CPPpriorElicit: Function to Set Hyperparameters of CPP Priors
Description
A function to set the hyperparameters of a CPP prior distribution, following the procedure described in La Rocca (2005).
Usage
CPPpriorElicit(r0 = 1, H = 1, T00 = 1, M00 = 1, extra = 0)
Arguments
r0
prior mean hazard rate ($r_0$)
H
corresponding coefficient of variation
T00
time-horizon of interest ($T_\infty$)
M00
number of extremes within the time-horizon in a "typical" hazard rate trajectory ($M_\infty$)
extra
number of additional CPP jumps (compared with default)
Value
A list with nine components:
r0prior mean hazard rate (copy of the input argument)
Hcorresponding coefficient of variation (copy of the input argument)
T00time-horizon of interest (copy of the input argument)
M00number of extremes within the time-horizon in a "typical" hazard rate trajectory (copy of the input argument)
ashape parameter of the jump-size distribution (always equal to 1)
sdstandard deviation of the Gaussian kernel (bandwidth)
qexpected number of CPP jumps per time unit
brate parameter of the jump-size distribution
Fmaximum number of jumps within the time-horizon (with high probability)
Details
A CPP prior hazard rate is defined, for $0
$\sigma_j$is the time of the$j$-th jump of a CPP process with gamma distributed jump-sizes
$\xi_j$is the$j$-th jump-size of the above process
$k$is a zero-mean Gaussian density (kernel)
$F$is a positive integer such that (with high probability)$\sigma_{F+1}$is much larger than$T_\infty$
$\xi_0$is an independent random variable with the same distribution as$\xi_j$
$k_0$is a suitable function such that the mean of$rho(t)$does not depend on$t$
The elicitation procedure makes the mean of $rho(t)$ identically equal to $r_0$
and its standard deviation approximately equal to $Hr_0$. An exponential distribution
is selected for the jump-sizes. The kernel bandwidth choice is based on $M_\infty$ (and $T_\infty$).
References
Luca La Rocca (2005). On Bayesian Nonparametric Estimation of Smooth Hazard Rates with a View to Seismic Hazard Assessment.
Research Report n. 38-05, Department of Social, Cognitive and Quantitative Sciences, Reggio Emilia, Italy.
# ten events per century with unit coefficient of variation# fifty year time horizon with a couple of extremes in a "typical" trajectoryhypars<-CPPpriorElicit(r0 = 0.1, H = 1, T00 = 50, M00 = 2)