Pareto_ML_Estimator_Alpha function

Maximum Likelihood Estimation of the Alpha of a Pareto distribution

Calculates the maximum likelihood estimator for the parameter alpha of a Pareto distribution with a known threshold and (if applicable) a known truncation

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).

Returns

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)