Package 'lognGPD'

Density of the lognormal-GPD mixture

Description

This function evaluates the lognormal-GPD mixture density function.

Usage

dlognGPD(x, p, mu, sigma, xi, beta)

Arguments

x	vector (nx1): points where the function is evaluated.
p	real, 0<p<1: prior probability
mu	real: log-mean of the truncated lognormal distribution.
sigma	positive real: log-standard deviation of the truncated lognormal distribution.
xi	real: shape parameter of the generalized Pareto distribution.
beta	positive real: scale parameter of the generalized Pareto distribution.

Value

ydens (n x 1) vector: numerical values of the lognormal - generalized Pareto mixture at x.

Examples

ydens <- dlognGPD(seq(0,20,length.out=500),.9,0,1,0.5,2)

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Density of the lognormal-Pareto spliced distribution

Description

This function evaluates the density of the continuous and differentiable version of the truncated lognormal-Pareto spliced distribution proposed by Scollnik (2007).

Usage

dlognPareto(x, sigma, xmin, alpha)

Arguments

x	vector (nx1): points where the function is evaluated.
sigma	positive real: log-standard deviation of the truncated lognormal distribution.
xmin	positive real: scale parameter of the Pareto distribution.
alpha	positive real: shape parameterof the Pareto distribution.

Details

To get a continuous and differentiable density, it is necessary to enforce constraints that reduce the number of free parameters of the model; in particular, the mixing weight and the log-mean of the lognormal distirbution are functions of the reamining parameters. See Scollnik (2007) for details.

Value

ysim (n x 1) vector: numerical values of the truncated lognormal-Pareto spliced distribution at x.

References

Scollnik DPM (2007). “On composite lognormal-Pareto models.” Scandinavian Actuarial Journal, 1, 20-33.

Examples

ysim <- dlognPareto(seq(0,20,length.out=500),1,5,2)

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Bootstrap standard errors for the MLEs of a lognormal-GPD mixture

Description

This function draws a bootstrap sample and uses it to estimate the parameters of a lognormal-Pareto mixture distribution. Since this is typically called by LPfitEM, see the help of LPfitEM for examples.

Usage

EMBoot(x, x0, y, maxiter)

Arguments

x	list: sequence of integers 1,...,K, where K is the mumber of datasets. Set x = 1 in case of a single dataset.
x0	numerical vector (5x1): initial values of the parameters p, $\mu$, $\sigma$, $\xi$, $\beta$.
y	numerical vector: observed sample.
maxiter	non-negative integer: maximum number of iterations of the EM algorithm.

Details

At each bootstrap replication, the mixture is estimated via the EM algorithm.

Value

Estimated parameters obtained from a bootstrap sample.

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Mixture estimation via EM

Description

This function estimates a static lognormal - generalized Pareto mixture by means of the EM algorithm. Optionally, bootstrap standard errors are computed via parallel computing.

Usage

EMlogngpdmix(x0, y, maxiter, nboot = 0)

Arguments

x0	numerical vector (5x1): initial values of the parameters p, $\mu$, $\sigma$, $\xi$, $\beta$.
y	vector: observed data.
maxiter	positive integer: maximum number of iterations of the EM algorithm.
nboot	positive integer: number of bootstrap replications for the computation of the standard errors (defaults to 0).

Value

A list with the following elements is returned: "p" = estimated value of p, "post" = posterior probabilities of all observations, "mu" = estimated value of $\mu$, "sigma " = estimated value of $\sigma$, "xi" = estimated value of $\xi$, "beta" = estimated value of $\beta$, "loglik" = maximimzed log-likelihood, "nit" = number of iterations, bootEst = matrix of parameter estimates at each bootstrap replications (only if nboot > 0). bootStd = bootstrap standard errors of each parameter (only if nboot > 0).

Examples

y <- rlognGPD(100,.9,0,1,0.5,2)
x0 <- c(.7,.2,1.3,.8,1.7)
res <- EMlogngpdmix(x0, y, 1000)

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Simulation of the lognormal-GPD mixture

Description

This function simulates a lognormal-GPD mixture.

Usage

rlognGPD(n, p, mu, sigma, xi, beta)

Arguments

n	positive integer: number of observations sampled.
p	real, 0<p<1: prior probability
mu	real: log-mean of the lognormal distribution.
sigma	positive real: log-standard deviation of the lognormal distribution.
xi	real: shape parameter of the generalized Pareto distribution.
beta	positive real: scale parameter of the generalized Pareto distribution.

Value

ysim (n x 1) vector: n random numbers from the lognormal - generalized Pareto mixture.

Examples

ysim <- rlognGPD(100,.9,0,1,0.5,2)

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Simulation of the lognormal-Pareto spliced distribution

Description

This function simulates the continuous and differentiable version of the truncated lognormal-Pareto spliced distribution proposed by Scollnik (2007).

Usage

rlognPareto(n, sigma, xmin, alphapar)

Arguments

n	positive integer: number of observations sampled.
sigma	positive real: log-standard deviation of the truncated lognormal distribution.
xmin	positive real: scale parameter of the Pareto distribution.
alphapar	positive real: shape parameterof the Pareto distribution.

Details

See Scollnik (2007) for details.

Value

ysim (nreps x 1) vector: nreps random numbers from the truncated lognormal-Pareto spliced distribution.

References

Scollnik DPM (2007). “On composite lognormal-Pareto models.” Scandinavian Actuarial Journal, 1, 20-33.

Examples

ysim <- rlognPareto(100,1,5,2)

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Weighted GPD log-likelihood

Description

This function evaluates the zero-mean generalized Pareto log-likelihood function computed with weighted observations.

Usage

weiGpdLik(x, y, post)

Arguments

x	numerical vector (2x1): values of the parameters $\xi$ and $\beta$.
y	numerical vector (nx1): observed data.
post	numerical vector (nx1) with elements in (0,1): weights of the observations (in the EM algorithm, posterior probabilities).

Value

llik real: numerical value of the log-likelihood function

Examples

y <- rlognGPD(100,.9,0,1,0.5,2)
x0 <- c(.7,.2,1.3,.8,1.7)
res <- EMlogngpdmix(x0, y, 1000)
llik <- weiGpdLik(c(res$beta,res$xi),y,res$post)

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