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

 Pareto_ML_Estimator_Alpha R Documentation

## Maximum Likelihood Estimation of the Alpha of a 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

``````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

``````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 April 18, 2023, 9:10 a.m.