# stability: Generalized Pareto parameter estimate stability In threshr: Threshold Selection and Uncertainty for Extreme Value Analysis

## Description

Uses maximum likelihood estimation to fit a Generalized Pareto (GP) model to threshold excesses over a range of thresholds. The threshold excesses are treated as independent and identically distributed (i.i.d.) observations. The resulting estimates and confidence intervals can be plotted, using `plot.stability`, to produce a crude graphical diagnostic for threshold choice.

## Usage

 ```1 2``` ```stability(data, u_vec, prof = FALSE, conf = 95, mult = 1:2, plot_prof = FALSE, ...) ```

## Arguments

 `data` A numeric vector of observations. `u_vec` A numeric vector of thresholds to be applied to the data. Any duplicated values will be removed. These could be set at sample quantiles of `data` using `quantile`. `prof` A logical scalar. Whether to calculate confidence intervals for the GP shape parameter ξ based on the profile-likelihood for ξ or using the MLE plus or minus a multiple of the estimated standard error (SE) of the MLE. The intervals produced by the former may be better but they take longer to calculate. Default: `FALSE`. `conf` A numeric scalar in (0, 100). Confidence level for the confidence intervals. Default: 95%. `mult` A numeric vector of length 2. The range of values over which the profile log-likelihood for ξ is calculated is (MLE - `mult[1]` c SE, MLE + `mult[2]` c SE), where MLE and SE are the MLE and estimated standard error of ξ and c is the constant for which this interval gives an approximate 100`conf`% level confidence interval for ξ when `mult = c(1, 1)`. The default, `mult = c(1, 2)`, works well in most cases. If the routine fails because the range of ξ is not sufficiently wide then the relevant components of `mult` should be increased. `plot_prof` A logical scalar. Only relevant if `prof = TRUE`. If `plot_prof = TRUE` then the profile log-likelihood is plotted for each threshold. If `FALSE` then nothing is plotted. `...` Further (optional) arguments to be passed to the `optim` function for the optimizations on which the profile-likelihood for xi is based.

## Details

For each threshold in `u_vec` a GP model is fitted by maximum likelihood estimation to the threshold excesses, i.e. the amounts by which the data exceed that threshold. The MLEs of the GP shape parameter \$ξ\$ and approximate `conf`% confidence intervals for ξ are stored for plotting (by `plot.stability`) to produce a simple graphical diagnostic to inform threshold selection. This plot is used to choose a threshold above which the underlying GP shape parameter may be approximately constant. See Chapter 4 of Coles (2001). See also the vignette "Introducing threshr".

## Value

An object (list) of class "stability" with components:

 `ests` MLEs of the GP shape parameter ξ. `ses` Estimated SEs of the MLEs of ξ. `lower` Lower limit of 100`conf`% confidence intervals for ξ. `upper` Upper limit of 100`conf`% confidence intervals for ξ. `nexc` The number of threshold excesses. `u_vec` The thresholds supplied by the user. `u_ps` The approximate sample quantiles to which the thresholds in `u_vec` correspond. `data` The input `data`. `conf` The input `conf`.

Each of these components is a numeric vector of length `length(u_vec)`.

## References

Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. http://dx.doi.org/10.1007/978-1-4471-3675-0_3

`ithresh` for threshold selection in the i.i.d. case based on leave-one-out cross-validation.

`plot.stability` for the S3 `plot` method for objects of class `stability`.

`quantile`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```# Set a vector of thresholds u_vec_gom <- quantile(gom, probs = seq(0, 0.95, by = 0.05)) # Symmetric confidence intervals gom_stab <- stability(data = gom, u_vec = u_vec_gom) plot(gom_stab) # Profile-likelihood-based confidence intervals gom_stab <- stability(data = gom, u_vec = u_vec_gom, prof = TRUE) plot(gom_stab) ```

threshr documentation built on Sept. 4, 2017, 9:03 a.m.