EM algorithm for Reliability Mixture Models (RMM) with right Censoring

Parametric EM algorithm for univariate finite mixture of exponentials distributions with randomly right censored data.

Usage

expRMM_EM(x, d=NULL, lambda = NULL, rate = NULL, k = 2, 
        complete = "tdz", epsilon = 1e-08, maxit = 1000, verb = FALSE)

Arguments

x: A vector of \(n\) real positive lifetime (possibly censored) durations. If d is not NULL then a vector of random censoring times c occurred, so that \(x= min(x,c)\) and \(d = I(x <= c)\).
d: The vector of censoring indication, where 1 means observed lifetime data, and 0 means censored lifetime data.
lambda: Initial value of mixing proportions. If NULL, then lambda is set to rep(1/k,k).
rate: Initial value of component exponential rates, all set to 1 if NULL.
k: Number of components of the mixture.
complete: Nature of complete data involved within the EM machinery, can be "tdz" for (t,d,z) (the default), or "xz" for (x,z) (see Bordes L. and Chauveau D. (2016) reference below).
epsilon: Tolerance limit for declaring algorithm convergence based on the change between two consecutive iterations.
maxit: The maximum number of iterations allowed, convergence may be declared before maxit iterations (see epsilon above).
verb: If TRUE, print updates for every iteration of the algorithm as it runs

Value

expRMM_EM returns a list of class "mixEM" with the following items:

x: The input data.
d: The input censoring indicator.
lambda: The estimates for the mixing proportions.
rate: The estimates for the component rates.
loglik: The log-likelihood value at convergence of the algorithm.
posterior: An \(n\times k\) matrix of posterior probabilities for observation, after convergence of the algorithm.
all.loglik: The sequence of log-likelihoods over iterations.
all.lambda: The sequence of mixing proportions over iterations.
all.rate: The sequence of component rates over iterations.
ft: A character vector giving the name of the function.

References

Bordes, L., and Chauveau, D. (2016), Stochastic EM algorithms for parametric and semiparametric mixture models for right-censored lifetime data, Computational Statistics, Volume 31, Issue 4, pages 1513-1538. https://link.springer.com/article/10.1007/s00180-016-0661-7

Author

Didier Chauveau

Examples

n <- 300 # sample size
m <- 2   # number of mixture components
lambda <- c(1/3,1-1/3); rate <- c(1,1/10) # mixture parameters
set.seed(1234)
x <- rexpmix(n, lambda, rate) # iid ~ exponential mixture
cs <- runif(n,0,max(x)) # Censoring (uniform) and incomplete data
t <- apply(cbind(x,cs),1,min) # observed or censored data
d <- 1*(x <= cs)              # censoring indicator

###### EM for RMM, exponential lifetimes
l0 <- rep(1/m,m); r0 <- c(1, 0.5) # "arbitrary" initial values
a <- expRMM_EM(t, d, lambda = l0, rate = r0, k = m)
#> number of iterations = 59 
summary(a)                 # EM estimates etc
#> summary of expRMM_EM object:
#>          comp 1    comp 2
#> lambda 0.351555 0.6484450
#> rate   0.969003 0.0857216
#> loglik at estimate:  -762.4311 
#> 11.33333 % of the data right censored
plotexpRMM(a, lwd=2) # default plot of EM sequences

plot(a, which=2) # or equivalently, S3 method for "mixEM" object