coxbctype1 {bccp} R Documentation

Computing the bias corrected maximum likelihood estimator.

Description

Computing the bias corrected maximum likelihood estimator (MLE) for the parameters of the general family of distributions under progressive type-I interval censoring scheme. Let y_1,y_2,\ldots,y_n represent the lifetimes of n items that independently follow the cumulative distribution function (cdf) F(.,\theta_{j}) under a progressive type-I interval censoring scheme. We use bctypei to compute the bias corrected ML estimator using the method of Cox and Snell (1968). Let [T_{0}-T_{1}),[T_{1}-T_{2}),...,[T_{m-1}-T_{m}) show a number of m censoring time intervals, {\bf{X}}=(X_{1},X_{2},\ldots,X_{m}) denotes the vector of failed items, and {\bf{R}}=(R_{1},R_{2},\ldots,R_{m}) represents the vector of removed alive items in each interval, from m\geq 1. A schematic, given by the following, displays the progressive type-I interval censoring scheme. We note that the sample size n is n=\sum_{i=1}^{m}X_{i}+\sum_{i=1}^{m}R_{i}. Furthermore, R_i can be determined by the pre-specified percentage of the remaining surviving items at T_i or equivalently R_i=\lfloor P_i X_i\rfloor, for i=1,\ldots,m. Here, \lfloor z\rfloor denotes the largest integer less than or equal to z.

 row Time interval {\bf{X}} {\bf{R}} 1 [T_{0}, T_{1}) X_1 R_1 2 [T_{1}, T_{2}) X_2 R_2 . . . . . . . . . . . . m-1 [T_{m-2}, T_{m-1}) X_{m-1} R_{m-1} m [T_{m-1},T_{m}) X_m R_m

Usage

coxbctype1(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 items X, and vector of removed items in each interval R. param Vector of the of the family parameter's names. mle A vector that contains MLE of the 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's 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

A list of the outputs including: a matric that represents the variance-covariance matrix of the MLE, a matrix that represents the variance-covariance matrix of the bias corrected MLE, a list of three outputs including MLE, bias of MLE, and bias corrected MLE, a list of goodness-of-fit measures consists of Anderson-Darling (AD), Cramer-von Misses (CVM), and Kolmogorov-Smirnov (KS) statistics.

Mahdi Teimouri

References

Z. Chen 2000. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics & Probability Letters, 49 (2), 155-161.

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.

D. R. Cox and E. J. Snell, 1968. A general definition of residuals. Journal of the Royal Statistical Society: Series B (Methodological), 30(2), 248-265.

M. Teimouri, 2020. Bias corrected maximum likelihood estimators under progressive type-I interval censoring scheme, https://doi.org/10.1080/03610918.2020.1819320

Examples

 data(plasma, package="bccp")
plan <- data.frame(T = plasma$upper, X = plasma$X, P = plasma$P, R = plasma$R)
param <- c("lambda","beta")
mle <- c(1.4, 0.05)
pdf <- quote( lambda*(1-exp( -(x*beta)))^(lambda-1)*beta*exp( -(x*beta)) )
cdf <- quote( (1-exp( -(x*beta)))^lambda )
lb <- 0
coxbctype1(plan = plan, param = param, mle = mle, cdf.expression = FALSE, pdf.expression = TRUE,
cdf = cdf, pdf = pdf, lb = lb)


[Package bccp version 0.5.0 Index]