# Fitting a GPD to Peaks Over a Threshold Maximum (Penalized) Likelihood, Unbiased Probability Weighted Moments,Biased Probability Weighted Moments, Moments, Pickands', Minimum Density Power Divergence, Medians, Likelihood Moment and Maximum Goodness-of-Fit Estimators to fit Peaks Over a Threshold to a GP distribution. ```r fitgpd(data, threshold, est = "mle", ...) ``` ## Arguments - `data`: A numeric vector. - `threshold`: A numeric value giving the threshold for the GPD. The `'mle'` estimator allows varying threshold; so that threshold could be for this case a numeric vector. Be careful, varying thresholds are used cyclically if length doesn't match with `data`. - `est`: A string giving the names of the estimator. It can be `'mle'` (the default), `'mple'`, `'moments'`, `'pwmu'`, `'pwmb'`, `'mdpd'`, `'med'`, `'pickands'`, `'lme'` and `'mgf'` for the maximum likelihood, maximum penalized likelihood, moments, unbiased probability weighted moments, biased probability weigthed moments, minimum density power divergence, medians, Pickands', likelihood moment and maximum goodness-of-fit estimators respectively. - ``...``: Other optional arguments to be passed to the `optim` function, allow hand fixed parameters (only for the `mle`, `mple` and `mgf` estimators) or passed several options to specific estimators - see the Note section. ## Returns This function returns an object of class `"uvpot"` with components: - **fitted.values**: A vector containing the estimated parameters. - **std.err**: A vector containing the standard errors. - **fixed**: A vector containing the parameters of the model that have been held fixed. - **param**: A vector containing all parameters (optimized and fixed). - **deviance**: The deviance at the maximum likelihood estimates. - **corr**: The correlation matrix. - **convergence, counts, message**: Components taken from the list returned by `optim` - for the `mle` method. - **threshold**: The threshold passed to argument `threshold`. - **nat, pat**: The number and proportion of exceedances. - **data**: The data passed to the argument `data`. - **exceed**: The exceedances, or the maxima of the clusters of exceedances. - **scale**: The scale parameter for the fitted generalized Pareto distribution. - **std.err.type**: The standard error type - for `'mle'` only. That is `Observed` or `Expected` Information matrix of Fisher. - **var.thresh**: Logical. Specify if the threshold is a varying one - `'mle'` only. For other methods, threshold is always constant i.e. `var.thresh = FALSE`. ## See Also The following usual generic functions are available `print`, `plot`, `confint` and `anova` as well as new generic functions `retlev`, `qq`, `pp`, `dens` and `convassess`. ## Note The Maximum Likelihood estimator is obtained through a slightly modified version of Alec Stephenson's `fpot.norm` function in the `evd` package. For the `'mple'` estimator, the likelihood function is penalized using the following function : $$ P(\xi) = \left\{\begin{array}{ll}1, & \xi \leq 0\\\exp\left[-\lambda \left(\frac{1}{1-\xi} - 1\right)^\alpha \right], &0 < \xi <1\\0, & \xi \geq 1\end{array}\right.P(xi)=1 if xi \le 0, exp[-lambda(1/(1-xi) -1)^alpha], if 0 < xi < 1, 0 if xi \ge 1 $$ where $alpha$ and $lambda$ are the penalizing constants. Coles and Dixon (1999) suggest the use of $alpha=lambda=1$. The `'lme'` estimator has a special parameter `'r'`. Zhang (2007) shows that a value of -0.5 should be accurate in most of the cases. However, other values such as `r < 0.5` can be explored. In particular, if `r` is approximatively equal to the opposite of the true shape parameter value, then the `lme` estimate is equivalent to the `mle` estimate. The `'pwmb'` estimator has special parameters `'a'` and `'b'`. These parameters are called the "plotting-position" values. Hosking and Wallis (1987) recommend the use of `a = 0.35` and `b = 0` (the default). However, different values can be tested. For the `'pwmu'` and `'pwmb'` approaches, one can pass the option `'hybrid = TRUE'` to use hybrid estimators as proposed by Dupuis and Tsao (1998). Hybrid estimators avoid to have no feasible points. The `mdpd` estimator has a special parameter `'a'`. This is a parameter of the "density power divergence". Juarez and Schucany (2004) recommend the use of `a = 0.1`, but any value of `a` such as `a > 0` can be used (small values are recommend yet). The `med` estimator admits two extra arguments `tol` and `maxit` to control the "stopping-rule" of the optimization process. The `mgf` approach uses goodness-of-fit statistics to estimate the GPD parameters. There are currently 8 different statitics: the Kolmogorov-Smirnov `"KS"`, Cramer von Mises `"CM"`, Anderson Darling `"AD"`, right tail Anderson Darling `"ADR"`, left tail Anderson Darling `"ADL"`, right tail Anderson Darling (second degree) `"AD2R"`, left tail Anderson Darling (second degree) `"AD2L"` and the Anderson Darling (second degree) `"AD2"` statistics. ## References Coles, S. (2001) **An Introduction to Statistical Modelling of Extreme Values**. Springer Series in Statistics. London. Coles, S. and Dixon, M. (1999) Likelihood-Based Inference for Extreme Value Models. Extremes 2 (1):5--23. Dupuis, D. and Tsao (1998) M. A hybrid estimator for generalized Pareto and extreme-value distributions. Communications in Statistics-Theory and Methods 27 :925--941. Hosking, J. and Wallis, J. (1987) Parameters and Quantile Estimation for the Generalized Pareto Distribution. Technometrics 29 :339--349. Juarez, S. and Schucany, W. (2004) Robust and Efficient Estimation for the Generalized Pareto Distribution. Extremes 7 :237--251. Luceno, A. (2006) Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators. Computational Statistics and Data Analysis 51 :904--917. Peng, L. and Welsh, A. (2001) Robust Estimation of the Generalized Pareto Distribution. Extremes 4 :53--65. Embrechts, P and Kluppelberg, C. and Mikosch, T (1997) **Modelling Extremal Events for Insurance and Finance.** Springers. Pickands, J. (1975) Statistical Inference Using Extreme Order Statistics. Annals of Statistics. 3 :119--131. Zhang, J. (2007) Likelihood Moment Estimation for the Generalized Pareto Distribution. Australian and New Zealand Journal of Statistics. 49 (1):69--77. ## Examples ```r x <- rgpd(200, 1, 2, 0.25) mle <- fitgpd(x, 1, "mle")$param pwmu <- fitgpd(x, 1, "pwmu")$param pwmb <- fitgpd(x, 1, "pwmb")$param pickands <- fitgpd(x, 1, "pickands")$param ##Check if Pickands estimates ##are valid or not !!! med <- fitgpd(x, 1, "med", ##Sometimes the fitting algo is not start = list(scale = 2, shape = 0.25))$param ##accurate. So specify ##good starting values is ##a good idea. mdpd <- fitgpd(x, 1, "mdpd")$param lme <- fitgpd(x, 1, "lme")$param mple <- fitgpd(x, 1, "mple")$param ad2r <- fitgpd(x, 1, "mgf", stat = "AD2R")$param print(rbind(mle, pwmu, pwmb, pickands, med, mdpd, lme, mple, ad2r)) ##Use PWM hybrid estimators fitgpd(x, 1, "pwmu", hybrid = FALSE) ##Now fix one of the GPD parameters ##Only the MLE, MPLE and MGF estimators are allowed ! fitgpd(x, 1, "mle", scale = 2) fitgpd(x, 1, "mple", shape = 0.25) ``` ## Author(s) Mathieu Ribatet

fitGPD function