The Piecewise Exponential Distribution

Distribution function, quantile function and random number generation for the piecewise exponential distribution.

getPiecewiseExponentialDistribution(
  time,
  ...,
  piecewiseSurvivalTime = NA_real_,
  piecewiseLambda = NA_real_,
  kappa = 1
)

ppwexp(t, ..., s = NA_real_, lambda = NA_real_, kappa = 1)

getPiecewiseExponentialQuantile(
  quantile,
  ...,
  piecewiseSurvivalTime = NA_real_,
  piecewiseLambda = NA_real_,
  kappa = 1
)

qpwexp(q, ..., s = NA_real_, lambda = NA_real_, kappa = 1)

getPiecewiseExponentialRandomNumbers(
  n,
  ...,
  piecewiseSurvivalTime = NA_real_,
  piecewiseLambda = NA_real_,
  kappa = 1
)

rpwexp(n, ..., s = NA_real_, lambda = NA_real_, kappa = 1)

Arguments

...

Ensures that all arguments (starting from the "...") are to be named and that a warning will be displayed if unknown arguments are passed.

kappa

A numeric value > 0. A kappa != 1 will be used for the specification of the shape of the Weibull distribution. Default is 1, i.e., the exponential survival distribution is used instead of the Weibull distribution. Note that the Weibull distribution cannot be used for the piecewise definition of the survival time distribution, i.e., only piecewiselambda (as a single value) and kappa can be specified. This function is equivalent to pweibull(t, shape = kappa, scale = 1 / lambda) of the stats package, i.e., the scale parameter is 1 / 'hazard rate'.
For example, getPiecewiseExponentialDistribution(time = 130, piecewiseLambda = 0.01, kappa = 4.2) and pweibull(q = 130, shape = 4.2, scale = 1 / 0.01) provide the same result.

t, time

Vector of time values.

s, piecewiseSurvivalTime

Vector of start times defining the "time pieces".

lambda, piecewiseLambda

Vector of lambda values (hazard rates) corresponding to the start times.

q, quantile

Vector of quantiles.

n

Number of observations.

Value

A numeric value or vector will be returned.

Details

getPiecewiseExponentialDistribution() (short: ppwexp()), getPiecewiseExponentialQuantile() (short: qpwexp()), and getPiecewiseExponentialRandomNumbers() (short: rpwexp()) provide probabilities, quantiles, and random numbers according to a piecewise exponential or a Weibull distribution. The piecewise definition is performed through a vector of starting times (piecewiseSurvivalTime) and a vector of hazard rates (piecewiseLambda). You can also use a list that defines the starting times and piecewise lambdas together and define piecewiseSurvivalTime as this list. The list needs to have the form, e.g., piecewiseSurvivalTime <- list( "0 - <6" = 0.025, "6 - <9" = 0.04, "9 - <15" = 0.015, ">=15" = 0.007) . For the Weibull case, you can also specify a shape parameter kappa in order to calculate probabilities, quantiles, or random numbers. In this case, no piecewise definition is possible, i.e., only piecewiseLambda (as a single value) and kappa need to be specified.

Examples

if (FALSE) { # \dontrun{
# Calculate probabilties for a range of time values for a
# piecewise exponential distribution with hazard rates
# 0.025, 0.04, 0.015, and 0.007 in the intervals
# [0, 6), [6, 9), [9, 15), [15, Inf), respectively,
# and re-return the time values:
piecewiseSurvivalTime <- list(
    "0 - <6"   = 0.025,
    "6 - <9"   = 0.04,
    "9 - <15"  = 0.015,
    ">=15"     = 0.01
)
y <- getPiecewiseExponentialDistribution(seq(0, 150, 15),
    piecewiseSurvivalTime = piecewiseSurvivalTime
)
getPiecewiseExponentialQuantile(y,
    piecewiseSurvivalTime = piecewiseSurvivalTime
)
} # }