# Maximum Likelihood Estimation of the Alphas of the Piecewise Pareto Distribution Calculates the maximum likelihood estimator of the parameter vector alpha for a a piecewise Pareto distribution with given vector t and (if applicable) a known truncation ```r PiecewisePareto_ML_Estimator_Alpha( losses, t, truncation = NULL, truncation_type = "lp", 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 vector. Thresholds of the piecewise Pareto distribution. - `truncation`: Numeric. If `truncation` is not `NULL` and `truncation > max(t)`, then the distribution is truncated at `truncation`. - `truncation_type`: Character. If `truncation_type = "wd"` then the whole distribution is truncated. If `truncation_type = "lp"` then a truncated Pareto is used for the last piece. - `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[1]`. - `is.censored`: Logical vector. `TRUE` indicates that a loss has been censored by the policy limit. The assumption is that the uncensored losses are piecewise 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 (only used in truncated case). - `alpha_max`: Numeric. Upper bound for the estimated alphas (only used in truncated case). ## Returns Maximum likelihood estimator for the parameter `alpha` of a Pareto distribution with threshold `t` given the observations `losses` ## Examples ```r losses <- rPiecewisePareto(10000, t = c(100,200,300), alpha = c(1,2,3)) PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300)) losses <- rPiecewisePareto(10000, t = c(100,200,300), alpha = c(1,2,3), truncation = 500, truncation_type = "wd") PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300)) PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), truncation = 500, truncation_type = "wd") reporting_thresholds <- rPareto(10000, 100, 3) index <- losses > reporting_thresholds losses <- losses[index] reporting_thresholds <- reporting_thresholds[index] PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), truncation = 500, truncation_type = "wd") PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), truncation = 500, truncation_type = "wd", reporting_thresholds = reporting_thresholds) losses <- c(140, 240, 490, 200, 110, 710, 120, 190, 210, 310) w <- rep(1, length(losses)) w[1] <- 2 losses2 <- c(losses, losses[1]) PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), weights = w) PiecewisePareto_ML_Estimator_Alpha(losses2, c(100,200,300)) ```

PiecewisePareto_ML_Estimator_Alpha function