EnvStats (version 2.3.1)

ebeta: Estimate Parameters of a Beta Distribution

Description

Estimate the shape parameters of a beta distribution.

Usage

ebeta(x, method = "mle")

Arguments

x

numeric vector of observations. All observations must be between greater than 0 and less than 1.

method

character string specifying the method of estimation. 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 for more information on these estimation methods.

Value

a list of class "estimate" containing the estimated parameters and other information. See estimate.object for details.

Details

If x contains any missing (NA), undefined (NaN) or infinite (Inf, -Inf) values, they will be removed prior to performing the estimation.

Let \(\underline{x} = (x_1, x_2, \ldots, x_n)\) be a vector of \(n\) observations from a beta distribution with parameters shape1=\(\nu\) and shape2=\(\omega\).

Maximum Likelihood Estimation (method="mle") The maximum likelihood estimators (mle's) of the shape parameters \(\nu\) and \(\omega\) are the solutions of the simultaneous equations: $$\Psi(\hat{\nu}) - \Psi(\hat{\nu} + \hat{\omega}) = (1/n) \sum_{i=1}^{n} log(x_i)$$ $$\Psi(\hat{\nu}) - \Psi(\hat{\nu} + \hat{\omega}) = (1/n) \sum_{i=1}^{n} log(1 - x_i)$$ where \(\Psi()\) is the digamma function (Forbes et al., 2011).

Method of Moments Estimators (method="mme") The method of moments estimators (mme's) of the shape parameters \(\nu\) and \(\omega\) are given by (Forbes et al., 2011): $$\hat{\nu} = \bar{x} \{ [ \bar{x}(1 - \bar{x}) / s_{m}^2] - 1 \}$$ $$\hat{\omega} = (1 - \bar{x}) \{ [ \bar{x}(1 - \bar{x}) / s_{m}^2] - 1 \}$$ where $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i; \, s_{m}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$$

Method of Moments Estimators Based on the Unbiased Estimator of Variance (method="mmue") These estimators are the same as the method of moments estimators except that the method of moments estimator of variance is replaced with the unbiased estimator of variance: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$

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.

See Also

Beta.

Examples

# NOT RUN {
  # Generate 20 observations from a beta distribution with parameters 
  # shape1=2 and shape2=4, then estimate the parameters via 
  # maximum likelihood. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(250) 
  dat <- rbeta(20, shape1 = 2, shape2 = 4) 
  ebeta(dat) 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Beta
  #
  #Estimated Parameter(s):          shape1 =  5.392221
  #                                 shape2 = 11.823233
  #
  #Estimation Method:               mle
  #
  #Data:                            dat
  #
  #Sample Size:                     20

  #==========

  # Repeat the above, but use the method of moments estimators:

  ebeta(dat, method = "mme")

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Beta
  #
  #Estimated Parameter(s):          shape1 =  5.216311
  #                                 shape2 = 11.461341
  #
  #Estimation Method:               mme
  #
  #Data:                            dat
  #
  #Sample Size:                     20

  #==========

  # Clean up
  #---------
  rm(dat)
# }