# GenPareto_ML_Estimator_Alpha: Maximum Likelihood Estimation of the Pareto Alphas of a... In Pareto: The Pareto, Piecewise Pareto and Generalized Pareto Distribution

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```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.

## Value

Maximum likelihood estimator for the parameters `alpha_ini` and `alpha_tail` of a generalized Pareto distribution with threshold `t` given the observations `losses`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```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 <- 3 losses2 <- c(losses, losses, losses) GenPareto_ML_Estimator_Alpha(losses, 100, weights = w) GenPareto_ML_Estimator_Alpha(losses2, 100) ```

Pareto documentation built on March 3, 2021, 5:07 p.m.