# confint_fisher: Fisher's Confidence Bounds for Quantiles and Probabilities In weibulltools: Statistical Methods for Life Data Analysis

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```confint_fisher(x, ...) ## S3 method for class '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` Object with classes `wt_model` and `wt_ml_estimation` returned from `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 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).

## 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.

## 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.

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67``` ```# 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" ) ```

weibulltools documentation built on Jan. 16, 2021, 5:21 p.m.