CPPpriorElicit: Function to Set Hyperparameters of CPP Priors
In BayHaz: R Functions for Bayesian Hazard Rate Estimation

Description Usage Arguments Details Value Note References See Also Examples

A function to set the hyperparameters of a CPP prior distribution, following the procedure described in La Rocca (2005).

1	CPPpriorElicit(r0 = 1, H = 1, T00 = 1, M00 = 1, extra = 0)

`r0`	prior mean hazard rate (r_0)
`H`	corresponding coefficient of variation
`T00`	time-horizon of interest (T_∞)
`M00`	number of extremes within the time-horizon in a "typical" hazard rate trajectory (M_∞)
`extra`	number of additional CPP jumps (compared with default)

A CPP prior hazard rate is defined, for 0<t<T_∞, by

ρ(t)=ξ_0 k_0(t)+∑_{j=1}^{F} ξ_j k(t-σ_j)

where:

σ_j is the time of the j-th jump of a CPP process with gamma distributed jump-sizes
ξ_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) σ_{F+1} is much larger than T_∞
ξ_0 is an independent random variable with the same distribution as ξ_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_∞ (and T_∞).

A list with nine components:

`r0`	prior mean hazard rate (copy of the input argument)
`H`	corresponding coefficient of variation (copy of the input argument)
`T00`	time-horizon of interest (copy of the input argument)
`M00`	number of extremes within the time-horizon in a "typical" hazard rate trajectory (copy of the input argument)
`a`	shape parameter of the jump-size distribution (always equal to 1)
`sd`	standard deviation of the Gaussian kernel (bandwidth)
`q`	expected number of CPP jumps per time unit
`b`	rate parameter of the jump-size distribution
`F`	maximum number of jumps within the time-horizon (with high probability)

As the default value of F is computed a priori, additional jumps may be needed a posteriori.

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.

BayHaz-package, CPPpriorSample, CPPpostSample

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# ten events per century with unit coefficient of variation
# fifty year time horizon with a couple of extremes in a "typical" trajectory
hypars<-CPPpriorElicit(r0 = 0.1, H = 1, T00 = 50, M00 = 2)