confint_fisher.default: Fisher's Confidence Bounds for Quantiles and Probabilities

Description

This function computes normal-approximation confidence intervals for quantiles and failure probabilities.

Usage

# S3 method for default
confint_fisher(
  x,
  status,
  dist_params,
  dist_varcov,
  distribution = c("weibull", "lognormal", "loglogistic", "normal", "logistic", "sev",
    "weibull3", "lognormal3", "loglogistic3"),
  b_lives = c(0.01, 0.1, 0.5),
  bounds = c("two_sided", "lower", "upper"),
  conf_level = 0.95,
  direction = c("y", "x"),
  ...
)

Value

A tibble with class wt_confint containing the following columns:

x : An ordered sequence of the lifetime characteristic regarding the failed units, starting at min(x) and ending up at max(x). With b_lives = c(0.01, 0.1, 0.5) the 1%, 10% and 50% quantiles are additionally included in x, but only if the specified probabilities are in the range of the estimated probabilities.
prob : An ordered sequence of probabilities with specified b_lives included.
std_err : Estimated standard errors with respect to direction.
lower_bound : Provided, if bounds is one of "two_sided" or "lower". Lower limit of the confidence region with respect to direction, i.e. quantiles or probabilities.
upper_bound : Provided, if bounds is one of "two_sided" or "upper". Upper limit of the confidence region with respect to direction, i.e. quantiles or probabilities.
distribution : Specified distribution (determined when calling ml_estimation).
bounds : Specified bound(s).
direction : Specified direction.
cdf_estimation_method : A character that is always NA_character. For the generic visualization functions this column has to be provided.

Arguments

x: A numeric vector which consists of lifetime data. Lifetime data could be every characteristic influencing the reliability of a product, e.g. operating time (days/months in service), mileage (km, miles), load cycles.
status: A vector of binary data (0 or 1) indicating whether unit i is a right censored observation (= 0) or a failure (= 1).
dist_params: A (named) numeric vector of (log-)location-scale parameters returned from ml_estimation.
dist_varcov: A (named) numeric matrix of estimated variances and covariances returned from ml_estimation.
distribution: Supposed distribution of the random variable. Has to be in line with the specification made in ml_estimation.
b_lives: A numeric vector indicating the probabilities p of the \(B_p\)-lives (quantiles) to be considered.
bounds: A character string specifying of which bounds have to be computed. One of "two_sided", "lower" or "upper".
conf_level: Confidence level of the interval.
direction: A character string specifying the direction of the confidence interval. One of "y" (failure probabilities) or "x" (quantiles).
...: Further arguments passed to or from other methods. Currently not used.

Details

The basis for the calculation of these confidence bounds are the standard errors determined by the delta method and hence the required (log-)location-scale parameters as well as the variance-covariance matrix of these have to be estimated with maximum likelihood.

The bounds on the probability are determined by the z-procedure. See 'References' for more information on this approach.

References

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

Examples

Run this code

# Vectors:
obs <- seq(10000, 100000, 10000)
status_1 <- c(0, 1, 1, 0, 0, 0, 1, 0, 1, 0)

cycles <- alloy$cycles
status_2 <- alloy$status


# Model estimation with ml_estimation():
ml <- ml_estimation(
  x = obs,
  status = status_1,
  distribution = "weibull",
  conf_level = 0.90
)

ml_2 <- ml_estimation(
  x = cycles,
  status = status_2,
  distribution = "lognormal3"
)

# Example 1 - Two-sided 95% confidence interval for probabilities ('y'):
conf_fisher_1 <- confint_fisher(
  x = obs,
  status = status_1,
  dist_params = ml$coefficients,
  dist_varcov = ml$varcov,
  distribution = "weibull",
  bounds = "two_sided",
  conf_level = 0.95,
  direction = "y"
)

# Example 2 - One-sided lower/upper 90% confidence interval for quantiles ('x'):
conf_fisher_2_1 <- confint_fisher(
  x = obs,
  status = status_1,
  dist_params = ml$coefficients,
  dist_varcov = ml$varcov,
  distribution = "weibull",
  bounds = "lower",
  conf_level = 0.90,
  direction = "x"
)

conf_fisher_2_2 <- confint_fisher(
  x = obs,
  status = status_1,
  dist_params = ml$coefficients,
  dist_varcov = ml$varcov,
  distribution = "weibull",
  bounds = "upper",
  conf_level = 0.90,
  direction = "x"
)

# Example 3 - Two-sided 90% confidence intervals for both directions using
# a three-parametric model:

conf_fisher_3_1 <- confint_fisher(
  x = cycles,
  status = status_2,
  dist_params = ml_2$coefficients,
  dist_varcov = ml_2$varcov,
  distribution = "lognormal3",
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "y"
)

conf_fisher_3_2 <- confint_fisher(
  x = cycles,
  status = status_2,
  dist_params = ml_2$coefficients,
  dist_varcov = ml_2$varcov,
  distribution = "lognormal3",
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "x"
)

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