Pareto_ML_Estimator_Alpha: Maximum Likelihood Estimation of the Alpha of a Pareto...

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

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[1] <- 3
losses2 <- c(losses, losses[1], losses[1])
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.