Estimate Parameters of a Beta Distribution
Estimate the shape parameters of a beta distribution.
ebeta(x, method = "mle")
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
|
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, …, x_n) be a vector of n observations
from a beta distribution with parameters
shape1=ν and shape2=ω.
Maximum Likelihood Estimation (method="mle")
The maximum likelihood estimators (mle's) of the shape parameters ν and
ω are the solutions of the simultaneous equations:
Ψ(\hat{ν}) - Ψ(\hat{ν} + \hat{ω}) = (1/n) ∑_{i=1}^{n} log(x_i)
Ψ(\hat{ν}) - Ψ(\hat{ν} + \hat{ω}) = (1/n) ∑_{i=1}^{n} log(1 - x_i)
where Ψ() 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 ν and
ω are given by (Forbes et al., 2011):
\hat{ν} = \bar{x} \{ [ \bar{x}(1 - \bar{x}) / s_{m}^2] - 1 \}
\hat{ω} = (1 - \bar{x}) \{ [ \bar{x}(1 - \bar{x}) / s_{m}^2] - 1 \}
where
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i; \, s_{m}^2 = \frac{1}{n} ∑_{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} ∑_{i=1}^n (x_i - \bar{x})^2
a list of class "estimate" containing the estimated parameters and other information.
See estimate.object for details.
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.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
Beta.
# 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)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.