mletype2 {bccp}

Computing the maximum likelihood estimator (MLE) for the parameters of the statistical model fitted to a progressive type-II censoring scheme.

Description

Computes the MLE of for the parameters of the model fitted to a progressive type-II censoring scheme with likelihood function

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

mletype2(plan, param, start, cdf, pdf, method = "Nelder-Mead", lb = 0, ub = Inf, N = 100,
        level = 0.05)

Arguments

plan

Censoring plan for progressive type-II censoring scheme. It must be given as a data.frame including: number of items placed on the test at time zero and a vector that contains number R, of the removed alive items.

param

Vector of the of the family parameter's names.

start

Vector of the initial values.

pdf

Expression of the probability density function.

cdf

Expression of the cumulative distribution function.

method

The method for the numerically optimization that includes one of CG, Nelder-Mead, BFGS, L-BFGS-B, SANN.

lb

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

ub

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

N

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

level

Significance level for constructing asymptotic confidence interval That is 0.05 by default for constructing a 95% confidence interval.

Value

MLE, standard error of MLE, and asymptotic confidence interval for MLE.

Author(s)

Mahdi Teimouri

References

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 <- 10
     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
 level <- 0.05
  plan <- rtype2(n = n, R = R, param = param, mle = mle, cdf = cdf, lb = lb, ub = ub)
mletype2(plan = plan, param = param, start = mle, cdf = cdf, pdf = pdf, method = "Nelder-Mead",
         lb = lb, ub = ub, N = N, level = level)