goftype1 {bccp}

Computing goodness-of-fit (GOF) measures under progressive type-I interval censoring scheme.

Description

The goodness-of-fit (GOF) measures consist of Anderson-Darling (AD) and Cram\'eer-von Misses (CVM) statistics for progressive type-I interval censoring scheme are given, respectively, by

AD=n\sum_{i=1}^{m}\gamma^{2}_{i}\log\left[\frac{A_{i+1}\bigl(1-A_i\bigr)}{A_i\bigl(1-A_{i+1}\bigr)}\right]+2n\sum_{i=1}^{m}\gamma_{i}\log\Bigl(\frac{1-A_{i+1}}{1-A_i}\Bigr)-n\bigl(A_{m+1}-A_1\bigr)

-n\log\Bigl(\frac{1-A_{m+1}}{1-A_1}\Bigr)+n\bigl(1-A_{m+1}-\log A_{m+1}\bigr),

{CVM}=n\sum_{i=1}^{m}\gamma^{2}_{i}\bigl(A_{i+1}-A_i\bigr)-n\sum_{i=1}^{m}\gamma_{i}\bigl(A^{2}_{i+1}-A^2_i\bigr)+\frac{n}{3}\bigl(A^{3}_{m+1}-A^{3}_{1}\bigr)+\frac{n}{3}\bigl(1-A_{m+1}\bigr)^3,

where R_0=0, \gamma_{i}=\bigl(\sum_{j=1}^{i}{X_j}+\sum_{j=1}^{i-1}{R_j}\bigr)/n, and A_i=G\bigl(T_{i-1}\big|\widehat{\Theta}\bigr), for i=1,\dots,m.

Usage

goftype1(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, and vector of removed subjects in each interval R.

param

Vector of the of the family parameter's names.

mle

Vector of the estimated parameters.

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

We note that for lifetime 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. Theoretically, for lifetime distribution, we have lb=T_{0}=0.

Value

A vector of goodness-of-fit measures consist of Anderson-Darling (AD) and Cramer-von Misses (CVM) statistics.

Author(s)

Mahdi Teimouri

References

M. Teimouri 2020. Bias corrected maximum likelihood estimators under progressive type-I interval censoring scheme, Communications in Statistics-Simulation and Computation, https://doi.org/10.10 80/03610918.2020.1819320.

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)
goftype1(plan = plan, param = param, mle = mle, cdf.expression=TRUE, pdf.expression = FALSE, cdf =
         cdf, pdf = pdf, lb = lb)