Maximum Likelihood Estimation under Censoring Schemes

Provides generalized functions to compute Maximum Likelihood Estimation (MLE) for any univariate distribution under various censoring and truncation schemes. Users supply the probability density function (PDF), cumulative distribution function (CDF), survival function, support bounds, and initial parameter values; the package constructs and maximizes the appropriate log-likelihood automatically. Supported schemes include right and left truncation, random, right, left, interval, and middle censoring, block random censoring, balanced joint progressive Type-II (BJPT-II), progressive first failure, joint Type-I, Type-I, Type-II, progressive Type-II, Type-II progressively hybrid, joint Type-II, hybrid, hybrid Type-I, doubly Type-II, Type-I hybrid, and hybrid Type-II censoring. Optimization methods include Newton-Raphson (NR), Broyden-Fletcher-Goldfarb-Shanno (BFGS), the BFGS algorithm implemented in R (BFGSR), Berndt-Hall-Hall-Hausman (BHHH), Simulated Annealing (SANN), Conjugate Gradients (CG), and Nelder-Mead (NM). Inference summaries provide the Akaike Information Criterion (AIC), estimated coefficients, log-likelihood, iteration count, standard errors, z-values, p-values, and the variance-covariance matrix. Methods are described in Nagar, Kumar, and Krishna (2026) , Goel, Kumar, and Krishna (2026, "Estimation in power Lindley distributions using balanced joint progressively Type-II censored data"), Wu and Kus (2009) , Goel and Krishna (2026) , Balakrishnan and Aggarwala (2000, ISBN:978-1-4612-1334-5), Mondal and Kundu (2020) , Ding and Gui (2023) , Prajapati, Mitra, and Kundu (2019) , Yadav, Jaiswal, and Yadav (2026) , Iyer, Jammalamadaka, and Kundu (2008) , Banerjee and Kundu (2008) , Kundu and Joarder (2006) , Berndt, Hall, Hall, and Hausman (1974) "Estimation and Inference in Nonlinear Structural Models" , Fletcher (1987, "Practical Methods of Optimization", ISBN:978-0-471-91547-8), Nelder and Mead (1965) , McKinnon (1999) "Convergence of the Nelder-Mead simplex method to a non-stationary point" , Kirkpatrick, Gelatt, and Vecchi (1983) , Fletcher and Reeves (1964) , and Nocedal and Wright (2006, "Numerical Optimization", ISBN:978-0-387-30303-1).

MleCensoR: Maximum Likelihood Estimation under Censoring Schemes

A comprehensive R package for finding Maximum Likelihood Estimation (MLE) for any univariate distribution under various censoring and truncation schemes.

Features

MleCensoR provides generalized functions to compute MLE under the following censoring schemes:

• Right Truncation (mle_right_truncation)
• Left Truncation (mle_left_truncation)
• Random Censoring (mle_random)
• Right Censoring (mle_right)
• Left Censoring (mle_left)
• Interval Censoring (mle_interval)
• Block Random Censoring (mle_block_random)
• Balanced Joint Progressive Type-II (BJPT-II) Censoring (mle_bjpt2)
• Progressive First Failure Censoring (mle_progressive_first_failure)
• Joint Type-I Censoring (mle_joint_type1)
• Type-I Censoring (mle_type1)
• Type-II Censoring (mle_type2)
• Progressive Type-II Censoring (mle_progressive_type2)
• Joint Type-II Censoring (mle_joint_type2)
• Hybrid Censoring (Type-I Hybrid) (mle_hybrid, mle_hybrid_type1, mle_type1_hybrid)
• Hybrid Type-II Censoring (mle_hybrid_type2)
• Doubly Type-II Censoring (mle_doubly_type2)
• Middle Censoring (mle_middle)
• Type-II Progressively Hybrid Censoring (mle_progressive_hybrid_type2)

Installation

# Install from local directory
install.packages("path/to/MleCensoR")

# Or install using devtools
devtools::install("path/to/MleCensoR")

Quick Start

Basic Usage Example

library(MleCensoR)

# Define distribution functions (e.g., Weibull distribution)
pdf_weibull <- function(x, theta) {
  shape <- theta[1]
  scale <- theta[2]
  dweibull(x, shape = shape, scale = scale)
}

cdf_weibull <- function(x, theta) {
  shape <- theta[1]
  scale <- theta[2]
  pweibull(x, shape = shape, scale = scale)
}

surv_weibull <- function(x, theta) {
  1 - cdf_weibull(x, theta)
}

# Example: Right Censoring
set.seed(123)
x <- rweibull(50, shape = 2, scale = 1)
status <- rep(1, 50)  # All observed

# MLE estimation
fit <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  method = "NR"
)

# View results
summary(fit)
coef(fit)
vcov(fit)
stdEr(fit)
logLik(fit)
AIC(fit)

Using Custom Log-Likelihood

You can also provide a custom log-likelihood function directly:

# Custom log-likelihood for right censoring
custom_loglik <- function(theta, x, status) {
  shape <- theta[1]
  scale <- theta[2]
  pdf_val <- dweibull(x, shape = shape, scale = scale)
  surv_val <- pweibull(x, shape = shape, scale = scale, lower.tail = FALSE)
  ll <- status * log(pdf_val) + (1 - status) * log(surv_val)
  return(ll)
}

fit <- mle_right(
  x = x,
  status = status,
  logLik = custom_loglik,
  start = c(shape = 1.5, scale = 0.8),
  method = "NR"
)

Optimization Methods

The package supports the following optimization methods:

• NR - Newton-Raphson (default for unconstrained problems)
• BFGS - Broyden-Fletcher-Goldfarb-Shanno
• BFGSR - BFGS algorithm implemented in R
• BHHH - Berndt-Hall-Hall-Hausman (recommended for observation-wise likelihood)
• SANN - Simulated Annealing
• CG - Conjugate Gradients
• NM - Nelder-Mead (default for constrained problems without gradient)

Example with Different Methods

# Using BHHH method
fit_bhhh <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  method = "BHHH"
)

# Using BFGS method
fit_bfgs <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  method = "BFGS"
)

Advanced Features

Parameter Constraints

# Constrained optimization
fit_constrained <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  method = "BFGS",
  constraints = list(
    ineqA = matrix(c(1, 0, 0, 1), nrow = 2, byrow = TRUE),
    ineqB = c(0, 0)  # theta > 0
  )
)

Parameter Ranges

# Specify parameter ranges
fit_range <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  param_range = list(shape = c(0.1, 10), scale = c(0.1, 10))
)

Custom Gradient and Hessian

# Provide analytic gradient
custom_grad <- function(theta, x, status) {
  shape <- theta[1]
  scale <- theta[2]
  # ... compute gradient ...
  grad <- c(d_shape, d_scale)
  return(grad)
}

fit_grad <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  grad = custom_grad,
  method = "NR"
)

Censoring Scheme Examples

Interval Censoring

# Interval censoring data
l <- c(0.5, 1.0, 1.5, 2.0)  # Lower bounds
r <- c(1.0, 1.5, 2.0, Inf)  # Upper bounds

fit_interval <- mle_interval(
  l = l,
  r = r,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 2, scale = 1)
)

Progressive Type-II Censoring

# Progressive Type-II censoring
x <- c(0.5, 0.8, 1.2, 1.5, 2.0)  # Observed failure times
r_removals <- c(1, 1, 2, 1, 0)   # Removals at each failure

fit_prog <- mle_progressive_type2(
  x = x,
  r_removals = r_removals,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 2, scale = 1)
)

Joint Type-I Censoring

# Joint Type-I censoring for two populations
x <- c(0.5, 0.8, 1.2, 1.5, 2.0)  # Failure times
z <- c(1, 1, 0, 0, 1)             # Group membership (1 = pop A, 0 = pop B)
tc <- 2.0                          # Censoring time
n_A <- 3                           # Initial units in pop A
n_B <- 2                           # Initial units in pop B

fit_joint1 <- mle_joint_type1(
  x = x,
  z = z,
  tc = tc,
  n_A = n_A,
  n_B = n_B,
  pdf_A = pdf_weibull,
  cdf_A = cdf_weibull,
  surv_A = surv_weibull,
  pdf_B = pdf_weibull,
  cdf_B = cdf_weibull,
  surv_B = surv_weibull,
  start = c(shape = 2, scale = 1)
)

Type-II Progressively Hybrid Censoring

# Suppose n = 20, m = 10, tc = 2.0
# 7 failures observed before tc: Case II
x <- c(0.12, 0.35, 0.62, 0.89, 1.15, 1.48, 1.82)
r_removals <- c(1, 0, 1, 0, 1, 0, 0, 1, 0, 0) # full plan for m = 10

fit_prog_hybrid <- mle_progressive_hybrid_type2(
  x = x,
  r_removals = r_removals,
  tc = 2.0,
  n = 20,
  m = 10,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 2, scale = 1)
)

Output Methods

The package provides S3 methods for extracting results:

• coef(fit) - Parameter estimates
• logLik(fit) - Log-likelihood value
• vcov(fit) - Variance-covariance matrix
• stdEr(fit) - Standard errors
• summary(fit) - Summary table with estimates, standard errors, z-values, and p-values
• AIC(fit) - Akaike Information Criterion
• nIter(fit) - Number of iterations

Control Parameters

Advanced control parameters can be passed via ...:

• tol - Stopping tolerance (default: 1e-8)
• reltol - Relative convergence tolerance (default: sqrt(.Machine$double.eps))
• gradtol - Gradient norm tolerance (default: 1e-6)
• steptol - Step size tolerance (default: 1e-10)
• iterlim - Maximum iterations (default: 100 for NR, 200 for BFGSR)
• printLevel - Printing level (0 = none, 1 = initial/final, 2 = all iterations)

fit <- mle_right(
  x = x,
  status = status,
  pdf = pdf_weibull,
  cdf = cdf_weibull,
  surv = surv_weibull,
  start = c(shape = 1.5, scale = 0.8),
  method = "NR",
  iterlim = 500,
  tol = 1e-10,
  printLevel = 2
)

License

GPL-3

Author

Shikhar Tyagi

Contact

Maintainer: Shikhar Tyagi [email protected]