ML Estimation for Parametric Lifetime Distributions — ml

This function estimates the parameters of a parametric lifetime distribution for complete and (multiple) right-censored data. The parameters are determined in the frequently used (log-)location-scale parameterization.

For the Weibull, estimates are additionally transformed such that they are in line with the parameterization provided by the stats package (see Weibull).

ml_estimation(x, ...)

# S3 method for wt_reliability_data
ml_estimation(
  x,
  distribution = c("weibull", "lognormal", "loglogistic", "sev", "normal", "logistic",
    "weibull3", "lognormal3", "loglogistic3", "exponential", "exponential2"),
  wts = rep(1, nrow(x)),
  conf_level = 0.95,
  start_dist_params = NULL,
  control = list(),
  ...
)

Arguments

x: A tibble with class wt_reliability_data returned by reliability_data.
...: Further arguments passed to or from other methods. Currently not used.
distribution: Supposed distribution of the random variable.
wts: Optional vector of case weights. The length of wts must be equal to the number of observations in x.
conf_level: Confidence level of the interval.
start_dist_params: Optional vector with initial values of the (log-)location-scale parameters.
control: A list of control parameters (see 'Details' and optim).

Value

A list with classes wt_model, wt_ml_estimation and wt_model_estimationwhich contains:

coefficients : A named vector of estimated coefficients (parameters of the assumed distribution). Note: The parameters are given in the (log-)location-scale-parameterization.
confint : Confidence intervals for the (log-)location-scale parameters.
shape_scale_coefficients : Only included if distribution is "weibull" or "weibull3" (parameterization used in Weibull).
shape_scale_confint : Only included if distribution is "weibull" or "weibull3". Confidence intervals for scale $\eta$ and shape $\beta$ (and threshold $\gamma$ if distribution = "weibull3").
varcov : Estimated variance-covariance matrix of (log-)location-scale parameters.
logL : The log-likelihood value.
aic : Akaike Information Criterion.
bic : Bayesian Information Criterion.
data : A tibble with class wt_reliability_data returned by
distribution : Specified distribution.

Details

Within ml_estimation, optim is called with method = "BFGS" and control$fnscale = -1 to estimate the parameters that maximize the log-likelihood (see loglik_function). For threshold models, the profile log-likelihood is maximized in advance (see loglik_profiling). Once the threshold parameter is determined, the threshold model is treated like a distribution without threshold (lifetime is reduced by threshold estimate) and the general optimization routine is applied.

Normal approximation confidence intervals for the parameters are computed as well.

References

Meeker, William Q; Escobar, Luis A., Statistical methods for reliability data, New York: Wiley series in probability and statistics, 1998

Examples

# Reliability data preparation:
## Data for two-parametric model:
data_2p <- reliability_data(
  shock,
  x = distance,
  status = status
)

## Data for three-parametric model:
data_3p <- reliability_data(
  alloy,
  x = cycles,
  status = status
)

# Example 1 - Fitting a two-parametric weibull distribution:
ml_2p <- ml_estimation(
  data_2p,
  distribution = "weibull"
)

# Example 2 - Fitting a three-parametric lognormal distribution:
ml_3p <- ml_estimation(
  data_3p,
  distribution = "lognormal3",
  conf_level = 0.99
)