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", "sev", "normal", "logistic",
    "weibull3", "lognormal3", "loglogistic3", "exponential", "exponential2"),
  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 confidence limits with respect to direction, i.e. limits for quantiles or probabilities.
upper_bound : Provided, if bounds is one of "two_sided" or "upper". Upper confidence limits with respect to direction, i.e. limits for quantiles or probabilities.
cdf_estimation_method : A character that is always NA_character. Only needed for internal use.

Further information is stored in the attributes of this tibble:

distribution : Distribution which was specified in ml_estimation.
bounds : Specified bound(s).
direction : Specified direction.

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 a unit is a right censored observation (= 0) or a failure (= 1).
dist_params: The parameters (coefficients) returned by ml_estimation.
dist_varcov: The variance-covariance-matrix (varcov) returned by 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 the bound(s) to be computed.
conf_level: Confidence level of the interval.
direction: A character string specifying the direction of the confidence interval. "y" for failure probabilities or "x" for 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 obtained by the delta method.

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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