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. A numeric vector indicating the probabilities $$p$$ of the $$B_p$$-lives (quantiles) to be considered. A character string specifying the bound(s) to be computed. Confidence level of the interval. 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"
)