Log-likelihood function for exponential series system with masked data (C1,C2,C3)

Returns a log-likelihood function for an exponential series system with masked component cause of failure under candidate sets satisfying conditions C1, C2, and C3.

Usage

md_loglike_exp_series_C1_C2_C3(
  md,
  sysvar = "t",
  setvar = "x",
  deltavar = "delta"
)

Arguments

md

Masked data frame containing:

• System lifetime column (default: "t")

• Candidate set columns (default: "x1", "x2", ..., "xm")

• Optional right-censoring indicator (default: "delta")

sysvar

Column name for system lifetime. Default is "t".

setvar

Column prefix for candidate set (Boolean matrix). Default is "x".

deltavar

Column name for right-censoring indicator. Default is "delta". If NULL or column doesn't exist, assumes no censoring.

Value

A function f(theta) that computes the log-likelihood given rate parameters theta = (lambda_1, ..., lambda_m).

Details

For a series system with m components having exponential lifetimes with rate parameters theta = (lambda_1, ..., lambda_m):

• System reliability: R(t) = exp(-sum(theta) * t)

• System hazard: h(t) = sum(theta)

For observation i with system lifetime s_i, right-censoring indicator delta_i, and candidate set C_i:

• If uncensored (delta_i = FALSE): L_i(theta) = R(s_i) * sum_{j in C_i} h_j(s_i)

• If censored (delta_i = TRUE): L_i(theta) = R(s_i)

The log-likelihood is: $$\ell(\theta) = \sum_i \log L_i(\theta)$$

For uncensored observations: $$\log L_i(\theta) = -s_i \sum_{j=1}^m \lambda_j + \log\left(\sum_{j \in C_i} \lambda_j\right)$$

For censored observations: $$\log L_i(\theta) = -s_i \sum_{j=1}^m \lambda_j$$

Examples

if (FALSE) { # \dontrun{
# Generate some masked data
md <- tibble::tibble(
  t = c(0.5, 1.2, 0.8),
  x1 = c(TRUE, TRUE, FALSE),
  x2 = c(TRUE, FALSE, TRUE),
  x3 = c(FALSE, TRUE, TRUE),
  delta = c(FALSE, FALSE, TRUE)
)
ll <- md_loglike_exp_series_C1_C2_C3(md)
ll(c(1, 1.5, 2))  # Evaluate at theta = (1, 1.5, 2)
} # }