Skip to contents

Beta Distribution

Source: R/02_Beta.R
Beta.Rd

The Beta distribution is a absolute continuous probability distribution with support \(S = [0,1]\), parameterized by two shape parameters, \(\alpha > 0\) and \(\beta > 0\).

Usage

Beta(shape1 = 1, shape2 = 1)

# S4 method for class 'Beta,numeric'
d(distr, x)

# S4 method for class 'Beta,numeric'
p(distr, x)

# S4 method for class 'Beta,numeric'
qn(distr, x)

# S4 method for class 'Beta,numeric'
r(distr, n)

# S4 method for class 'Beta'
mean(x)

# S4 method for class 'Beta'
median(x)

# S4 method for class 'Beta'
mode(x)

# S4 method for class 'Beta'
var(x)

# S4 method for class 'Beta'
sd(x)

# S4 method for class 'Beta'
skew(x)

# S4 method for class 'Beta'
kurt(x)

# S4 method for class 'Beta'
entro(x)

# S4 method for class 'Beta'
finf(x)

llbeta(x, shape1, shape2)

# S4 method for class 'Beta,numeric'
ll(distr, x)

ebeta(x, type = "mle", ...)

# S4 method for class 'Beta,numeric'
mle(distr, x, par0 = "same", method = "L-BFGS-B", lower = 1e-05, upper = Inf)

# S4 method for class 'Beta,numeric'
me(distr, x)

# S4 method for class 'Beta,numeric'
same(distr, x)

vbeta(shape1, shape2, type = "mle")

# S4 method for class 'Beta'
avar_mle(distr)

# S4 method for class 'Beta'
avar_me(distr)

# S4 method for class 'Beta'
avar_same(distr)

Arguments

shape1, shape2

numeric. The distribution parameters (positive real numbers).

distr, x

If both arguments coexist, distr is an object of class Beta and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Beta instead.

n

numeric. The sample size.

type

character, case ignored. The estimator type (mle, me, or same).

...

extra arguments.

par0, method, lower, upper

arguments passed to optim for the mle optimization.

Value

Each type of function returns a different type of object:

  • Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

  • Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

  • Estimation: Returns a list. The estimator of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

  • Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Details

The probability density function (PDF) of the Beta distribution is given by: $$ f(x; \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \quad \alpha\in\mathbb{R}_+, \, \beta\in\mathbb{R}_+$$ for \(x \in S = [0, 1]\), where \(B(\alpha, \beta)\) is the Beta function: $$ B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.$$

Moments

The Beta distribution has the following known moments:

  • Mean: \(E[X] = \frac{\alpha}{\alpha + \beta}.\)

  • Median: Approximated by \(\frac{\alpha - 1/3}{\alpha + \beta - 2/3}\), for \(\alpha, \beta > 1\)

  • Mode: \(\frac{\alpha - 1}{\alpha + \beta - 2}\), for \(\alpha, \beta > 1.\)

  • Variance: \(Var(X) = \frac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}.\)

  • Skewness: \(\frac{2(\beta - \alpha)\sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2)\sqrt{\alpha\beta}}\)

  • Kurtosis: \(\frac{6[(\alpha - \beta)^2 (\alpha + \beta + 1) - \alpha\beta (\alpha + \beta + 2)]}{\alpha\beta (\alpha + \beta + 2) (\alpha + \beta + 3)}\)

  • Entropy: \(H(X) = \log B(\alpha, \beta) - (\alpha - 1)\psi(\alpha) - (\beta - 1)\psi(\beta) + (\alpha + \beta - 2)\psi(\alpha + \beta)\)

  • Fisher Information: The Fisher information matrix for \((\alpha, \beta)\) is given by: $$ I(\alpha, \beta) = \begin{bmatrix} \psi' (\alpha) - \psi' (\alpha + \beta) & -\psi' (\alpha + \beta) \\ -\psi' (\alpha + \beta) & \psi' (\beta) - \psi' (\alpha + \beta) \end{bmatrix} $$

Estimation

The parameters \((\alpha, \beta)\) can be estimated using:

  • Method of Moments: Solving for \(\alpha\) and \(\beta\) using sample mean and variance.

  • Maximum Likelihood Estimation (MLE): Numerical methods are typically used to maximize the log-likelihood function.

References

  • Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

  • Papadatos, N. (2022), On point estimators for gamma and beta distributions, arXiv preprint arXiv:2205.10799.

  • Oikonomidis, I. & Trevezas, S. (2025), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See also

Functions from the stats package: dbeta(), pbeta(), qbeta(), rbeta()

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 5

# Density function
df <- d(D) ; df(x) # df is a function itself
#> [1] 2.268945 0.107520 1.640625
d(D, x) # alternative way to use the function
#> [1] 2.268945 0.107520 1.640625

# Distribution function
pf <- p(D) ; pf(x)
#> [1] 0.3529305 0.9953280 0.7734375
p(D, x)
#> [1] 0.3529305 0.9953280 0.7734375

# Inverse distribution function
qf <- qn(D) ; qf(x)
#> [1] 0.2763397 0.5167578 0.3641161
qn(D, x)
#> [1] 0.2763397 0.5167578 0.3641161

# Random Generator function
rf <- r(D) ; rf(n)
#> [1] 0.14417973 0.42618777 0.04439554 0.37391245 0.50406517
r(D, n)
#> [1] 0.62327883 0.09493516 0.32794119 0.32852028 0.32141440

# List of all available moments
mom <- moments(D)

# Expectation
mean(D)
#> [1] 0.375
mom$mean
#> [1] 0.375

# Variance and Standard Deviation
var(D)
#> [1] 0.02604167
sd(D)
#> [1] 0.1613743

# Skewness and Excess Kurtosis
skew(D)
#> [1] 0.3098387
kurt(D)
#> [1] 0.04

# Entropy
entro(D)
#> [1] -0.4301508

# Fisher Information Matrix
finf(D)
#>            shape1      shape2
#> shape1  0.2617971 -0.13313701
#> shape2 -0.1331370  0.08818594

# Point Estimation - The e Functions

ebeta(x, type = "mle")
#> $shape1
#> [1] 2.942949
#> 
#> $shape2
#> [1] 2.535325
#> 
ebeta(x, type = "me")
#> $shape1
#> [1] 2.610526
#> 
#> $shape2
#> [1] 2.284211
#> 
ebeta(x, type = "same")
#> $shape1
#> [1] 2.819973
#> 
#> $shape2
#> [1] 2.467476
#> 

mle(D, x)
#> $shape1
#> [1] 2.942949
#> 
#> $shape2
#> [1] 2.535325
#> 
me(D, x)
#> $shape1
#> [1] 2.610526
#> 
#> $shape2
#> [1] 2.284211
#> 
same(D, x)
#> $shape1
#> [1] 2.819973
#> 
#> $shape2
#> [1] 2.467476
#> 
e(D, x, type = "mle")
#> $shape1
#> [1] 2.942949
#> 
#> $shape2
#> [1] 2.535325
#> 

mle("beta", x) # the distr argument can be a character
#> $shape1
#> [1] 2.942949
#> 
#> $shape2
#> [1] 2.535325
#> 

# Asymptotic Variance - The v Functions

vbeta(a, b, type = "mle")
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039
vbeta(a, b, type = "me")
#>          shape1   shape2
#> shape1 17.64848 26.56970
#> shape2 26.56970 51.39394
vbeta(a, b, type = "same")
#>          shape1   shape2
#> shape1 16.57719 24.96198
#> shape2 24.96198 49.01071

avar_mle(D)
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039
avar_me(D)
#>          shape1   shape2
#> shape1 17.64848 26.56970
#> shape2 26.56970 51.39394
avar_same(D)
#>          shape1   shape2
#> shape1 16.57719 24.96198
#> shape2 24.96198 49.01071

avar(D, type = "mle")
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039