Prior simulation of GEV parameters - prior on probability scale

Simulates from the prior distribution for GEV parameters based on Crowder (1992), in which independent beta priors are specified for ratios of probabilities (which is equivalent to a Dirichlet prior on differences between these probabilities).

rprior_prob(n, quant, alpha, exc = FALSE, lb = NULL, lb_prob = 0.001)

Arguments

n

A numeric scalar. The size of sample required.

quant

A numeric vector of length 3. Contains quantiles \(q_1, q_2, q_3\). A prior distribution is placed on the non-exceedance (exc = FALSE) or exceedance (exc = TRUE) probabilities corresponding to these quantiles. The values should increase with the index of the vector. If not, the values in quant will be sorted into increasing order without warning.

alpha

A numeric vector of length 4. Parameters of the Dirichlet distribution for the exceedance probabilities.

exc

A logical scalar. Let \(M\) be the GEV variable, \(r_q = P(M \leq q)\), \(p_q = P(M > q) = 1 - r_q\) and quant = (\(q_1, q_2, q_3\)). If exc = FALSE then a Dirichlet(alpha) distribution is placed on \((r_{q_1}, r_{q_2} - r_{q_1}, r_{q_3} - r_{q_2}, 1 - r_{q_3})\), as in Northrop et al. (2017). If exc = TRUE then a Dirichlet(alpha) distribution is placed on \((1 - p_{q_1}, p_{q_1} - p_{q_2}, p_{q_2} - p_{q_3}, p_{q_3})\), where \(p_q = P(M > q)\), as in Stephenson (2016).

lb

A numeric scalar. If this is not NULL then the simulation is constrained so that lb is an approximate lower bound on the GEV variable. Specifically, only simulated GEV parameter values for which the 100lb_prob% quantile is greater than lb are retained.

lb_prob

A numeric scalar. The non-exceedance probability involved in the specification of lb. Must be in (0,1). If lb=NULL then lb_prob is not used.

Value

An n by 3 numeric matrix.

Details

The simulation is based on the way that the prior is constructed. See Stephenson (1996) the evdbayes user guide or Northrop et al. (2017) Northrop et al. (2017) for details of the construction of the prior. First, differences between probabilities are simulated from a Dirichlet distribution. Then the GEV location, scale and shape parameters that correspond to these quantile values are found, by solving numerically a set of three non-linear equations in which the GEV quantile function evaluated at the simulated probabilities is equated to the quantiles in quant. This is reduced to a one-dimensional optimisation over the GEV shape parameter.

References

Crowder, M. (1992) Bayesian priors based on parameter transformation using the distribution function. Ann. Inst. Statist. Math., 44(3), 405-416. https://link.springer.com/article/10.1007/BF00050695

Stephenson, A. 2016. Bayesian Inference for Extreme Value Modelling. In Extreme Value Modeling and Risk Analysis: Methods and Applications, edited by D. K. Dey and J. Yan, 257-80. London: Chapman and Hall. doi:10.1201/b19721

Northrop, P. J., Attalides, N. and Jonathan, P. (2017) Cross-validatory extreme value threshold selection and uncertainty with application to ocean storm severity. Journal of the Royal Statistical Society Series C: Applied Statistics, 66(1), 93-120. doi:10.1111/rssc.12159

See also

rpost and rpost_rcpp for sampling from an extreme value posterior distribution.

Examples

quant <- c(85, 88, 95)
alpha <- c(4, 2.5, 2.25, 0.25)
x <- rprior_prob(n = 1000, quant = quant, alpha = alpha, exc = TRUE)
x <- rprior_prob(n = 1000, quant = quant, alpha = alpha, exc = TRUE, lb = 0)