Example data: Attachment Points

Description

Example data: Attachment Points

Usage

Example1_AP

Format

An object of class numeric of length 5.

Example data: Expected Losses

Description

Example data: Expected Losses

Usage

Example1_EL

Format

An object of class numeric of length 5.

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a collective model

Usage

Excess_Frequency(CollectiveModel, x = 0)

Arguments

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Excess_Frequency(CollectiveModel, x = 0)

Arguments

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Excess_Frequency(PGPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Excess_Frequency(CollectiveModel, x = 0)

Arguments

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Fits a Collective Model to a PML Curve

Description

Fits a PPP_Model that matches the values of a PML curve

Usage

Fit_PML_Curve(
  return_periods,
  amounts,
  tail_alpha = 2,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1
)

Arguments

Value

A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

• FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

• t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

• alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

• truncation Numerical. 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.

• dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

• Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

• Comment Character. Information on whether the fit was successful

Examples

return_periods <- c(1, 5, 10, 20, 50, 100)
amounts <- c(1000, 4000, 7000, 10000, 13000, 14000)

fit <- Fit_PML_Curve(return_periods, amounts)
1 / Excess_Frequency(fit, amounts)

fit <- Fit_PML_Curve(return_periods, amounts, tail_alpha = 1.5,
                     truncation = 20000, truncation_type = "wd")
1 / Excess_Frequency(fit, amounts)

Fit a Collective Model to a Wishlist of References

Description

The function fits a collective model to a wishlist of references (expected layer losses and excess frequencies). The function allows to specify the family of the severity distribution that is used. Depending on this distribution family the function works slightly differently:

• For the severity distribution PiecewisePareto the function returns a PPP_Model that satisfies all the references

• For the severity distribution Pareto the function returns a PPP_Model that minimizes the squared relative deviations from the references

• For the severity distribution GenPareto the function returns a PGP_Model that minimizes the squared relative deviations from the references

Usage

Fit_References(
  Covers = NULL,
  Attachment_Points = NULL,
  Expected_Layer_Losses = NULL,
  Thresholds = NULL,
  Frequencies = NULL,
  model_threshold = min(c(Attachment_Points, Thresholds)),
  default_alpha = 2,
  dispersion = 1,
  alpha_max = 100,
  severity_distribution = "PiecewisePareto",
  ignore_inconsistent_references = FALSE
)

Arguments

Value

For severity_distribution = "PiecewisePareto" or "Pareto": A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

• FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

• t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

• alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

• truncation Numerical. 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.

• dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

• Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

• Comment Character. Information on whether the fit was successful

For severity_distribution = "GenPareto": A PGP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

• FQ Expected claim count of the collective model.

• t Numeric. Threshold of the Pareto distribution.

• alpha_ini Numeric. Initial Pareto alpha (at t).

• alpha_tail Numeric. Tail Pareto alpha.

• truncation If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

• dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

• Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

• Comment Character. Information on whether the fit was successful

Examples

covers <- c(1000, 1000, 1000)
att_points <- c(1000, 2000, 5000)
exp_losses <- c(100, 50, 10)
thresholds <- c(4000, 10000)
fqs <- c(0.04, 0.005)
fit <- Fit_References(covers, att_points, exp_losses, thresholds, fqs)
Layer_Mean(fit, covers, att_points)
Excess_Frequency(fit, thresholds)
fit <- Fit_References(covers, att_points, exp_losses, thresholds, fqs,
                      severity_distribution = "Pareto")
Layer_Mean(fit, covers, att_points)
Excess_Frequency(fit, thresholds)
fit <- Fit_References(covers, att_points, exp_losses,
                      severity_distribution = "GenPareto")
Layer_Mean(fit, covers, att_points)

Layer Mean of the generalized Pareto Distribution

Description

Calculates the expected loss of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_Mean(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Value

The expected loss of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_Mean(4000, 1000, 1000, 1, 3)
GenPareto_Layer_Mean(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Mean(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Mean(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_Mean(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

Second Layer Moment of the Generalized Pareto Distribution

Description

Calculates the second moment of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_SM(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Value

The second moment of the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_SM(4000, 1000, 1000, 1, 2)
GenPareto_Layer_SM(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_SM(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_SM(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_SM(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

Layer Variance of the Generalized Pareto Distribution

Description

Calculates the variance of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_Var(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Value

Variance of the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_Var(4000, 1000, 1000, 1, 2)
GenPareto_Layer_Var(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Var(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Var(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_Var(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

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

Description

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

Usage

GenPareto_ML_Estimator_Alpha(
  losses,
  t,
  truncation = NULL,
  reporting_thresholds = NULL,
  is.censored = NULL,
  weights = NULL,
  alpha_min = 0.001,
  alpha_max = 10
)

Arguments

Value

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)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a collective model

Usage

Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PGP_Model

Usage

## S3 method for class 'PGP_Model'
Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PGPM, Example1_Cov, Example1_AP)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PPP_Model

Usage

## S3 method for class 'PPP_Model'
Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a collective model

Usage

Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PGPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a collective model

Usage

Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PGPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Local Pareto Alpha

Description

Calculates the local Pareto alpha of the normal, lognormal and gamma distribution

Usage

Local_Pareto_Alpha(x, distribution, ...)

Arguments

Value

Local Pareto alpha of the selected distribution at x

References

Riegel, U. (2008) Generalizations of common ILF models. Blaetter der DGVFM 29: 45–71

Examples

x <- 1:10 * 1e6
Local_Pareto_Alpha(x, "norm", mean = 5e6, sd = 2e6)
Local_Pareto_Alpha(x, "lnorm", meanlog = 0, sdlog = 4)
Local_Pareto_Alpha(x, "gamma", shape = 5, rate = 1e-6)
Local_Pareto_Alpha(x, "weibull", shape = 0.5, scale = 1e6)
Local_Pareto_Alpha(x, "exp", rate = 1e-6)
Local_Pareto_Alpha(x, "Pareto", t = 1e6, alpha = 1, truncation = 20e6)
Local_Pareto_Alpha(x, "GenPareto", t = 1e6, alpha_ini = 1, alpha_tail = 2)
Local_Pareto_Alpha(x, "PiecewisePareto", t = c(1e6, 3e6, 5e6), alpha = c(1, 2, 3),
                       truncation = 20e6, truncation_type = "wd")

PGP_Model (Collective Panjer & Generalized Pareto Model) Object

Description

Constructor function for the PGP_Model object

Usage

PGP_Model(
  FQ = NULL,
  t = NULL,
  alpha_ini = NULL,
  alpha_tail = NULL,
  truncation = NULL,
  dispersion = 1,
  Status = 0,
  Comment = "OK"
)

Arguments

Examples

PGPM <- PGP_Model(2, t = 1000, alpha_ini = 1, alpha_tail = 2 , dispersion = 2)
PGPM

PPP_Model (Collective Panjer & Piecewise Pareto Model) Object

Description

Constructor function for the PPP_Model object

Usage

PPP_Model(
  FQ = NULL,
  t = NULL,
  alpha = NULL,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1,
  Status = 0,
  Comment = "OK"
)

Arguments

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PPP_Model

Usage

PPP_Model_Excess_Frequency(x, PPP_Model)

Arguments

Value

The expected frequency in excess of x for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PPP_Model. This function is deprecated. Use Layer_Mean instead.

Usage

PPP_Model_Exp_Layer_Loss(Cover, AttachmentPoint, PPP_Model)

Arguments

Value

The expected loss of the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PPP_Model

Usage

PPP_Model_Layer_Sd(Cover, AttachmentPoint, PPP_Model)

Arguments

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PPP_Model. This function is deprecated. Use Layer_Var instead.

Usage

PPP_Model_Layer_Var(Cover, AttachmentPoint, PPP_Model)

Arguments

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Simulate Losses with a PPP_Model

Description

Simulates losses of a PPP_Model

Usage

PPP_Model_Simulate(n, PPP_Model)

Arguments

Value

A matrix where row k contains the simulated losses of the kth simulation.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)

Distribution Function of the Pareto Distribution

Description

Calculates the cumulative distribution function of a Pareto distribution. This function is deprecated. Use pPareto instead.

Usage

Pareto_CDF(x, t, alpha, truncation = NULL)

Arguments

Value

Distribution function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
pPareto(x, 1000, 2)
pPareto(x, 1000, 2, truncation = 5000)

Pareto Extrapolation

Description

Uses a Pareto distribution to derive the expected loss of a layer from the expected loss of another layer

Usage

Pareto_Extrapolation(
  Cover_1,
  AttachmentPoint_1,
  Cover_2,
  AttachmentPoint_2,
  alpha,
  ExpLoss_1 = NULL,
  truncation = NULL
)

Arguments

Value

The expected loss of the layer Cover_2 xs AttachmentPoint_2 given that Cover_1 xs AttachmentPoint_1 has expected loss ExpLoss_1 and assuming a (truncated) Pareto distribution with parameters t and alpha. If missing then ExpLoss_1 == 1 is assumed.

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, ExpLoss_1 = 100)
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2) * 100
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, truncation = 5000, ExpLoss_1 = 100)
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, truncation = 5000) * 100

Pareto Alpha Between a Frequency and a Layer

Description

Finds the Pareto alpha between an excess frequency and the expected loss of a layer

Usage

Pareto_Find_Alpha_btw_FQ_Layer(
  Threshold,
  Frequency,
  Cover,
  AttachmentPoint,
  ExpLoss,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Value

The Pareto alpha between the expected number of claims Frequency excess Threshold and the layer Cover xs AttachmentPoint with expected loss ExpLoss

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_FQ_Layer(1000, 1, 1000, 1000, 500)
Pareto_Find_Alpha_btw_FQ_Layer(1000, 1, 1000, 1000, 500, truncation = 5000)

Pareto Alpha Between Two Frequencies

Description

Finds the Pareto alpha between two excess frequencies

Usage

Pareto_Find_Alpha_btw_FQs(
  Threshold_1,
  Frequency_1,
  Threshold_2,
  Frequency_2,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Value

The Pareto alpha between the expected number of claims Frequency_1 excess Threshold_1 and the expected number of claims Frequency_2 excess Threshold_2

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_FQs(1000, 1, 2000, 0.5)
Pareto_Find_Alpha_btw_FQs(1000, 1, 2000, 0.5, truncation = 5000)

Pareto Alpha Between Two Layers

Description

Finds the Pareto alpha between two layers

Usage

Pareto_Find_Alpha_btw_Layers(
  Cover_1,
  AttachmentPoint_1,
  ExpLoss_1,
  Cover_2,
  AttachmentPoint_2,
  ExpLoss_2,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Value

The Pareto alpha between the layer Cover_1 xs AttachmentPoint_1 with expected loss ExpLoss_1 and the layer Cover_2 xs AttachmentPoint_2 with expected loss ExpLoss_2

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_Layers(100, 100, 100, 200, 200, 50)
Pareto_Find_Alpha_btw_Layers(100, 100, 100, 200, 200, 50, truncation = 500)

Layer Mean of the Pareto Distribution

Description

Calculates the expected loss of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_Mean(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Value

The expected loss of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_Mean(4000, 1000, 2)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_Mean(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

Second Layer Moment of the Pareto Distribution

Description

Calculates the second moment of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_SM(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Value

The second moment of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_SM(4000, 1000, 2)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_SM(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

Layer Variance of the Pareto Distribution

Description

Calculates the variance of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_Var(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Value

The variance of the(truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_Var(4000, 1000, 2)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_Var(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

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

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)

Density of the Pareto Distribution

Description

Calculates the density function of the Pareto distribution. This function is deprecated. Use dPareto instead.

Usage

Pareto_PDF(x, t, alpha, truncation = NULL)

Arguments

Value

Density function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
dPareto(x, 1000, 2)
dPareto(x, 1000, 2, truncation = 5000)

Distribution Function of the Piecewise Pareto Distribution

Description

Calculates the cumulative distribution function of a Piecewise Pareto Distribution. This function is deprecated. Use pPiecewisePareto instead.

Usage

PiecewisePareto_CDF(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

Value

Distribution function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
pPiecewisePareto(x, t, alpha)
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Layer Mean of the Piecewise Pareto Distribution

Description

Calculates the expected loss of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_Mean(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Value

The expected loss of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

Second Layer Moment of the Piecewise Pareto Distribution

Description

Calculates the second moment of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_SM(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Value

The second moment of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

Layer Variance of the Piecewise Pareto Distribution

Description

Calculate the variance of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_Var(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Value

The variance of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_Var(4000, 1000, t, alpha)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha) - PiecewisePareto_Layer_Mean(4000, 1000, t, alpha)^2
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

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

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))

Match a Tower of Expected Layers Losses

Description

Matches the expected losses of a tower of reinsurance layers using a piecewise Pareto severity

Usage

PiecewisePareto_Match_Layer_Losses(
  Attachment_Points,
  Expected_Layer_Losses,
  Unlimited_Layers = FALSE,
  Frequencies = NULL,
  FQ_at_lowest_AttPt = NULL,
  FQ_at_highest_AttPt = NULL,
  TotalLoss_Frequencies = NULL,
  minimize_ratios = TRUE,
  Use_unlimited_Layer_for_FQ = TRUE,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1,
  tolerance = 1e-10,
  alpha_max = 100,
  merge_tolerance = 1e-06,
  RoL_tolerance = 1e-06
)

Arguments

Value

A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

• FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

• t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

• alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

• truncation Numerical. 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.

• dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

• Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

• Comment Character. Information on whether the fit was successful

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

AP <- Example1_AP
EL <- Example1_EL
PiecewisePareto_Match_Layer_Losses(AP, EL)
EL_unlimited <- rev(cumsum(rev(Example1_EL)))
PiecewisePareto_Match_Layer_Losses(AP, EL_unlimited, Unlimited_Layers = TRUE)
PiecewisePareto_Match_Layer_Losses(AP, EL, FQ_at_lowest_AttPt = 0.5)
Example1_FQ <- c(0.3, 0.15, 0.08, 0.02, 0.005)
PiecewisePareto_Match_Layer_Losses(AP, EL, Frequencies = Example1_FQ)

Density of the Piecewise Pareto Distribution

Description

Calculates the density function of the piecewise Pareto distribution. This function is deprecated. Use dPiecewisePareto instead.

Usage

PiecewisePareto_PDF(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

Value

Density function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
dPiecewisePareto(x, t, alpha)
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Simulate Losses with a Collective Model

Description

Simulates losses with a collective model

Usage

Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)

Simulate Losses with a PGP_Model

Description

Simulates losses with a PGP_Model

Usage

## S3 method for class 'PGP_Model'
Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Simulate_Losses(PGPM, 100)

Simulate Losses with a PPP_Model

Description

Simulates losses with a PPP_Model

Usage

## S3 method for class 'PPP_Model'
Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)

Density of the generalized Pareto Distribution

Description

Calculates the density function of the generalized Pareto distribution

Usage

dGenPareto(x, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

Value

Density function of the Pareto distribution with parameters t, alpha_ini and alpha_tail evaluated at x

Examples

x <- 0:10 * 1000
dGenPareto(x, 1000, 1, 3)
dGenPareto(x, 1000, 1, 3, truncation = 5000)

Density of the Pareto Distribution

Description

Calculates the density function of the Pareto distribution

Usage

dPareto(x, t, alpha, truncation = NULL)

Arguments

Value

Density function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
dPareto(x, 1000, 2)
dPareto(x, 1000, 2, truncation = 5000)

Density of the Piecewise Pareto Distribution

Description

Calculates the density function of the piecewise Pareto distribution

Usage

dPiecewisePareto(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

Value

Density function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
dPiecewisePareto(x, t, alpha)
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Check if an object is a PGP_Model

Description

Checks if the class of an object is 'PGP_Model'

Usage

is.PGP_Model(x)

Arguments

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

PGPM$alpha_tail <- -2
is.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

Check if an object is a PPP_Model

Description

Checks if the class of an object is 'PPP_Model'

Usage

is.PPP_Model(x)

Arguments

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM
is.valid.PPP_Model(PPPM)

PPPM$alpha <- 2
is.valid.PPP_Model(PPPM)
is.PPP_Model(PPPM)

Check if an object is a valid PGP_Model

Description

Checks if an object is a PGP_Model object and whether it is valid for the use in functions like Layer_Mean

Usage

is.valid.PGP_Model(x, comment = FALSE)

Arguments

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

PGPM$alpha_tail <- -2
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

Check if an object is a valid PPP_Model

Description

Checks if an object is a PPP_Model object and whether it is valid for the use in functions like Layer_Mean

Usage

is.valid.PPP_Model(x, comment = FALSE)

Arguments

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM
is.valid.PPP_Model(PPPM)
is.valid.PPP_Model(PPPM, comment = TRUE)

PPPM$alpha <- 2
is.valid.PPP_Model(PPPM)
is.valid.PPP_Model(PPPM, comment = TRUE)

Distribution Function of the generalized Pareto Distribution

Description

Calculates the cumulative distribution function of a generalized Pareto distribution

Usage

pGenPareto(x, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

Value

Distribution function of the generalized Pareto distribution with parameters t, alpha_ini and alpha_tail evaluated at x

Examples

x <- 0:10 * 1000
pGenPareto(x, 1000, 1, 3)
pGenPareto(x, 1000, 1, 3, truncation = 5000)

Distribution Function of the Pareto Distribution

Description

Calculates the cumulative distribution function of a Pareto distribution

Usage

pPareto(x, t, alpha, truncation = NULL)

Arguments

Value

Distribution function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
pPareto(x, 1000, 2)
pPareto(x, 1000, 2, truncation = 5000)

Distribution Function of the Piecewise Pareto Distribution

Description

Calculates the cumulative distribution function of a Piecewise Pareto Distribution

Usage

pPiecewisePareto(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

Value

Distribution function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
pPiecewisePareto(x, t, alpha)
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Print a PGP_Model Object (Collective Panjer & Generalized Pareto Model) Object

Description

Print method for PGP_Model objects

Usage

## S3 method for class 'PGP_Model'
print(x, ...)

Arguments

Print a PPP_Model Object (Collective Panjer & Piecewise Pareto Model) Object

Description

Print method for PPP_Model objects

Usage

## S3 method for class 'PPP_Model'
print(x, ...)

Arguments

Quantile Function of the generalized Pareto Distribution

Description

Calculates the quantile function of a generalized Pareto distribution

Usage

qGenPareto(p, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

Value

Quantile function of the Pareto distribution with parameters t, alpha_ini and alpha_tail, evaluated at p

Examples

p <- 0:10 * 0.1
qGenPareto(p, 1000, 2, 3)
qGenPareto(p, 1000, 2, 3, truncation = 5000)

Quantile Function of the Pareto Distribution

Description

Calculates the quantile function of a Pareto distribution

Usage

qPareto(p, t, alpha, truncation = NULL)

Arguments

Value

Quantile function of the Pareto distribution with parameters t and alpha, evaluated at p

Examples

p <- 0:10 * 0.1
qPareto(p, 1000, 2)
qPareto(p, 1000, 2, truncation = 5000)

Quantile Function of the Piecewise Pareto Distribution

Description

Calculates the quantile function of a piecewise Pareto distribution

Usage

qPiecewisePareto(p, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

Value

Quantile function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at p

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
p <- 0:10 * 0.1
qPiecewisePareto(p, t, alpha)
qPiecewisePareto(p, t, alpha, truncation = 5000, truncation_type = "lp")
qPiecewisePareto(p, t, alpha, truncation = 5000, truncation_type = "wd")

Simulation of the generalized Pareto Distribution

Description

Generates random deviates of a generalized Pareto distribution

Usage

rGenPareto(n, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

Value

A vector of n samples from the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail

Examples

rGenPareto(100, 1000, 2, 3)
rGenPareto(100, 1000, 2, 3, truncation = 2000)
rGenPareto(100, t = c(1, 10, 100, 1000), alpha_ini = 1, alpha_tail = c(2, 5))

Simulation of the Pareto Distribution

Description

Generates random deviates of a Pareto distribution

Usage

rPareto(n, t, alpha, truncation = NULL)

Arguments

Value

A vector of n samples from the (truncated) Pareto distribution with parameters t and alpha

Examples

rPareto(100, 1000, 2)
rPareto(100, 1000, 2, truncation = 2000)
rPareto(100, t = c(1, 10, 100, 1000, 10000), alpha = c(1, 2, 4, 8, 16))

Simulation of the Piecewise Pareto Distribution

Description

Generates random deviates of a piecewise Pareto distribution

Usage

rPiecewisePareto(
  n,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp",
  scale_pieces = NULL
)

Arguments

Value

A vector of n samples from the (truncated) piecewise Pareto distribution with parameter vectors t and alpha

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
rPiecewisePareto(100, t, alpha)
rPiecewisePareto(100, t, alpha, truncation = 5000)
rPiecewisePareto(100, t, alpha, truncation = 5000, truncation_type = "lp")
rPiecewisePareto(100, t, alpha, truncation = 5000, truncation_type = "wd")