Beta {joker}R Documentation

Beta Distribution

Description

The Beta distribution is an 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 signature 'Beta,numeric'
d(distr, x, log = FALSE)

## S4 method for signature 'Beta,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)

## S4 method for signature 'Beta,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)

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

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

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

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

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

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

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

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

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

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

llbeta(x, shape1, shape2)

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

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

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

## S4 method for signature 'Beta,numeric'
me(distr, x, na.rm = FALSE)

## S4 method for signature 'Beta,numeric'
same(distr, x, na.rm = FALSE)

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

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

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

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

Arguments

shape1, shape2

numeric. The non-negative distribution parameters.

distr

an object of class Beta.

x

For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Beta. For the log-likelihood and the estimation functions, x is the sample of observations.

log, log.p

logical. Should the logarithm of the probability be returned?

q

numeric. Vector of quantiles.

lower.tail

logical. If TRUE (default), probabilities are P(X \leq x), otherwise P(X > x).

p

numeric. Vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

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. See Details.

na.rm

logical. Should the NA values be removed?

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.

The MLE of the beta distribution parameters is not available in closed form and has to be approximated numerically. This is done with optim(). Specifically, instead of solving a bivariate optimization problem w.r.t (\alpha, \beta), the optimization can be performed on the parameter sum \alpha_0:=\alpha + \beta \in(0,+\infty). The default method used is the L-BFGS-B method with lower bound 1e-5 and upper bound Inf. The par0 argument can either be a numeric (satisfying ⁠lower <= par0 <= upper⁠) or a character specifying the closed-form estimator to be used as initialization for the algorithm ("me" or "same" - the default value).

Value

Each type of function returns a different type of object:

References

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)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 0.8, 0.5)) # density function
p(D, c(0.3, 0.8, 0.5)) # distribution function
qn(D, c(0.4, 0.8)) # inverse distribution function
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

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

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# Estimator Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

v(D, type = "mle")

[Package joker version 0.14.2 Index]