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

Description

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

Usage

 `1` ```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_∞) `M00` number of extremes within the time-horizon in a "typical" hazard rate trajectory (M_∞) `extra` number of additional CPP jumps (compared with default)

Details

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_∞).

Value

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)

Note

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

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.

`BayHaz-package`, `CPPpriorSample`, `CPPpostSample`
 ```1 2 3``` ```# 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) ```