)} = \sum _{m=\max {\{0,k+l-n\}}}^{\min {\{k,l\}}} \left( {\begin{array}{c}l\\ m\end{array}}\right) \left( {\begin{array}{c}n-l\\ k-m\end{array}}\right) \alpha _h^m (1-\alpha _h)^{l-m} \beta _h^{k-m} (1-\beta _h)^{n-l+m-k}, \end{aligned}$$
(A.1)

see Weiß and Pollett (2012), and conditional mean and variance are both linear in \(X_{t-h}\):

$$\begin{aligned} \begin{array}{@{}l} E[X_t\ |\ X_{t-h}]\ = \rho ^h\, X_{t-h} + n\beta _h,\\ V[X_t\ |\ X_{t-h}]\ = \rho ^h(1-\rho ^h)(1-2\pi )\, X_{t-h}\ +\ n\beta _h(1-\beta _h). \end{array} \end{aligned}$$
(A.2)

The ACF equals \(\rho _X(k)=\rho ^k\).

1.1 Proof of Theorem 1

We have \(E[{\varvec{Y}}_t]={\varvec{0}}\), and with analogous arguments as in “Appendix A.6” in Weiß and Kim (2013), we conclude that \(T^{-1/2}\cdot \sum _{t=1}^T {\varvec{Y}}_t\) is asymptotically normally distributed with covariance matrix \({\varvec{\varSigma }}= (\sigma _{ij})\) given by

$$\begin{aligned} \sigma _{ij}\ =\ E[Y_{0,i}\cdot Y_{0,j}]\ +\ \sum _{k=1}^{\infty }\ \big (E[Y_{0,i}\cdot Y_{k,j}]+E[Y_{k,i} \cdot Y_{0,j}]\big ). \end{aligned}$$

Note that a BAR(1) process is time-reversible (McKenzie 1985), \(E[Y_{0,i}\cdot Y_{k,j}] = E[Y_{k,i} \cdot Y_{0,j}]\) always holds. So we compute \(\sigma _{11}\) as

$$\begin{aligned} \sigma _{11}&= E[Y_{0,1} \cdot Y_{0,1}] + 2 \sum _{k=1}^\infty E[Y_{0,1} \cdot Y_{k,1}]. \end{aligned}$$

To get \( E[Y_{0,1} \cdot Y_{0,1}]\) and \(E[Y_{0,1} \cdot Y_{k,1}]\), we evaluate

$$\begin{aligned} E[Y_{t,1}^2]&= E\big [(\mathbb {1}_{\{X_t=0\}}-p_0)^2\big ]\\&= E\big [\mathbb {1}_{\{X_t=0\}} -2p_0 \mathbb {1}_{\{X_t=0\}} +p_0^2\big ]=p_0(1-p_0),\\ E[Y_{0,1} \cdot Y_{k,1}]&= E[ \mathbb {1}_{\{X_t=0\}}\cdot \mathbb {1}_{\{X_{t-k}=0\}}] - p_0^2 = P(X_t = X_{t-k} = 0) - p_0^2 \\&= p_{0|0}^{(k)} \cdot p_0 - p_0^2 = p_0 ( p_{0|0}^{(k)} -p_{0} ). \end{aligned}$$

Hence,

$$\begin{aligned} \sigma _{11}&= p_0(1-p_0) + 2 \sum _{k=1}^{\infty } p_0 (p_{0|0}^{(k)}-p_{0}), \quad \text {where}~p_{0|0}^{(k)}=\big (1-\pi (1-\rho ^k )\big )^n \text { according to}\, (\text {A.1}),\\&= p_0(1-p_0) + 2 \sum _{k=1}^{\infty } p_0 \big [(1-\pi +\pi \rho ^k )^n -p_0 \big ]\\&= p_0(1-p_0) + 2 \sum _{k=1}^{\infty } p_0 \bigg [ \sum _{j=0}^{n} \left( {\begin{array}{c}n\\ j\end{array}}\right) (1-\pi )^{n-j} (\pi \rho ^{k})^{j} -p_0 \bigg ],\quad \text {where}\quad p_0 = (1-\pi )^n,\\&= p_0(1-p_0) + 2 p_0^2 \sum _{j=1}^{n} \left( {\begin{array}{c}n\\ j\end{array}}\right) \bigg ( \frac{\pi \rho }{1-\pi }\bigg )^{j} \frac{1}{1-\rho ^{j}}. \end{aligned}$$

For \(\sigma _{12}\), it follows that

$$\begin{aligned} \sigma _{12}= E[Z_{0,1} \cdot Z_{0,2}] + 2 \sum _{k=1}^\infty E[Z_{0,1} \cdot Z_{k,2}]. \end{aligned}$$

To compute \(E[Z_{0,1} \cdot Z_{0,2}]\) and \(E[Z_{0,1} \cdot Z_{k,2}]\), note that

$$\begin{aligned} E[Z_{t,1} \cdot Z_{t,2}]&=E[(\mathbb {1}_{\{X_{t}=0\}}- p_0)(X_t - n\pi )]\\&=E[\mathbb {1}_{\{X_{t}=0\}} X_t - n \pi \mathbb {1}_{\{X_{t}=0\}} -p_0 X_t + p_0 n \pi ]\ =\ -n\pi p_0,\\ E[Z_{t,2} \cdot Z_{t-k,1}]&= E[X_t \cdot \mathbb {1}_{\{X_{t-k}=0\}}] - p_0 n\pi \\&= p_0 \cdot E[X_t|X_{t-k}=0] - p_0 n \pi ,\quad \text {where}\quad \\&\qquad E[X_t|X_{t-k}=l]\overset{(\text {A.2})}{=}\rho ^k \cdot l + n \pi (1-\rho ^k), \\&= p_0 \big (\rho ^k \cdot 0 + n \pi (1-\rho ^k)\big ) - p_0 n\pi = -\,n\pi p_0 \rho ^k. \end{aligned}$$

Hence,

$$\begin{aligned} \sigma _{12}&=-\,n\pi p_0 - 2 \sum _{k=1}^\infty n\pi p_0 \rho ^k \ =\ -\,n \pi p_0 \frac{1+\rho }{1-\rho }. \end{aligned}$$

For

$$\begin{aligned} \sigma _{13}= E[Z_{0,1} \cdot Z_{0,3}] + 2 \, \sum _{k=1}^\infty E[Z_{0,1} \cdot Z_{k,3}] \end{aligned}$$

we need to calculate

$$\begin{aligned} E[Z_{t,1} \cdot Z_{t,3}]&= E\big [( \mathbb {1}_{\{X_t=0\}}- p_0)\big (X_{t}^2 - n \pi (n\pi + 1-\pi )\big )\big ]\\&= E[\mathbb {1}_{\{X_t=0\}}\, X_t^2] -n\pi (n\pi +1-\pi )\, E[\mathbb {1}_{\{X_t=0\}}] \\&\qquad - p_0\, E[X_t^2] + p_0 n\pi (n\pi +1-\pi )\\&= -\,n\pi (n \pi +1-\pi )p_0 . \end{aligned}$$

Next

$$\begin{aligned} E[Z_{t,3} \cdot Z_{t-k,1}]&= E\big [\big (X_t^2 - n \pi (n \pi +1-\pi )\big )(\mathbb {1}_{\{X_{t-k}=0\}} - p_0)\big ]\\&= E[X_t^2\, \mathbb {1}_{\{X_{t-k}=0\}}]- p_0 n \pi (n \pi +1-\pi )\\&= p_0 E[X_t^2\ |\ X_{t-k}=0]- p_0 n \pi (n \pi +1-\pi ). \end{aligned}$$

Since

$$\begin{aligned} E[X_t^2\ |\ X_{t-k}=x]&= V[X_t\ |\ X_{t-k}=x]+ E^2[X_t\ |\ X_{t-k}=x] \\&=\rho ^k (1-\rho ^k)(1-2\pi ) x + n \beta _k (1-\beta _k)+ (\rho ^k x + n \beta _k)^2 , \end{aligned}$$

we obtain

$$\begin{aligned} E[X_t^2\ |\ X_{t-k}=0]&= n \beta _k ( 1-\beta _k + n \beta _k )\ =\ n \pi (1-\rho ^k) \big ( 1 + (n-1)\, \pi (1-\rho ^k)\big )\\&=n \pi \big (n \pi +1-\pi - (2n \pi +1-2\pi )\rho ^k + (n-1)\, \pi \rho ^{2k}\big ). \end{aligned}$$

Hence,

$$\begin{aligned} E[Z_{t,3} \cdot Z_{t-k,1}]&= p_0 n \pi \big ((2\pi -2n \pi -1)\rho ^k + (n-1)\, \pi \rho ^{2k}\big ). \end{aligned}$$

By inserting the corresponding terms, we obtain

$$\begin{aligned} \sigma _{13}&= -\,n\pi (n \pi + 1-\pi )p_0 + 2\sum _{k=1}^\infty p_0 n \pi \bigg ((2\pi -2n \pi -1)\rho ^k + (n-1)\, \pi \rho ^{2k}\bigg )\\&= -\,n\pi p_0(1-2\pi +2 n \pi ) \frac{1+\rho }{1-\rho } + n(n-1)\pi ^2 p_0 \frac{1+\rho ^2}{1-\rho ^2}. \end{aligned}$$

The remaining entries of \({\varvec{\varSigma }}\) are available from Section 3 in Weiß and Kim (2014).

1.2 Proof of Theorem 2

Define the function \({\varvec{g}}:\mathbb {R}^3\rightarrow \mathbb {R}^2\) with components

$$\begin{aligned} g_1(x_1,x_2,x_3) := f_{\text {z}}(x_1,x_2),\qquad g_2(x_1,x_2,x_3) := f_{\text {d}}(x_2,x_3), \end{aligned}$$

where

$$\begin{aligned} f_{\text {z}}(x_1,x_2): = x_1 \Big (1-\frac{x_2}{n}\Big )^{-n},\quad f_{\text {d}}(x_2,x_3):=\frac{n(x_3 -x_2^2)}{x_2(n-x_2)}. \end{aligned}$$

The gradient of \(f_{\text {z}}\), \(f_{\text {d}}\) equals

$$\begin{aligned}&\frac{\partial }{\partial x_1}f_{\text {z}}(x_1, x_2) =\Big (1-\frac{x_2}{n}\Big )^{-n},&\frac{\partial }{\partial x_2}f_{\text {z}}(x_1, x_2) = x_1\Big (1-\frac{x_2}{n}\Big )^{-n-1},\\&\frac{\partial }{\partial x_2}f_{\text {d}}(x_2, x_3) =-n\,\frac{n x_2^2 + x_3(n-2x_2)}{x_2^2(n-x_2)^2},&\frac{\partial }{\partial x_3}f_{\text {d}}(x_2, x_3) =\frac{n}{x_2(n-x_2)}. \end{aligned}$$

Evaluated for \((x_1, x_2, x_3)=\big ((1-\pi )^n, n\pi , n\pi (n \pi +1-\pi )\big )\), we obtain

$$\begin{aligned}&\frac{\partial }{\partial x_1}f_{\text {z}}(x_1, x_2) =(1-\pi )^{-n}:=d_{11},&\frac{\partial }{\partial x_2}f_{\text {z}}(x_1, x_2) =(1-\pi )^{-1}:=d_{12},\\&\frac{\partial }{\partial x_2}f_{\text {d}}(x_2, x_3) =\frac{-(1-2\pi + 2n \pi )}{n\pi (1-\pi )}:=d_{22},&\frac{\partial }{\partial x_3}f_{\text {d}}(x_2, x_3) =\frac{1}{n \pi (1-\pi )}:=d_{23}. \end{aligned}$$

Theorem 2 follows from Theorem 1 and the Delta method. The matrix \(\tilde{{\varvec{\varSigma }}} := {\mathbf{D }}{\varvec{\varSigma }}{\mathbf{D }}^{\top }\) has entries

$$\begin{aligned} \tilde{{\varvec{\varSigma }}} = \begin{pmatrix} d_{11} &{}\quad d_{12} &{}\quad 0 \\ 0 &{}\quad d_{22} &{}\quad d_{23}\\ \end{pmatrix} \begin{pmatrix} \sigma _{11} &{}\quad \sigma _{12} &{}\quad \sigma _{13} \\ \sigma _{12} &{}\quad \sigma _{22} &{}\quad \sigma _{23} \\ \sigma _{13} &{}\quad \sigma _{23} &{}\quad \sigma _{33} \\ \end{pmatrix} \begin{pmatrix} d_{11} &{}\quad 0 \\ d_{12} &{}\quad d_{22} \\ 0 &{}\quad d_{23}\\ \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \tilde{\sigma }_{11}&= d_{11}^2 \sigma _{11} + 2 d_{11} d_{12} \sigma _{12} + d_{12}^2 \sigma _{22}, \\ \tilde{\sigma }_{12}&= d_{11} d_{22} \sigma _{12} + d_{11} d_{23} \sigma _{13} + d_{12} d_{22} \sigma _{22} + d_{12} d_{23} \sigma _{23}, \\ \tilde{\sigma }_{22}&= d_{22}^2 \sigma _{22} + 2 d_{22} d_{23} \sigma _{23} + d_{23}^2 \sigma _{33}. \end{aligned}$$

1.3 Proof of Theorem 3

The approximate bias of is obtained in analogy to the approach of Weiß et al. (2016), by using the second-order Taylor expansion of \(f_{\text {z}}\) in “Appendix A.2”. The derivatives of \(f_{\text {z}}\) are

$$\begin{aligned} \frac{\partial ^2}{\partial x_1^2} f_{\text {z}}(x_1, x_2)&=0,\\ \frac{\partial ^2}{\partial x_1 \partial x_2} f_{\text {z}}(x_1, x_2)&=\Big (1- \frac{x_2}{n}\Big )^{-n-1},\\ \frac{\partial ^2}{\partial x_2^2 } f_{\text {z}}(x_1, x_2)&=\frac{n+1}{n} x_1 \Big (1-\frac{x_2}{n}\Big )^{-n-2}. \end{aligned}$$

So the Hessian of \(f_{\text {z}}\) evaluated at \(\big ((1-\pi )^n , n \pi \big )\) is given by

$$\begin{aligned} {\mathbf{H }}_{f_{\text {z}}}\big ((1-\pi )^n , n \pi \big ) = \begin{pmatrix} 0 &{}\quad (1-\pi )^{-n-1} \\ (1-\pi )^{-n-1} &{}\quad \frac{n+1}{n}(1-\pi )^{-2} \\ \end{pmatrix} :=\begin{pmatrix} h_{11} &{}\quad h_{12} \\ h_{21} &{}\quad h_{22} \\ \end{pmatrix}. \end{aligned}$$

Therefore, we obtain , with \({\varvec{Z}}_T=\frac{1}{\sqrt{T}}\sum _{t=1}^T{\varvec{Y}}_t\) satisfying \(E[{\varvec{Z}}_T]={\varvec{0}}\), and

$$\begin{aligned} E\left[ \frac{1}{2}{\varvec{Z}}_T^{'}{\mathbf{H }}_{f_{\text {z}}}{\varvec{Z}}_T\right]&= \frac{1}{2} E\left[ h_{11}Z_1^2 + 2 h_{12}Z_1 Z_2 + h_{22}Z_2^2\right] \\&= \frac{1}{2T} \left( h_{11}\sigma _{11} + 2h_{12}\sigma _{12} + h_{22}\sigma _{22}\right) {=} -\frac{1}{2T}\frac{1+\rho }{1-\rho }\frac{\pi }{1-\pi } (n-1). \end{aligned}$$

In order to get the approximate bias of , by analogous computation, we obtain the Hessian of \(f_{\text {d}}\) evaluated at \(\big (n \pi , n\pi (n\pi +1-\pi )\big )\):

$$\begin{aligned} {\mathbf{H }}_{f_{\text {d}}}\big (n \pi , n\pi (n\pi +1-\pi )\big ) :=\begin{pmatrix} g_{11} &{}\quad g_{12} \\ g_{21} &{}\quad g_{22} \\ \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} g_{11}= & {} \frac{2(1-n)}{n^2 \pi (1-\pi )} +\frac{4(1-2\pi )}{n\pi (1-\pi )^2} + \frac{2(1-2\pi )^2}{n^2 \pi ^2(1-\pi )^2}, \\ \quad g_{12}= & {} \frac{2\pi -1}{n^2\pi ^2(1-\pi )^2},\quad g_{21} =g_{12},\quad g_{22} = 0. \end{aligned}$$

Therefore,

B Summary of models used for power study in Sect. 3.3

The BB-AR(1) model used for DGP1 has been proposed by Weiß and Kim (2014) and extends the BAR(1) model to account for extra-binomial variation in the time series. This model is based on beta-binomial thinning: let \(\alpha _{\phi }\) be a random variable being independent of X, which follows the beta distribution BETA\(\big (\frac{1-\phi }{\phi }\cdot \alpha ,\ \frac{1-\phi }{\phi }\cdot (1-\alpha )\big )\), where \(\alpha ,\phi \in (0;1)\), then the random variable \(\alpha _{\phi }\circ X\) is obtained from X by beta-binomial thinning if the operator “\(\circ \)” is the binomial thinning operator, performed independently of X and \(\alpha _{\phi }\).

$$\begin{aligned} X_t\ =\ \alpha _{\phi }\circ X_{t-1}\ +\ \beta _{\phi }\circ (n-X_{t-1}), \end{aligned}$$
(B.1)

where all \(\alpha _{\phi },\beta _{\phi }\) and all thinnings are performed independently of each other, and where \(\alpha _{\phi },\beta _{\phi }\) and the thinnings at time t are independent of \((X_s)_{s<t}\). In analogy to the interpretation of (1.1), “\(\alpha _{\phi }\circ X_{t-1}\)” expresses a survival mechanism and “\(\beta _{\phi }\circ (n-X_{t-1})\)” a revival mechanism.

The models used for DGP2–DGP5 have been proposed by Möller et al. (2018). These four extensions of the BAR(1) model can accommodate a broad variety of zero inflation patterns. The RZ-BAR(1) process (DGP2),

$$\begin{aligned} Z_t\ =\ \kappa _t\cdot X_t\quad \text {with } \kappa _t \sim \text {Bin}(1, 1-\omega ),\quad X_t\ =\ \alpha \circ X_{t-1}\ +\ \beta \circ (n-X_{t-1}),\nonumber \\ \end{aligned}$$
(B.2)

and the IZ-BAR(1) process (DGP3),

$$\begin{aligned} Z_t\ =\ \kappa _t\cdot X_t\quad \text {with } \kappa _t \sim \text {Bin}(1, 1-\omega ),\quad X_t\ =\ \alpha \circ Z_{t-1}\ +\ \beta \circ (n-Z_{t-1}),\nonumber \\ \end{aligned}$$
(B.3)

are defined by distinguishing between the underlying BAR(1) process (kernel) and the resulting zero-inflated processes, by denoting the BAR(1) kernel by \((X_t)_{\mathbb {N}}\) and the zero-inflated process by \((Z_t)_{\mathbb {N}}\).

The ZIB-AR(1) process (DGP4) uses the concept of zero-inflated binomial thinning (ZIB thinning) “\(\odot \)”, which is defined as \((\alpha ,\omega )\odot Z | Z\ \sim {\text {ZIB}}(Z,\alpha , \omega )\). It follows the recursion

$$\begin{aligned} Z_t\ =\ (\alpha , \omega _{\alpha })\odot Z_{t-1}\ +\ (\beta , \omega _{\beta })\odot (n-Z_{t-1}), \end{aligned}$$
(B.4)

where we considered the special case \(\omega _{\alpha }=\omega _{\beta }\) for the simulations. In analogy to the interpretation of (1.1), “\((\alpha , \omega _{\alpha })\odot Z_{t-1}\)” expresses a survival mechanism and “\((\beta , \omega _{\beta })\odot (n-Z_{t-1})\)” a revival mechanism.

The ZT-BAR(1) process (DGP5) with the additional model parameter \(\beta _{0} \in (0;1)\) is defined by a self-exciting threshold mechanism with threshold value 0:

$$\begin{aligned} Z_t\ =\ {\left\{ \begin{array}{ll} \beta _{0} \circ n &{}\quad \text { if }\ Z_{t-1}=0,\\ \alpha \circ Z_{t-1}\ +\ \beta \circ (n-Z_{t-1}) &{}\quad \text { if }\ Z_{t-1}> 0. \end{array}\right. } \end{aligned}$$
(B.5)

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Kim, HY., Weiß, C.H. & Möller, T.A. Testing for an excessive number of zeros in time series of bounded counts. Stat Methods Appl 27, 689–714 (2018). https://doi.org/10.1007/s10260-018-00431-z

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