FTG Density Function

Description

This function computes the density of the full-tail gamma with the input sample data. The expression for the density used is:

g(x; \alpha, \theta, \rho) = \frac{\rho^{\alpha}}{\sigma}\left(\rho + \frac{x}{\sigma}\right)^{\alpha - 1}\exp\left(-\left(\rho + \frac{x}{\sigma}\right)\right)/\Gamma(\alpha, \rho).

Usage

dFTG(x, threshold, scale, shape)

Arguments

Value

Gives the density of the FTG. The length of the result is determined by the length of x.

References

del Castillo, Joan & Daoudi, Jalila & Serra, Isabel. (2012). The full-tails gamma distribution applied to model extreme values. ASTIN Bulletin. <doi:10.1017/asb.2017.9>.

Examples

a <- 0.3
t <- 0.3
r <- 0.8
n <- 1000
sample <- rFTG(n, a, t, r)
x <- seq(min(sample), max(sample), length.out = 200)
d <- dFTG(x, a, t, r)
hist(sample, breaks = "FD", probability = TRUE)
lines(x, d, col = "red")

TailW Density function

Description

This function computes the density of the tailW with the input sample data. The expression for the density used is:

f(x, \alpha, \beta, \nu) = \alpha \beta (x + \nu)^{\beta -1} \exp(-\alpha(x + \nu)^{\beta} + \alpha \nu^{\beta})

Usage

dtailw(x, threshold, scale, shape)

Arguments

Value

Gives the density of the TailW. The length of the result is determined by the length of x.

Examples

# Generate random deviates from a weibull tail and plot the theoretical density.
scale <- 2
shape <- 1
threshold <- 1
x_seq <- seq(threshold, 5, length.out = 500)
theo_density <- dtailw(x_seq, threshold = threshold, scale = scale, shape = shape)
sample <- rtailw(500, threshold = threshold, scale = scale, shape = shape)
hist(sample, probability = TRUE)
lines(x = x_seq, y = theo_density, col = "red")

TailW Maximum Likelihood Estimation

Description

Maximum Likelihood Estimation of the tails by fitting a tailW or a FTG.

Usage

fittail(sample, dist = "TailW")

Arguments

Value

Gives a list of the estimated parameters fo the function fitted. For the TailW it returns, scale and shape. Fot the FTG it returns the parameters scale, shape, and threshold.

Examples

scale <- 2
shape <- 1
threshold <- 1
s <- rtailw(1000, threshold = threshold , scale = scale, shape = shape)
fits <- fittail(s, dist = "TailW")
x_seq <- seq(threshold, max(s), length.out = 500)
theo_density <- dtailw(x_seq, threshold = threshold, scale = fits$scale, shape = fits$shape)
hist(s, probability = TRUE, breaks = "FD")
lines(x = x_seq, y = theo_density, col = "red")

FTG Log-likelihood Function

Description

This function computes the loglikelihood of the full-tail gamma with the input sample data. The expression used is:

l(x; \alpha, \sigma, \rho) = -n\left(\log\Gamma(\alpha, \rho) + \log(\sigma) -\alpha \log(\rho) - \frac{\alpha - 1}{n}\sum_{i = 1}^{n}\log\left(1 + \frac{x_{i}}{\sigma}\right) + \frac{\rho}{n} \sum_{i = 1}^{n}\left(1 + \frac{x_{i}}{\sigma}\right)\right)

Usage

lFTG(x, threshold, scale, shape)

Arguments

Value

Gives the log-likelihood of the FTG. The length of the result is determined by the length of x.

References

del Castillo, Joan & Daoudi, Jalila & Serra, Isabel. (2012). The full-tails gamma distribution applied to model extreme values. ASTIN Bulletin. <doi:10.1017/asb.2017.9>.

Examples

lFTG(1,1,1,1)

TailW Log-likelihood function

Description

This function computes the log-likelihood of the tailW with the input sample data.

l(x;\alpha,\beta) = n(\log(\alpha)+\log(\beta))+(\beta-1)\sum_{i=1}^{n}\log(x+\nu)-\alpha\sum_{i=1}^{n}((x+\nu)^\beta-\nu^\beta)

Usage

ltailw(x, threshold, scale, shape)

Arguments

Value

Gives the log-likelihood of the TailW. The length of the result is determined by the length of x.

Examples

ltailw(1,1,1,1)

FTG Probability Function

Description

This function computes the probability of the full-tail gamma with the input sample data. The expression for the probability used is:

G(x; \alpha, \theta, \rho) = 1 - \Gamma\left(\alpha, \rho\left(1 + \frac{x}{\sigma}\right)\right)/\Gamma(\alpha, \rho).

Usage

pFTG(x, threshold, scale, shape)

Arguments

Value

Gives the distribution function of the FTG. The length of the result is determined by the length of x.

References

del Castillo, Joan & Daoudi, Jalila & Serra, Isabel. (2012). The full-tails gamma distribution applied to model extreme values. ASTIN Bulletin. <doi:10.1017/asb.2017.9>.

Examples

pFTG(1,1,1,1)

TailW Probability Function

Description

This function computes the cumulative density function of the tailW with the input sample data.

F(x,\alpha,\beta,\nu)=1-\exp(-\alpha(x+\nu)^\beta+\alpha\nu^\beta).

Usage

ptailw(x, threshold, scale, shape)

Arguments

Value

Gives the distribution function of the TailW. The length of the result is determined by the length of x.

Examples

# Using the probability function to show the fitting.
samp <- rtailw(1000, 1, 2, 3)
emp_cdf <- ecdf(samp)(samp)
pars <- fittail(samp, dist = "TailW")
x_seq <- seq(min(samp), max(samp), length.out = 250)
p <- ptailw(x_seq, threshold = 1, scale = pars$scale, shape = pars$shape)
plot(samp, 1-emp_cdf, log = "y")
lines(x_seq, 1-p, col = "red")

FTG Quantile function

Description

This function computes the quantiles of the full-tail gamma with the input sample data.

Usage

qFTG(p, threshold, scale, shape, interval)

Arguments

Value

Gives the quantiles of the FTG. The length of the result is determined by the length of x.

References

del Castillo, Joan & Daoudi, Jalila & Serra, Isabel. (2012). The full-tails gamma distribution applied to model extreme values. ASTIN Bulletin. <doi:10.1017/asb.2017.9>.

Examples

qFTG(0.5,1,1,1, c(0,10))

Quantile function

Description

This function computes the quantile function of the tailW.

Q(p,\alpha, \beta, \nu) = \left(\frac{-\log(1 - p)}{\alpha} + \nu^\beta\right)^{1 / \beta}

Usage

qtailw(p, threshold, scale, shape)

Arguments

Value

Gives the quantiles of the TailW. The length of the result is determined by the length of x.

Examples

qtailw(0.5, 1, 1, 1)

FTG Random Sample Generation

Description

This function computes n random variates from full-tail gamma with a rejection method.

Usage

rFTG(n, threshold, scale, shape)

Arguments

Value

Gives random deviates of the FTG. The length of the result is determined by n.

References

del Castillo, Joan & Daoudi, Jalila & Serra, Isabel. (2012). The full-tails gamma distribution applied to model extreme values. ASTIN Bulletin. <doi:10.1017/asb.2017.9>.

Examples

x <- rFTG(100, 1, 1, 1)
hist(x, breaks = "FD")

TailW Random Sample Generation

Description

This function generates random deviates for the tailW distribution.

Usage

rtailw(n, threshold, scale, shape)

Arguments

Value

Gives random deviates of the TailW. The length of the result is determined by n.

Examples

x <- rtailw(1000, 1, 2, 3)
hist(x, breaks = "FD")