# eqweibull: Estimate Quantiles of a Weibull Distribution In EnvStats: Package for Environmental Statistics, Including US EPA Guidance

## Description

Estimate quantiles of a Weibull distribution.

## Usage

 `1` ``` eqweibull(x, p = 0.5, method = "mle", digits = 0) ```

## Arguments

 `x` a numeric vector of observations, or an object resulting from a call to an estimating function that assumes a Weibull distribution (e.g., `eweibull`). If `x` is a numeric vector, missing (`NA`), undefined (`NaN`), and infinite (`Inf`, `-Inf`) values are allowed but will be removed. `p` numeric vector of probabilities for which quantiles will be estimated. All values of `p` must be between 0 and 1. The default value is `p=0.5`. `method` character string specifying the method of estimating the distribution parameters. Possible values are `"mle"` (maximum likelihood; the default), `"mme"` (methods of moments), and `"mmue"` (method of moments based on the unbiased estimator of variance). See the DETAILS section of the help file for `eweibull` for more information. `digits` an integer indicating the number of decimal places to round to when printing out the value of `100*p`. The default value is `digits=0`.

## Details

The function `eqweibull` returns estimated quantiles as well as estimates of the shape and scale parameters.

Quantiles are estimated by 1) estimating the shape and scale parameters by calling `eweibull`, and then 2) calling the function `qweibull` and using the estimated values for shape and scale.

## Value

If `x` is a numeric vector, `eqweibull` returns a list of class `"estimate"` containing the estimated quantile(s) and other information. See `estimate.object` for details.

If `x` is the result of calling an estimation function, `eqweibull` returns a list whose class is the same as `x`. The list contains the same components as `x`, as well as components called `quantiles` and `quantile.method`.

## Note

The Weibull distribution is named after the Swedish physicist Waloddi Weibull, who used this distribution to model breaking strengths of materials. The Weibull distribution has been extensively applied in the fields of reliability and quality control.

The exponential distribution is a special case of the Weibull distribution: a Weibull random variable with parameters `shape=`1 and `scale=`β is equivalent to an exponential random variable with parameter `rate=`1/β.

The Weibull distribution is related to the Type I extreme value (Gumbel) distribution as follows: if X is a random variable from a Weibull distribution with parameters `shape=`α and `scale=`β, then

Y = -log(X) \;\;\;\; (10)

is a random variable from an extreme value distribution with parameters `location=`-log(β) and `scale=`1/α.

## Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

## References

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

## See Also

`eweibull`, Weibull, Exponential, EVD, `estimate.object`.

## 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``` ``` # Generate 20 observations from a Weibull distribution with parameters # shape=2 and scale=3, then estimate the parameters via maximum likelihood, # and estimate the 90'th percentile. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rweibull(20, shape = 2, scale = 3) eqweibull(dat, p = 0.9) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Weibull # #Estimated Parameter(s): shape = 2.673098 # scale = 3.047762 # #Estimation Method: mle # #Estimated Quantile(s): 90'th %ile = 4.163755 # #Quantile Estimation Method: Quantile(s) Based on # mle Estimators # #Data: dat # #Sample Size: 20 #---------- # Clean up #--------- rm(dat) ```

EnvStats documentation built on Oct. 23, 2020, 6:41 p.m.