Erratum to: Alternated estimation in semi-parametric space-time branching-type point processes with application to seismic catalogs

1 Erratum to: Stoch Environ Res Risk Assess DOI 10.1007/s00477-014-0873-8

Equations 4, 5 and 6 are correct as they appear in this erratum.

$$ \begin{aligned}\log L(\hat{{\varvec{\psi}}}(H_{{t_{k} }} );H_{{t_{k} }} ) & = \sum\limits_{i = 1}^{k} {\log \,} \lambda ({\mathbf{z}}_{i} ;\hat{{\varvec{\psi}}}(H_{{t_{k} }} )) \\ & \quad - \int\limits_{{T_{0} }}^{{T_{max} }} {\int\limits_{{\Omega_{{\mathbf{s}}} }} {\kern 1pt} } \lambda ({\mathbf{z}};\hat{{\varvec{\psi}}}(H_{{t_{k} }} )){\kern 1pt} d{\mathbf{s}}\,{\kern 1pt} dt \\ \end{aligned}$$

(4)

$$ \begin{aligned} \log L(\hat{{\varvec{\psi}}}(H_{{t_{k} }} );H_{{t_{k + 1} }} ) & = \sum\limits_{i = 1}^{k + 1} {\log } \,\lambda ({\mathbf{z}}_{i} ;\hat{{\varvec{\psi}}}(H_{{t_{k} }} )) \\ & \quad - \int\limits_{{T_{0} }}^{{t_{k + 1} }} {\int\limits_{{\Omega_{{\mathbf{s}}} }} {\kern 1pt} } \lambda ({\mathbf{z}};\hat{{\varvec{\psi}}}(H_{{t_{k} }} )){\kern 1pt} d{\mathbf{s}}{\kern 1pt} dt \\ \end{aligned}$$

(5)

$$ \begin{aligned} & \delta_{k,k + 1} (\hat{{\varvec{\psi}}}(H_{{t_{k} }} );H_{{t_{k + 1} }} ) \equiv \\ & \quad { = }\log L(\hat{{\varvec{\psi}}}(H_{{t_{k} }} );H_{{t_{k + 1} }} ) - \log L(\hat{{\varvec{\psi}}}(H_{{t_{k} }} );H_{{t_{k} }} ) \\ & \quad = \left(\sum\limits_{i = 1}^{k + 1} {\log } \lambda ({\mathbf{z}}_{i} ;\hat{{\varvec{\psi}}}(H_{{t_{k} }} )) - \int\limits_{{T_{0} }}^{{t_{k + 1} }} {\int\limits_{{\Omega_{{\mathbf{s}}} }} {\kern 1pt} } \lambda ({\mathbf{z}};\hat{{\varvec{\psi}}}(H_{{t_{k} }} ))d{\mathbf{s}}dt\right) - \\ &\qquad \left(\sum\limits_{i = 1}^{k} {\log } \lambda ({\mathbf{z}}_{i} ;\hat{{\varvec{\psi}}}(H_{{t_{k} }} )) - \int\limits_{{T_{0} }}^{{t_{k} }} {\int\limits_{{\Omega_{{\mathbf{s}}} }} {\kern 1pt} } \lambda ({\mathbf{z}};\hat{{\varvec{\psi}}}(H_{{t_{k} }} ))d{\mathbf{s}}dt\right) \\ & \quad = \log \lambda ({\mathbf{z}}_{k + 1} ;\hat{{\varvec{\psi}}}(H_{{t_{k} }} )) - \int\limits_{{t_{k} }}^{{t_{k + 1} }} {\int\limits_{{\Omega_{{\mathbf{s}}} }} {\kern 1pt} } \lambda ({\mathbf{z}};\hat{{\varvec{\psi}}}(H_{{t_{k} }} ))d{\mathbf{s}}dt. \\ \end{aligned} $$

(6)