Computation of Davies bound p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order \(p\) (PNAR(\(p\))) versus the non-linear Smooth Transition alternative (ST-PNAR(\(p\))).
score_test_stnarpq_DV(b, y, W, p, d, Z = NULL, gama_L = NULL,
gama_U = NULL, len = 100)A list including:
The Davies bound of p-values for sup test.
The value of the sup test statistic in the sample \(y\).
The estimated parameters from the linear model, in the following order: (intercept, network parameters, autoregressive parameters, covariates). The dimension of the vector should be \(2p + 1 + q\), where \(q\) denotes the number of covariates.
A \(TT \times N\) time series object or a \(TT \times N\) numerical matrix with the \(N\) multivariate count time series over \(TT\) time periods.
The \(N \times N\) row-normalized non-negative adjacency matrix describing the network. The main diagonal entries of the matrix should be zeros, all the other entries should be non-negative and the maximum sum of elements over the rows should equal one. The function row-normalizes the matrix if a non-normalized adjacency matrix is provided.
The number of lags in the model.
The lag parameter of non-linear variable (should be between 1 and \(p\)).
An \(N \times q\) matrix of covariates (one for each column), where \(q\) is the number of covariates in the model. Note that they must be non-negative.
The lower value of the nuisance parameter \(\gamma\) to consider. Use NULL if there is not information about its value. See the details for default computation.
The upper value of the nuisance parameter \(\gamma\) to consider. Use NULL if there is not information about its value. See the details for default computation.
The length of the grid of values of \(\gamma\) values to consider.
Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.
The function computes an upper-bound for the p-value of the sup-type test for testing linearity of Poisson Network Autoregressive model of order \(p\) (PNAR(\(p\))) versus the following Smooth Transition alternative (ST-PNAR(\(p\))). For each node of the network \(i=1,...,N\) over the time sample \(t=1,...,TT\) $$ \lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h}+\alpha_{h}e^{-\gamma X_{i,t-d}^{2}}X_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l} $$ where \(X_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t}\) is the network effect, i.e. the weighted average impact of node \(i\) connections, with the weights of the mean being \(W_{ij}\), the single element of the network matrix \(W\). The sequence \(\lambda_{i,t}\) is the expectation of \(Y_{i,t}\), conditional to its past values.
The null hypothesis of the test is defined as \(H_{0}: \alpha_{1}=...=\alpha_{p}=0\), versus the alternative that at least one among \(\alpha_{h}\) is not 0. The test statistic has the form $$ LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma), $$ where $$ S(\hat{\theta},\gamma)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left(\frac{Y_{i,t}}{\lambda_{i,t}(\hat{\theta},\gamma)}-1\right)\frac{\partial\lambda_{i,t}(\hat{\theta},\gamma)}{\partial\alpha} $$ is the partition of the quasi score related to the vector of non-linear parameters \(\alpha=(\alpha_{1},...,\alpha_{p})\), evaluated at the estimated parameters \(\hat{\theta}\) under the null assumption \(H_{0}\) (linear model), and \(\Sigma(\hat{\theta},\gamma)\) is the variance of \(S(\hat{\theta},\gamma)\). Since the test statistic depends on an unknown nuisance parameter (\(\gamma\)), the supremum of the statistic is considered in the test, \(\sup_{\gamma}LM(\gamma)\). The function computes the bound of the p-value, suggested by Davies (1987), for the test statistic \(\sup_{\gamma}LM(\gamma)\), with scalar nuisance parameter \(\gamma\), as follows. $$ P(\chi^{2}_{k} \geq M)+V M^{1/2(k-1)}\frac{e^{-M/2}2^{-k/2}}{\Gamma(k/2)} $$ where \(M\) is the maximum of the test statistic \(LM(\gamma)\), computed by the available sample, over a grid of values for the nuisance parameter \(\gamma_{F}=(\gamma_{L},\gamma_{1},...,\gamma_{l},\gamma_{U})\); \(k\) is the number of non-linear parameters tested. So the first summand of the bound is just the p-value of a chi-square test with \(k\) degrees of freedom. The second summand is a correction term depending on \(V\), which is the approximated total variation computed as $$ V=|LM^{1/2}(\gamma_{1})-LM^{1/2}(\gamma_{L})|+|LM^{1/2}(\gamma_{2})-LM^{1/2}(\gamma_{1})|+...+|LM^{1/2}(\gamma_{U})-LM^{1/2}(\gamma_{l})|. $$ The feasible bound allows to approximate the p-values of the sup-type test in a straightforward way, by adding to the tail probability of a chi-square distribution a correction term which depends on the total variation of the process. For details see Armillotta and Fokianos (2023, Sec. 5).
The values of gama_L and gama_U are computed internally as gama_L \(=-\log(0.9)/X^{2}\) and gama_U \(=-\log(0.1)/X^{2}\), where \(X\) is the overall mean of \(X_{i,t}\) over the nodes \(i=1,...,N\) and times \(t=1,...,TT\). Since the non-linear function \(e^{-\gamma X_{i,t-d}^{2}}\) ranges between 0 and 1, by considering \(X\) to be a representative value for the network mean, gama_U and gama_L would be the values of \(\gamma\) leading the non-linear switching function to be 0.1 and 0.9, respectively, so that in the optimization procedure the extremes of the function domain are excluded. Alternatively, their values can be supplied by the user.
Armillotta, M. and K. Fokianos (2023). Nonlinear network autoregression. Annals of Statistics, 51(6): 2526--2552.
Armillotta, M. and K. Fokianos (2024). Count network autoregression. Journal of Time Series Analysis, 45(4): 584--612.
Armillotta, M., Tsagris, M. and Fokianos, K. (2024). Inference for Network Count Time Series with the R Package PNAR. The R Journal, 15/4: 255--269.
Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33--43.
score_test_stnarpq_j, global_optimise_LM_stnarpq
data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 1)
ca <- mod1$coefs[, 1]
score_test_stnarpq_DV(ca, crime, crime_W, p = 1, d = 1)
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