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

## Description

Calculates the maximum likelihood estimator for the parameter alpha of a Pareto distribution with a known threshold and (if applicable) a known truncation

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```Pareto_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. Threshold of the Pareto distribution. `truncation` Numeric. If `truncation` is not `NULL`, then the 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 Pareto distributed with the alpha 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 alpha (only used in truncated case). `alpha_max` Numeric. Upper bound for alpha (only used in truncated case).

## Value

Maximum likelihood estimator for the parameter `alpha` of a 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``` ```losses <- rPareto(100, 1000, 2) Pareto_ML_Estimator_Alpha(losses, 1000) losses <- rPareto(100, 1000, 2, truncation = 2000) Pareto_ML_Estimator_Alpha(losses, 1000) Pareto_ML_Estimator_Alpha(losses, 1000, truncation = 2000) t <- 100 alpha <- 2 losses <- rPareto(10000, t, alpha) reporting_thresholds <- rPareto(10000, t, 5) index <- losses > reporting_thresholds losses <- losses[index] reporting_thresholds <- reporting_thresholds[index] Pareto_ML_Estimator_Alpha(losses, t) Pareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds) losses <- rPareto(10, 1000, 2) w <- rep(1, 10) w <- 3 losses2 <- c(losses, losses, losses) Pareto_ML_Estimator_Alpha(losses, 1000, weights = w) Pareto_ML_Estimator_Alpha(losses2, 1000) ```

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