R: Fit parameters of truncated normal distribution based on a...

paramtnormci_fit {decisionSupport}

R Documentation

Fit parameters of truncated normal distribution based on a confidence interval.

Description

This function fits the distribution parameters, i.e. mean and sd, of a truncated normal distribution from an arbitrary confidence interval and, optionally, the median.

Usage

paramtnormci_fit(
  p,
  ci,
  median = mean(ci),
  lowerTrunc = -Inf,
  upperTrunc = Inf,
  relativeTolerance = 0.05,
  fitMethod = "Nelder-Mead",
  ...
)

Arguments

`p`	`numeric` 2-dimensional vector; probabilities of upper and lower bound of the corresponding confidence interval.
`ci`	`numeric` 2-dimensional vector; lower, i.e `ci[[1]]`, and upper bound, i.e `ci[[2]]`, of the confidence interval.
`median`	if `NULL`: truncated normal is fitted only to lower and upper value of the confidence interval; if `numeric`: truncated normal is fitted on the confidence interval and the median simultaneously. For details cf. below.
`lowerTrunc`	`numeric`; lower truncation point of the distribution (>= `-Inf`).
`upperTrunc`	`numeric`; upper truncation point of the distribution (<= `Inf`).
`relativeTolerance`	`numeric`; the relative tolerance level of deviation of the generated probability levels from the specified confidence interval. If the relative deviation is greater than `relativeTolerance` a warning is given.
`fitMethod`	optimization method used in `constrOptim`.
`...`	further parameters to be passed to `constrOptim`.

Details

For details of the truncated normal distribution see tnorm.

The cumulative distribution of a truncated normal F_{\mu, \sigma}(x) gives the probability that a sampled value is less than x. This is equivalent to saying that for the vector of quantiles q=(q(p_1), \ldots, q(p_k)) at the corresponding probabilities p=(p_1, \ldots, p_k) it holds that

p_i = F_{\mu, \sigma}(q_{p_i}),~i = 1, \ldots, k

In the case of arbitrary postulated quantiles this system of equations might not have a solution in \mu and \sigma. A least squares fit leads to an approximate solution:

\sum_{i=1}^k (p_i - F_{\mu, \sigma}(q_{p_i}))^2 = \min

defines the parameters \mu and \sigma of the underlying normal distribution. This method solves this minimization problem for two cases:

ci[[1]] < median < ci[[2]]: The parameters are fitted on the lower and upper value of the confidence interval and the median, formally:
k=3
p_1=p[[1]], p_2=0.5 and p_3=p[[2]];
q(p_1)=ci[[1]], q(0.5)=median and q(p_3)=ci[[2]]
median=NULL: The parameters are fitted on the lower and upper value of the confidence interval only, formally:
k=2
p_1=p[[1]], p_2=p[[2]];
q(p_1)=ci[[1]], q(p_2)=ci[[2]]

The (p[[2]]-p[[1]]) - confidence interval must be symmetric in the sense that p[[1]] + p[[2]] = 1.

Value