Pareto_ML_Estimator_Alpha: Maximum Likelihood Estimation of the Alpha of a Pareto...
In Pareto: The Pareto, Piecewise Pareto and Generalized Pareto Distribution

Description Usage Arguments Value Examples

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
)

`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).

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

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)