R/Distributions.R

In TReynkens/ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

# This file contains implementations of several distributions that are useful for EVT:
#
# The Burr distribution (type XII).
# The Extended Pareto Distribution (EPD).
# The Fréchet distribution.
# The Generalised Pareto Distribution (GPD).
# The Pareto distribution.
# 
# The truncated Burr distribution.
# The truncated exponential distribution.
# The truncated Fréchet distribution.
# The truncated GPD.
# The truncated log-normal distribution.
# The truncated Pareto distribution.
# The truncated Weibull distribution.

###########################################################################

# Pareto

dpareto <- function(x, shape, scale = 1, log = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  d <- ifelse(x >= scale, shape/scale*(scale/x)^(shape+1), 0)
  
  if (log) d <- log(d)
  
  return(d)
  
}

ppareto <- function(x, shape, scale = 1, lower.tail = TRUE, log.p = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  p <- ifelse(x >= scale, 1-(scale/x)^shape, 0)
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  
  return(p)
  
}

qpareto <- function(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (all(p >= 0 & p <= 1)) {
    
    return(scale*(1-p)^(-1/shape))
    
  } else {
    stop("p should be between 0 and 1.")
  }
  
}

rpareto <- function(n, shape, scale = 1) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  return(qpareto(runif(n), scale=scale, shape=shape))
}


###############################################################
# Truncated Pareto

dtpareto <- function(x, shape, scale=1, endpoint=Inf, log = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= scale) {
    stop("endpoint should be strictly larger than scale.")
  }
  
  d <- ifelse(x <= endpoint, 
              dpareto(x, shape=shape, scale=scale)/ppareto(endpoint, shape=shape, scale=scale),
              0)
  if (log) d <- log(d)
  
  return(d)
}

ptpareto <- function(x, shape, scale=1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= scale) {
    stop("endpoint should be strictly larger than scale.")
  }
 
  p <- ifelse(x <= endpoint, 
              ppareto(x, shape=shape, scale=scale)/ppareto(endpoint, shape=shape, scale=scale),
              1)
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
  
  #return(ifelse(x >= scale,(1-(scale/x)^shape)/(1-(scale/endpoint)^shape),0))
}

qtpareto <- function(p, shape, scale=1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= scale) {
    stop("endpoint should be strictly larger than scale.")
  }
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (all(p >= 0 & p <= 1)) {

    return(qpareto(p*ppareto(endpoint, shape=shape, scale=scale), shape=shape, scale=scale))
    #return(scale*(1-p*(1-(scale/endpoint)^shape))^(-1/shape))
  } else {
    stop("p should be between 0 and 1.")
  }
  
}

rtpareto <- function(n, shape, scale=1, endpoint=Inf) {
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= scale) {
    stop("endpoint should be strictly larger than scale.")
  }
  
  return(qtpareto(runif(n), scale=scale, shape=shape, endpoint=endpoint))
}


######################
# Generalised Pareto distribution (GPD)

eps <- 10^(-14)


dgpd <- function(x, gamma, mu = 0, sigma, log = FALSE) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  if (abs(gamma) < eps) {
    d <- ifelse(x >= mu, 1/sigma * exp(-(x-mu)/sigma), 0)
  } else {
    d <- ifelse(x >= mu, 1/sigma * (1+gamma*(x-mu)/sigma)^(-1/gamma-1), 0)
  }
 
  if (gamma < -eps) {
    d[x > mu-sigma/gamma] <- 0
  } 
  
  if (log) d <- log(d)
  
  return(d)
}

pgpd <- function(x, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  if (abs(gamma) < eps) {
    p <- ifelse(x >= mu, 1-exp(-(x-mu)/sigma), 0)
  } else {
    p <- ifelse(x >= mu, 1-(1+gamma*(x-mu)/sigma)^(-1/gamma), 0)
  }
  
  if (gamma < -eps) {
    p[x >= mu-sigma/gamma] <- 1
  }
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  
  return(p)
}


qgpd <- function(p, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE) {
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (!all(p >= 0 & p <= 1)) {
    stop("p should be between 0 and 1.")
  }
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  

  if (abs(gamma) < eps) {
    q <- mu - sigma * log(1-p)
  } else {
    q <- mu + sigma/gamma * ((1-p)^(-gamma) - 1)
  }
  
  return(q)
}


rgpd <- function(n, gamma, mu = 0, sigma) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  qgpd(runif(n), gamma=gamma, mu=mu, sigma=sigma)
}

######################
# Truncated GPD

dtgpd <- function(x, gamma, mu = 0, sigma, endpoint=Inf, log = FALSE) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  d <- ifelse(x <= endpoint, 
              dgpd(x, gamma=gamma, mu=mu, sigma=sigma)/pgpd(endpoint, gamma=gamma, mu=mu, sigma=sigma),
              0)
  
  if (log) d <- log(d)
  
  return(d)
  
}

ptgpd <- function(x, gamma, mu = 0, sigma, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  if (endpoint <= mu) {
    stop("endpoint should be strictly larger than mu.")
  }
  
  p <- ifelse(x <= endpoint, 
              pgpd(x, gamma=gamma, mu=mu, sigma=sigma)/pgpd(endpoint, gamma=gamma, mu=mu, sigma=sigma),
              1)
  
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  
  return(p)
}


qtgpd <- function(p, gamma, mu = 0, sigma, endpoint = Inf, lower.tail = TRUE, log.p = FALSE) {
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (!all(p >= 0 & p <= 1)) {
    stop("p should be between 0 and 1.")
  }
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  if (endpoint <= mu) {
    stop("endpoint should be strictly larger than mu.")
  }

  q <- qgpd(p*pgpd(endpoint, gamma=gamma, mu=mu, sigma=sigma), gamma=gamma, mu=mu, sigma=sigma)
  
  return(q)
   
}


rtgpd <- function(n, gamma, mu = 0, sigma, endpoint = Inf) {
  
  if (sigma <= 0) {
    stop("sigma should be strictly positive.")
  }
  
  if (endpoint <= mu) {
    stop("endpoint should be strictly larger than mu.")
  }
  
  
  qtgpd(runif(n), gamma=gamma, mu=mu, sigma=sigma, endpoint=endpoint)
}




###############################################################
# Extended Pareto distribution

# Check input for EPD distribution
.EPDinput <- function(y, gamma, kappa, tau, kappaTau = TRUE) {
  
  # Check if arguments are numeric
  if (!is.numeric(gamma)) {
    stop("gamma should be numeric.")
  }
  
  if (!is.numeric(kappa)) {
    stop("kappa should be numeric.")
  }
  
  if (!is.numeric(tau)) {
    stop("tau should be numeric.")
  }
  
  # Check if right sign
  if (any(tau >= 0)) {
    stop("tau should be strictly negative.")
  }
  
  if (any(gamma <= 0)) {
    stop("gamma should be strictly positive.")
  }
  
  if (kappaTau) {
    if (any(kappa <= pmax(-1, 1/tau))) {
      stop("kappa should be larger than max(-1,1/tau).")
    }
  }
  
  # Check if correct length
  ly <- length(y)
  lg <- length(gamma)
  lk <- length(kappa)
  lt <- length(tau)
  
  l <- c(ly, lg, lk, lt)
  
  # Indices of lengths larger than 1
  ind <- which(l > 1)
  
  if (length(ind) > 1) {
    # Check that lengths larger than 1 are equal
    if (!length(unique(l[ind])) == 1) {
      stop("All input arguments should have length 1 or equal length.")
    }
  }
  
  
}
# Density of an extended Pareto distribution
depd <- function(x, gamma, kappa, tau = -1, log = FALSE) {
  
  # Check input
  .EPDinput(x, gamma, kappa, tau, kappaTau = TRUE)
  
  # Compute density
  d <- 1 / (gamma*x^(1/gamma+1)) * (1+kappa*(1-x^tau))^(-1/gamma-1) * 
    (1+kappa*(1-(1+tau)*x^tau))
  # Formula is not valid for values below 1
  d[x <= 1] <- 0
  
  if (log) d <- log(d)
  
  return(d)
}

# CDF of an extended Pareto distribution
pepd <- function(x, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE) {
  
  # Check input
  .EPDinput(x, gamma, kappa, tau, kappaTau = FALSE)
  
  # Compute probabilities
  p <- 1 - (x * (1+kappa*(1-x^tau)))^(-1/gamma) 
  # Formula is not valid for values below 1
  p[x <= 1] <- 0
  
  # Problems when condition not satisfied
  if (any(kappa <= pmax(-1, 1/tau))) {
    if (length(kappa) > 1 | length(tau) > 1) {
      p[kappa <= pmax(-1, 1/tau)] <- NA
    } else {
      p <- NA
    }
  }
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
}

# Quantile function of EPD
qepd <-  function(p, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE) {
  
  # Check input
  .EPDinput(p, gamma, kappa, tau, kappaTau = TRUE)
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  if (any(p < 0 | p > 1)) {
    stop("p should be between 0 and 1.")
  }
  
  # Compute quantiles numerically
  l <- length(p)
  Q <- numeric(l)
  
  # Take 10 as endpoint for interval to search over unless not large enough
  endpoint <- 10
  
  if (any(p < 1)) {
    
    mx <- max(p[p < 1])
    
    while (pepd(endpoint, gamma, kappa, tau) <= mx) {
      endpoint <- endpoint*10
    }
    
  }
 
  
  
  for (i in 1:l) {

    if (p[i] < .Machine$double.eps) {
      # p=0 case
      Q[i] <- 1
      
    } else if (abs(p[i]-1) > .Machine$double.eps) {
      # 0<p<1 case
      
      # Function to minimise
      f <- function(x) {
        ((1-p[i])^(-gamma) - x*(1+kappa*(1-x^tau)))^2
      }
      # If minimising fails return NA
      Q[i] <- tryCatch(optimise(f, lower=1, upper=endpoint)$minimum, error=function(e) NA) 

    } else {
      # p=1 case
      Q[i] <- Inf
    }
    
  }

  return(Q)
}



# Random number generation for EPD
repd <-  function(n, gamma, kappa, tau = -1) {
  
  # Rely on input checking in qepd
  
  # Generate random numbers
  return(qepd(runif(n), gamma=gamma, kappa=kappa, tau=tau))
}


###############################################################
# Burr (type XII)

dburr <- function(x, alpha, rho, eta = 1, log = FALSE) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  d <- ifelse(x > 0, alpha / eta * ((eta+x^(-rho*alpha))/eta)^(1/rho-1) * x^(-rho*alpha-1), 0)
  
  if (log) d <- log(d)
  
  return(d)
  
}


pburr <- function(x, alpha, rho, eta = 1, lower.tail = TRUE, log.p = FALSE) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  p <- ifelse(x > 0, 1-((eta+x^(-rho*alpha))/eta)^(1/rho), 0)
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
  
}

qburr <- function(p, alpha, rho, eta = 1, lower.tail = TRUE, log.p = FALSE) {
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  if (all(p >= 0 & p <= 1)) {
    return( (eta*((1-p)^rho-1))^(-1/(rho*alpha)) )
    
  } else {
    stop("p should be between 0 and 1.")
  }
  
}

rburr <- function(n, alpha, rho, eta = 1) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  return(qburr(runif(n), rho=rho, alpha=alpha, eta=eta))
}


###############################################################
# Truncated Burr


dtburr <- function(x, alpha, rho, eta = 1, endpoint=Inf, log = FALSE) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  d <- ifelse(x <= endpoint, 
              dburr(x, alpha=alpha, rho=rho, eta=eta)/pburr(endpoint, alpha=alpha, rho=rho, eta=eta), 
              0)
  
  if (log) d <- log(d)
  
  return(d)
  
}


ptburr <- function(x, alpha, rho, eta = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  #   A = 1-(1+endpoint^(-rho*alpha))^(1/rho)
  #   return(ifelse(x > 0 & x < endpoint,(1-(1+x^(-rho*alpha))^(1/rho))/A,0))
  
  p <- ifelse(x <= endpoint, 
              pburr(x, alpha=alpha, rho=rho, eta=eta)/pburr(endpoint, alpha=alpha, rho=rho, eta=eta), 
              1)
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
  
}


qtburr <- function(p, alpha, rho, eta = 1, endpoint = Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  if (all(p >= 0 & p <= 1)) {
    #     A = 1-(1+endpoint^(-rho*alpha))^(1/rho)
    #     return(((1-p*A)^rho-1)^(-1/(rho*alpha)))
    return( qburr(p*pburr(endpoint, alpha=alpha, rho=rho, eta=eta), alpha=alpha, rho=rho, eta=eta) )
    
  } else {
    stop("p should be between 0 and 1.")
  }
  
}

rtburr <- function(n, alpha, rho, eta=1, endpoint=Inf) {
  
  if (alpha <= 0) {
    stop("alpha should be strictly positive.")
  }
  
  if (rho >= 0) {
    stop("rho should be strictly negative")
  }
  
  if (eta <= 0) {
    stop("eta should be strictly positive.")
  }
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly larger than 0.")
  }
  
  return(qtburr(runif(n), alpha=alpha, rho=rho, eta=eta, endpoint=endpoint))
}

###############################################################
# Truncated log-normal

dtlnorm <- function(x, meanlog = 0, sdlog = 1, endpoint=Inf, log = FALSE) {
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  d <- ifelse(x <= endpoint, 
              dlnorm(x, meanlog=meanlog, sdlog=sdlog)/plnorm(endpoint, meanlog=meanlog, sdlog=sdlog), 
              0)
  
  if (log) d <- log(d)
  
  return(d)
  #return(dnorm((log(x)-meanlog)/sdlog)/pnorm((log(endpoint)-meanlog)/sdlog))
}

ptlnorm <- function(x, meanlog = 0, sdlog = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {

  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  p <- ifelse(x <= endpoint, 
              plnorm(x, meanlog=meanlog, sdlog=sdlog)/plnorm(endpoint, meanlog=meanlog, sdlog=sdlog), 
              1)

  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
  #ifelse(x <= 0, 0, pnorm((log(x)-meanlog)/sdlog)/pnorm((log(endpoint)-meanlog)/sdlog))
}

qtlnorm <- function(p, meanlog = 0, sdlog = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE)  {
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  if (all(p >= 0 & p <= 1)) {

    return(qlnorm(p*plnorm(endpoint, meanlog=meanlog, sdlog=sdlog), meanlog=meanlog, sdlog=sdlog))
#     A = pnorm((log(endpoint)-meanlog)/sdlog)
#     return(exp(meanlog+sdlog*qnorm(p*A)))
  } else {
    stop("p should be between 0 and 1.")
  }
  
}

rtlnorm <- function(n, meanlog = 0, sdlog = 1, endpoint=Inf) {
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  return(qtlnorm(runif(n), meanlog=meanlog, sdlog=sdlog, endpoint=endpoint))
}

# ###############################################################
# # Weibull 
# 
# dweibull <- function(x, lambda, tau = 1, log = FALSE) {
#   
#   if (lambda <= 0) {
#    stop("lambda should be strictly positive.")
#   }
#   
#   if (tau <= 0) {
#    stop("tau should be strictly positive.")
#   }
#   
# 
#   d <- ifelse(x <= 0, 0, lambda*tau*x^(tau-1)*exp(-lambda*x^tau))
#   
#   if (log) d <- log(d)
#   
#   return(d)
# }
# 
# 
# pweibull <- function(x, lambda, tau = 1, lower.tail = TRUE, log.p = FALSE) {
#   
#   if (lambda <= 0) {
#    stop("lambda should be strictly positive.")
#   }
#   
#   if (tau <= 0) {
#    stop("tau should be strictly positive.")
#   }
#   
#   
#   p <- ifelse(x <= 0, 0, 1-exp(-lambda*x^tau))
#   
#   if (!lower.tail) p <- 1-p
#   
#   if (log.p) p <- log(p)
#   
#   return(p)
# }
# 
# 
# qweibull <- function(p, lambda, tau = 1, lower.tail = TRUE, log.p = FALSE) {
#   
#   if (lambda <= 0) {
#    stop("lambda should be strictly positive.")
#   }
#   
#   if (tau <= 0) {
#    stop("tau should be strictly positive.")
#   }
#   
#   if (log.p) p <- exp(p)
#   
#   if (!lower.tail) p <- 1-p
#   
#   
#   if (all(p >= 0 & p <= 1)) {
# 
#     return((-log(1-p)/lambda)^(1/tau))
#   } else {
#     stop("p should be between 0 and 1.")
#   }
# 
# }
# 
# rweibull <- function(n, lambda, tau = 1) {
#   
#   if (lambda <= 0) {
#    stop("lambda should be strictly positive.")
#   }
#   
#   if (tau <= 0) {
#    stop("tau should be strictly positive.")
#   }
#   
#   return(qweibull(runif(n),lambda=lambda,tau=tau))
#   
# }

###############################################################
#Truncated Weibull

dtweibull <- function(x, shape, scale = 1, endpoint=Inf, log = FALSE) {
  
  if (shape <= 0) {
   stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
   stop("scale should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  d <- ifelse(x <= endpoint,
              dweibull(x, shape=shape, scale=scale)/pweibull(endpoint, shape=shape, scale=scale), 
              0)
  
  if (log) d <- log(d)
  
  return(d)
}

ptweibull <- function(x, shape, scale = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (shape <= 0) {
   stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
   stop("scale should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  p <- ifelse(x <= endpoint, 
              pweibull(x, shape=shape, scale=scale)/pweibull(endpoint, shape=shape, scale=scale), 
              1)
  #ifelse(x <= 0, 0, (1-exp(-(x/beta)^alpha))/1-exp(-(endpoint/beta)^alpha))

  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
}

qtweibull <- function(p, shape, scale = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (shape <= 0) {
   stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
   stop("scale should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (all(p >= 0 & p <= 1)) {

    return(qweibull(p*pweibull(endpoint, shape=shape, scale=scale), shape=shape, scale=scale))
  } else {
    stop("p should be between 0 and 1.")
  }
}

rtweibull <- function(n, shape, scale = 1, endpoint=Inf) {
  
  if (shape <= 0) {
   stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
   stop("scale should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  return(qtweibull(runif(n), shape=shape, scale=scale, endpoint=endpoint))
}

###############################################################
#Truncated Exponential

dtexp <- function(x, rate = 1, endpoint=Inf, log = FALSE) {
  
  if (rate <= 0) {
    stop("rate should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  d <- ifelse(x <= endpoint, dexp(x, rate=rate)/pexp(endpoint, rate=rate), 0)
  
  if (log) d <- log(d)
  
  return(d)
}


ptexp <- function(x, rate = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (rate <= 0) {
    stop("rate should be strictly positive.")
  }
  
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  p <- ifelse(x <= endpoint, pexp(x, rate=rate)/pexp(endpoint, rate=rate), 1)

  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  return(p)
}

qtexp <- function(p, rate = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (rate <= 0) {
    stop("rate should be strictly positive.")
  }
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  

  if (all(p >= 0 & p <= 1)) {
    
    return( qexp(p*pexp(endpoint, rate=rate), rate=rate))
    
  } else {
    stop("p should be between 0 and 1.")
  }
}

rtexp <- function(n, rate = 1, endpoint=Inf) {
  
  if (rate <= 0) {
    stop("rate should be strictly positive.")
  }
  
  
  
  if (endpoint <= 0) {
    stop("endpoint should be strictly positive.")
  }
  

  return(qtexp(runif(n), rate=rate, endpoint=endpoint))
}


###############################################################
# Fréchet

dfrechet <- function(x, shape, loc = 0, scale = 1, log = FALSE) {
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  d <- ifelse(x >= loc, shape/scale * ((x-loc)/scale)^(-shape-1) * exp(-((x-loc)/scale)^(-shape)), 0)
  
  if (log) d <- log(d)
  
  return(d)
}

pfrechet <- function(x, shape, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) {
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  p <- ifelse(x >= loc, exp(-((x-loc)/scale)^(-shape)), 0)
  
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  
  return(p)
}


qfrechet <- function(p, shape, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) {
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (!all(p >= 0 & p <= 1)) {
    stop("p should be between 0 and 1.")
  }
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  q <- loc + scale * (-log(p))^(-1/shape)
  
  return(q)
}


rfrechet <- function(n, shape, loc = 0, scale = 1) {
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  qfrechet(runif(n), shape=shape, loc=loc, scale=scale)
}

######################
# Truncated Fréchet

dtfrechet <- function(x, shape, loc = 0, scale = 1, endpoint=Inf, log = FALSE) {
  
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  d <- ifelse(x <= endpoint, 
              dfrechet(x, shape=shape, loc=loc, scale=scale)/pfrechet(endpoint, shape=shape, loc=loc, scale=scale),
              0)
  
  if (log) d <- log(d)
  
  return(d)
  
}

ptfrechet <- function(x, shape, loc = 0, scale = 1, endpoint=Inf, lower.tail = TRUE, log.p = FALSE) {
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= loc) {
    stop("endpoint should be strictly larger than loc.")
  }
  
  p <- ifelse(x <= endpoint, 
              pfrechet(x, shape=shape, loc=loc, scale=scale)/pfrechet(endpoint, shape=shape, loc=loc, scale=scale),
              1)
  
  
  if (!lower.tail) p <- 1-p
  
  if (log.p) p <- log(p)
  
  
  return(p)
}


qtfrechet <- function(p, shape, loc = 0, scale = 1, endpoint = Inf, lower.tail = TRUE, log.p = FALSE) {
  
  
  if (log.p) p <- exp(p)
  
  if (!lower.tail) p <- 1-p
  
  
  if (!all(p >= 0 & p <= 1)) {
    stop("p should be between 0 and 1.")
  }
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  
  if (endpoint <= loc) {
    stop("endpoint should be strictly larger than loc.")
  }
  
  q <- qfrechet(p*pfrechet(endpoint, shape=shape, loc=loc, scale=scale), shape=shape, loc=loc, scale=scale)
  
  return(q)
  
}


rtfrechet <- function(n, shape, loc = 0, scale = 1, endpoint = Inf) {
  
  if (shape <= 0) {
    stop("shape should be strictly positive.")
  }
  
  
  if (scale <= 0) {
    stop("scale should be strictly positive.")
  }
  
  if (endpoint <= loc) {
    stop("endpoint should be strictly larger than loc.")
  }
  
  
  qtfrechet(runif(n), shape=shape, loc=loc, scale=scale, endpoint=endpoint)
}