Estimate Parameters of a Beta Distribution

Estimate the shape parameters of a beta distribution.

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.

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: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$

Value

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

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. 
  # (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
#> 

  #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
#> 

  #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)