Compute count probabilities using convolution

Compute count probabilities using one of several convolution methods. dCount_conv_bi does the computations for the distributions with builtin support in this package.

dCount_conv_user does the same using a user defined survival function.

dCount_conv_bi(
  x,
  distPars,
  dist = c("weibull", "gamma", "gengamma", "burr"),
  method = c("dePril", "direct", "naive"),
  nsteps = 100,
  time = 1,
  extrap = TRUE,
  log = FALSE
)

dCount_conv_user(
  x,
  distPars,
  extrapolPars,
  survR,
  method = c("dePril", "direct", "naive"),
  nsteps = 100,
  time = 1,
  extrap = TRUE,
  log = FALSE
)

Arguments

x: integer (vector), the desired count values.
distPars: Rcpp::List with distribution specific slots, see details.
dist: character name of the built-in distribution, see details.
method: character string, the method to use, see Details.
nsteps: unsiged integer, number of steps used to compute the integral.
time: double, time at wich to compute the probabilities. Set to 1 by default.
extrap: logical, if TRUE, Richardson extrapolation will be applied to improve accuracy.
log: logical, if TRUE the log-probability will be returned.
extrapolPars: vector of length 2, the extrapolation values.
survR: function, user supplied survival function; should have signature function(t, distPars), where t is a positive real number (the time where the survival function is evaluated) and distPars is a list of distribution parameters. It should return a double value.

Value

vector of probabilities \(P(x(i),\ i = 1, ..., n)\) where \(n\)

is the length of x.

Details

dCount_conv_bi computes count probabilities using one of several convolution methods for the distributions with builtin support in this package.

The following convolution methods are implemented: "dePril", "direct", and "naive".

The builtin distributions currently are Weibull, gamma, generalised gamma and Burr.

Examples

x <- 0:10
lambda <- 2.56
p0 <- dpois(x, lambda)
ll <- sum(dpois(x, lambda, TRUE))

err <- 1e-6
## all-probs convolution approach
distPars <- list(scale = lambda, shape = 1)
pmat_bi <- dCount_conv_bi(x, distPars, "weibull", "direct",
                          nsteps = 200)

## user pwei
pwei_user <- function(tt, distP) {
    alpha <- exp(-log(distP[["scale"]]) / distP[["shape"]])
    pweibull(q = tt, scale = alpha, shape = distP[["shape"]],
             lower.tail = FALSE)
}

pmat_user <- dCount_conv_user(x, distPars, c(1, 2), pwei_user, "direct",
                              nsteps = 200)
max((pmat_bi - p0)^2 / p0)
#> [1] 3.879293e-16
max((pmat_user - p0)^2 / p0)
#> [1] 4.753405e-15

## naive convolution approach
pmat_bi <- dCount_conv_bi(x, distPars, "weibull", "naive",
                          nsteps = 200)
pmat_user <- dCount_conv_user(x, distPars, c(1, 2), pwei_user, "naive",
                              nsteps = 200)
max((pmat_bi- p0)^2 / p0)
#> [1] 3.879293e-16
max((pmat_user- p0)^2 / p0)
#> [1] 4.753405e-15

## dePril conv approach
pmat_bi <- dCount_conv_bi(x, distPars, "weibull", "dePril",
                          nsteps = 200)
pmat_user <- dCount_conv_user(x, distPars, c(1, 2), pwei_user, "dePril",
                              nsteps = 200)
max((pmat_bi- p0)^2 / p0)
#> [1] 4.105397e-15
max((pmat_user- p0)^2 / p0)
#> [1] 4.753405e-15