Constructs a likelihood model for wei_series_homogeneous_md_c1_c2_c3.

Likelihood model for Weibull series systems with homogeneous shape parameter and masked component cause of failure with candidate sets that satisfy conditions C1, C2, and C3.

Usage

wei_series_homogeneous_md_c1_c2_c3(
  shape = NULL,
  scales = NULL,
  lifetime = "t",
  lifetime_upper = "t_upper",
  omega = "omega",
  candset = "x"
)

Arguments

shape

common shape parameter for all Weibull component lifetimes

scales

scale parameters for Weibull component lifetimes (optional)

lifetime

column name for system lifetime, defaults to "t"

lifetime_upper

column name for interval upper bound, defaults to "t_upper". Only used for interval-censored observations.

omega

column name for observation type, defaults to "omega". Must contain character values: "exact", "right", "left", or "interval".

candset

column prefix for candidate set indicators, defaults to "x"

Value

likelihood model object with class c("wei_series_homogeneous_md_c1_c2_c3", "series_md", "likelihood_model")

Details

This is a reduced model where all components share a common shape parameter k, while retaining individual scale parameters. The parameter vector is (k, scale_1, ..., scale_m), giving m+1 parameters instead of 2m.

A key property of this model is that the series system lifetime is itself Weibull distributed with shape k and scale lambda_s = (sum(scale_j^-k))^-1/k.

This model satisfies the concept of a likelihood_model in the likelihood.model package by providing the following methods:

(1) loglik.wei_series_homogeneous_md_c1_c2_c3 (2) score.wei_series_homogeneous_md_c1_c2_c3 (3) hess_loglik.wei_series_homogeneous_md_c1_c2_c3

In this likelihood model, masked component data approximately satisfies:

C1: Pr{K[i] in C[i]} = 1 C2: Pr{C[i]=c[i] | K[i]=j, T[i]=t[i]} = Pr{C[i]=c[i] | K[i]=j', T[i]=t[i]} for any j, j' in c[i]. C3: masking probabilities are independent of theta

See also

wei_series_md_c1_c2_c3() for the full model with heterogeneous shapes

Examples

# Create model and fit to data using generic dispatch
model <- wei_series_homogeneous_md_c1_c2_c3()
# solver <- fit(model)
# theta: (shape, scale1, scale2, ...)
# mle <- solver(data, par = c(1.2, 1000, 900, 850))