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

 PiecewisePareto_ML_Estimator_Alpha R Documentation

## Maximum Likelihood Estimation of the Alphas of the Piecewise Pareto Distribution

### Description

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

### Usage

``````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`. `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).

### Value

Maximum likelihood estimator for the parameter `alpha` of a Pareto distribution with threshold `t` given the observations `losses`

### Examples

``````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 <- 2
losses2 <- c(losses, losses)
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), weights = w)
PiecewisePareto_ML_Estimator_Alpha(losses2, c(100,200,300))
``````

Pareto documentation built on April 18, 2023, 9:10 a.m.