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Observed Fisher information matrix for exponential series system (C1,C2,C3)

Source: R/md_candidate_set_models.R
md_fim_exp_series_C1_C2_C3.Rd

Returns a function that computes the observed Fisher information matrix (FIM) for an exponential series system with masked data under C1, C2, C3 conditions.

Usage

md_fim_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 I(theta) that computes the m x m observed Fisher information matrix at parameter values theta.

Details

The observed FIM is the negative Hessian of the log-likelihood, which under regularity conditions converges to the expected FIM and can be inverted to obtain the asymptotic variance-covariance matrix of the MLE.

The (j,k) element of the observed FIM is: $$I_{jk}(\theta) = \sum_{i: \delta_i=0} \frac{I(j \in C_i) I(k \in C_i)}{(\sum_{l \in C_i} \lambda_l)^2}$$

Note: The survival contribution to the Hessian is zero (second derivative of linear term is zero), so only uncensored observations contribute.

Examples

if (FALSE) { # \dontrun{
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)
)
fim <- md_fim_exp_series_C1_C2_C3(md)
fim(c(1, 1.5, 2))
} # }