fitype1 {bccp}

Computing the Fisher information matrix under progressive type-I interval censoring scheme

Description

Computes the Fisher information matrix under progressive type-I interval censoring scheme. The Fisher information matrix is given by

I_{rs}=-E\Bigl(\frac{\partial^2 l(\Theta)}{\partial \theta_r \partial \theta_s}\Bigr),

where

l(\Theta)=\log L(\Theta) \propto \sum_{i=1}^{m}X_i \log \bigl[F(t_{i}{{;}}\Theta)-F(t_{i-1}{{;}}\Theta)\bigr]+\sum_{i=1}^{m}R_i\bigl[1-F(t_{i}{{;}}\Theta)\bigr],

in which F(.;\Theta) is the family cumulative distribution function for \Theta=(\theta_1,\dots,\theta_k)^T and r,s=1,\dots,k.

Usage

fitype1(plan, param, mle, cdf.expression = FALSE, pdf.expression = TRUE, cdf, pdf
        , lb = 0)

Arguments

plan

Censoring plan for progressive type-I interval censoring scheme. It must be given as a data.frame that includes vector of upper bounds of the censoring times T, vector of number of failed subjects X, vector of removed subjects in each interval R, and percentage of the removed alive items in each interval P.

param

Vector of the of the family parameter's names.

mle

Vector of the maximum likelihood estimators.

cdf.expression

Logical. That is TRUE, if there is a closed form expression for the cumulative distribution function.

pdf.expression

Logical. That is TRUE, if there is a closed form expression for the probability density function.

cdf

Expression of the cumulative distribution function.

pdf

Expression of the probability density function.

lb

Lower bound of the family support. That is zero by default.

Details

For some families of distributions whose support is the positive semi-axis, i.e., x>0, the cumulative distribution function (cdf) may not be differentiable. In this case, the lower bound of the support of random variable, i.e., lb that is zero by default, must be chosen some positive small value to ensure the differentiability of the cdf.

Value

Matrices that represent the expected and observed Fisher information matrices.

Author(s)

Mahdi Teimouri

References

N. Balakrishnan and E. Cramer. 2014. The art of progressive censoring. New York: Springer.

D. G. Chen and Y. L. Lio 2010. Parameter estimations for generalized exponential distribution under progressive type-I interval censoring, Computational Statistics and Data Analysis, 54, 1581-1591.

M. Teimouri 2020. Bias corrected maximum likelihood estimators under progressive type-I interval censoring scheme, Communications in Statistics-Simulation and Computation, doi.org/10.1080/036 10918.2020.1819320

Examples

  data(plasma)
      n <- 20
  param <- c("alpha","beta")
    mle <- c(0.4, 0.05)
    cdf <- quote( 1-exp( beta*(1-exp( x^alpha )) ) )
       pdf <- quote( exp( beta*(1-exp( x^alpha )) )*( beta*(exp( x^alpha )*(x^(alpha-1)*alpha) )) )
     lb <- 0
     plan <- rtype1(n = n, P = plasma$P, T = plasma$upper, param = param, mle = mle, cdf.expression
                  = FALSE, pdf.expression = TRUE, cdf = cdf, pdf = pdf, lb = lb)
fitype1(plan = plan, param = param, mle = mle, cdf.expression = FALSE, pdf.expression = TRUE, cdf =
        cdf, pdf = pdf, lb = lb)