fitype2 {bccp}

Computing the Fisher information matrix under progressive type-II 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 C \sum_{i=1}^{m} \log f(x_{i:m:n}{{;}}\Theta) + \sum_{i=1}^{m} R_i \log \bigl[1-F(x_{i:m:n}{{;}}\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, and C=n(n-R_1-1)(n-R_1-R_2-2)\dots (n-R_1-R_2-\dots R_{m-1}-m+1).

Usage

fitype2(plan, param, mle, cdf, pdf, lb = 0, ub = Inf, N = 100)

Arguments

plan

Censoring plan for progressive type-II censoring scheme. It must be given as a data.frame that includes number of failed items X, and vector of removed items R.

param

Vector of the of the family parameter's names.

mle

Vector of the maximum likelihood estimators.

cdf

Expression for the cumulative distribution function.

pdf

Expression for the probability density function.

lb

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

ub

Upper bound of the family support. That is Inf by default.

N

An even integer value indicating the number of subdivisions for applying Simpson's integration method.

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 AHMED Hossain 2007. Inference for the Type II generalized logistic distribution under progressive Type II censoring, Journal of Statistical Computation and Simulation, 77(12), 1013-1031.

M. Teimouri and S. Nadarajah 2016. Bias corrected MLEs under progressive type-II censoring scheme, Journal of Statistical Computation and Simulation, 86 (14), 2714-2726.

Examples

     n <- 20
     R <- c(5, rep(0, n-6) )
 param <- c("alpha","beta")
   mle <- c(2,6)
   pdf <- quote( alpha/beta*(x/beta)^(alpha-1)*exp( -(x/beta)^alpha ) )
   cdf <- quote( 1-exp( -(x/beta)^alpha ) )
    lb <- 0
    ub <- Inf
     N <- 100
  plan <- rtype2(n = n, R = R, param = param, mle = mle, cdf = cdf, lb = lb, ub = ub)
  fitype2(plan = plan, param = param, mle = mle, cdf = cdf, pdf = pdf, lb = lb, ub = ub, N = N)