Estimate Quantiles of a Beta Distribution

Estimate quantiles of a beta distribution.

eqbeta(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 beta distribution (e.g., ebeta). 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 to use to estimate the shape and scale parameters of the distribution. The possible values are "mle" (maximum likelihood; the default), "mme" (method of moments), and "mmue" (method of moments based on the unbiased estimator of variance). See the DETAILS section of the help file for ebeta 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 eqbeta returns estimated quantiles as well as estimates of the shape1 and shape2 parameters.

Quantiles are estimated by 1) estimating the shape1 and shape2 parameters by calling ebeta, and then 2) calling the function qbeta and using the estimated values for shape1 and shape2.

Value

If x is a numeric vector, eqbeta 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, eqbeta 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.

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. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

The beta distribution takes real values between 0 and 1. Special cases of the beta are the Uniform[0,1] when shape1=1 and shape2=1, and the arcsin distribution when shape1=0.5 and
shape2=0.5. The arcsin distribution appears in the theory of random walks. The beta distribution is used in Bayesian analyses as a conjugate to the binomial distribution.

Examples

  # Generate 20 observations from a beta distribution with parameters 
  # shape1=2 and shape2=4, 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 <- rbeta(20, shape1 = 2, shape2 = 4) 
  eqbeta(dat, p = 0.9)
#> 
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#> 
#> Assumed Distribution:            Beta
#> 
#> Estimated Parameter(s):          shape1 =  5.392221
#>                                  shape2 = 11.823233
#> 
#> Estimation Method:               mle
#> 
#> Estimated Quantile(s):           90'th %ile = 0.4592796
#> 
#> Quantile Estimation Method:      Quantile(s) Based on
#>                                  mle Estimators
#> 
#> Data:                            dat
#> 
#> Sample Size:                     20
#> 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Beta
  #
  #Estimated Parameter(s):          shape1 =  5.392221
  #                                 shape2 = 11.823233
  #
  #Estimation Method:               mle
  #
  #Estimated Quantile(s):           90'th %ile = 0.4592796
  #
  #Quantile Estimation Method:      Quantile(s) Based on
  #                                 mle Estimators
  #
  #Data:                            dat
  #
  #Sample Size:                     20

  #----------
  # Clean up

  rm(dat)