gpd_init: Initial estimates for Generalized Pareto parameters

Description Usage Arguments Details Value References See Also Examples

View source: R/gp_example.R

Description

Calculates initial estimates and estimated standard errors (SEs) for the generalized Pareto parameters sigma and xi based on an assumed random sample from this distribution. Also, calculates initial estimates and estimated standard errors for phi1 = sigma and phi2 = xi + sigma / xm.

Usage

1
2
gpd_init(gpd_data, m, xm, sum_gp = NULL, xi_eq_zero = FALSE,
  init_ests = NULL)

Arguments

gpd_data

A numeric vector containing positive sample values.

m

A numeric scalar. The sample size, i.e. the length of gpd_data.

xm

A numeric scalar. The sample maximum.

sum_gp

A numeric scalar. The sum of the sample values.

xi_eq_zero

A logical scalar. If TRUE assume that the shape parameter xi = 0.

init_ests

A numeric vector. Initial estimate of theta = (sigma, xi). If supplied gpd_init() just returns the corresponding initial estimate of phi = (phi1, phi2).

Details

The main aim is to calculate an admissible estimate of theta, i.e. one at which the log-likelihood is finite (necessary for the posterior log-density to be finite) at the estimate, and associated estimated SEs. These are converted into estimates and SEs for phi. The latter can be used to set values of min_phi and max_phi for input to find_lambda.

In the default setting (xi_eq_zero = FALSE and init_ests = NULL) the methods tried are Maximum Likelihood Estimation (MLE) (Grimshaw, 1993), Probability-Weighted Moments (PWM) (Hosking and Wallis, 1987) and Linear Combinations of Ratios of Spacings (LRS) (Reiss and Thomas, 2007, page 134) in that order.

For xi < -1 the likelihood is unbounded, MLE may fail when xi is not greater than -0.5 and the observed Fisher information for (sigma, xi) has finite variance only if xi > -0.25. We use the ML estimate provided that the estimate of xi returned from gpd_mle is greater than -1. We only use the SE if the MLE of xi is greater than -0.25.

If either the MLE or the SE are not OK then we try PWM. We use the PWM estimate only if is admissible, and the MLE was not OK. We use the PWM SE, but this will be c(NA, NA) is the PWM estimate of xi is > 1/2. If the estimate is still not OK then we try LRS. As a last resort, which will tend to occur only when xi is strongly negative, we set xi = -1 and estimate sigma conditional on this.

Value

If init_ests is not supplied by the user, a list is returned with components

init

A numeric vector. Initial estimates of sigma and xi.

se

A numeric vector. Estimated standard errors of sigma and xi.

init_phi

A numeric vector. Initial estimates of phi1 = sigma and phi2 = xi + sigma / xm, where xm is the maximum of gpd_data.

se_phi

A numeric vector. Estimated standard errors of phi1 and phi2.

If init_ests is supplied then only the numeric vector init_phi is returned.

References

Grimshaw, S. D. (1993) Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution. Technometrics, 35(2), 185-191. and Computing (1991) 1, 129-133. http://dx.doi.org/10.1007/BF01889987.

Hosking, J. R. M. and Wallis, J. R. (1987) Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics, 29(3), 339-349. http://dx.doi.org/10.2307/1269343.

Reiss, R.-D., Thomas, M. (2007) Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields.Birkhauser. http://dx.doi.org/10.1007/978-3-7643-7399-3.

See Also

gpd_sum_stats to calculate summary statistics for use in gpd_loglik.

rgpd for simulation from a generalized Pareto

find_lambda to produce (somewhat) automatically a list for the argument lambda of ru.

Examples

1
2
3
4
5
6
7
8
9
## Not run: 
# Sample data from a GP(sigma, xi) distribution
gpd_data <- rgpd(m = 100, xi = 0, sigma = 1)
# Calculate summary statistics for use in the log-likelihood
ss <- gpd_sum_stats(gpd_data)
# Calculate initial estimates
do.call(gpd_init, ss)

## End(Not run)

paulnorthrop/rust documentation built on Jan. 6, 2019, 3:09 a.m.