GenPareto_ML_Estimator_Alpha function

Maximum Likelihood Estimation of the Pareto Alphas of a Generalized Pareto Distribution

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

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.

Returns

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[1] <- 3
losses2 <- c(losses, losses[1], losses[1])
GenPareto_ML_Estimator_Alpha(losses, 100, weights = w)
GenPareto_ML_Estimator_Alpha(losses2, 100)