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

 GenPareto_ML_Estimator_Alpha R Documentation

## Maximum Likelihood Estimation of the Pareto Alphas of a 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

``````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

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