PiecewisePareto_ML_Estimator_Alpha: Maximum Likelihood Estimation of the Alphas of the Piecewise...

In Pareto: The Pareto, Piecewise Pareto and Generalized Pareto Distribution

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[1].
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[1] <- 2
losses2 <- c(losses, losses[1])
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.