# Maximum Likelihood Estimation of the Pareto Alphas of a Generalized Pareto Distribution Calculates the maximum likelihood estimators of the parameters alpha_ini and alpha_tail of a generalized Pareto distribution with known threshold and (if applicable) known truncation ```r GenPareto_ML_Estimator_Alpha( losses, t, truncation = NULL, reporting_thresholds = NULL, is.censored = NULL, weights = NULL, alpha_min = 0.001, alpha_max = 10 ) ``` ## Arguments - `losses`: Numeric vector. Losses that are used for the ML estimation. - `t`: Numeric or numeric vector. Threshold of the generalized Pareto distribution. Alternatively, `t` can be a vector of same length as `losses`. In this case `t[i]` is the reporting threshold of `losses[i]`. - `truncation`: Numeric. If `truncation` is not `NULL` and `truncation > t`, then the generalized Pareto distribution is truncated at `truncation`. - `reporting_thresholds`: Numeric vector. Allows to enter loss specific reporting thresholds. If `NULL` then all reporting thresholds are assumed to be less than or equal to `t`. - `is.censored`: Logical vector. `TRUE` indicates that a loss has been censored by the policy limit. The assumption is that the uncensored losses are Generalized Pareto distributed with the alphas we are looking for. `is.censored = NULL` means that no losses are censored. - `weights`: Numeric vector. Weights for the losses. For instance `weights[i] = 2` and `weights[j] = 1` for `j != i` has the same effect as adding another loss of size `loss[i]`. - `alpha_min`: Numeric. Lower bound for the estimated alphas. - `alpha_max`: Numeric. Upper bound for the estimated alphas. ## Returns Maximum likelihood estimator for the parameters `alpha_ini` and `alpha_tail` of a generalized Pareto distribution with threshold `t` given the observations `losses` ## Examples ```r losses <- rGenPareto(1000, 1000, 2,3) GenPareto_ML_Estimator_Alpha(losses, 1000) losses <- rGenPareto(1000, 1000, 2, 1, truncation = 10000) GenPareto_ML_Estimator_Alpha(losses, 1000) GenPareto_ML_Estimator_Alpha(losses, 1000, truncation = 10000) t <- 1000 alpha_ini <- 1 alpha_tail <- 3 losses <- rGenPareto(5000, t, alpha_ini, alpha_tail) reporting_thresholds <- rPareto(5000, 1000, 3) reported <- losses > reporting_thresholds losses <- losses[reported] reporting_thresholds <- reporting_thresholds[reported] GenPareto_ML_Estimator_Alpha(losses, t) GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds) limit <- 3000 censored <- losses > limit losses[censored] <- limit reported <- losses > reporting_thresholds losses <- losses[reported] censored <- censored[reported] reporting_thresholds <- reporting_thresholds[reported] GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds) GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds, is.censored = censored) losses <- c(190, 600, 120, 270, 180, 120) w <- rep(1, length(losses)) w[1] <- 3 losses2 <- c(losses, losses[1], losses[1]) GenPareto_ML_Estimator_Alpha(losses, 100, weights = w) GenPareto_ML_Estimator_Alpha(losses2, 100) ```

GenPareto_ML_Estimator_Alpha function