Fisher's Confidence Bounds for Quantiles and Probabilities

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

confint_fisher(x, ...)

# S3 method for wt_model
confint_fisher(
  x,
  b_lives = c(0.01, 0.1, 0.5),
  bounds = c("two_sided", "lower", "upper"),
  conf_level = 0.95,
  direction = c("y", "x"),
  ...
)

Arguments

x: A list with classes wt_model and wt_ml_estimation returned by ml_estimation.
...: Further arguments passed to or from other methods. Currently not used.
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.

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.
model_estimation : Input list with classes wt_model and wt_ml_estimation.

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

# 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
)

# Model estimation with ml_estimation():
ml_2p <- ml_estimation(
  data_2p,
  distribution = "weibull"
)

ml_3p <- ml_estimation(
  data_3p,
  distribution = "lognormal3",
  conf_level = 0.90
)


# Example 1 - Two-sided 95% confidence interval for probabilities ('y'):
conf_fisher_1 <- confint_fisher(
  x = ml_2p,
  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 = ml_2p,
  bounds = "lower",
  conf_level = 0.90,
  direction = "x"
)

conf_fisher_2_2 <- confint_fisher(
  x = ml_2p,
  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 = ml_3p,
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "y"
)

conf_fisher_3_2 <- confint_fisher(
  x = ml_3p,
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "x"
)