joker (version 0.14.2)

Beta: 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 Beta,numeric
d(distr, x, log = FALSE)
# S4 method for Beta,numeric
p(distr, q, lower.tail = TRUE, log.p = FALSE)
# S4 method for Beta,numeric
qn(distr, p, lower.tail = TRUE, log.p = FALSE)
# S4 method for Beta,numeric
r(distr, n)
# S4 method for Beta
mean(x)
# S4 method for Beta
median(x)
# S4 method for Beta
mode(x)
# S4 method for Beta
var(x)
# S4 method for Beta
sd(x)
# S4 method for Beta
skew(x)
# S4 method for Beta
kurt(x)
# S4 method for Beta
entro(x)
# S4 method for Beta
finf(x)
llbeta(x, shape1, shape2)
# S4 method for Beta,numeric
ll(distr, x)
ebeta(x, type = "mle", ...)
# S4 method for Beta,numeric
mle(
  distr,
  x,
  par0 = "same",
  method = "L-BFGS-B",
  lower = 1e-05,
  upper = Inf,
  na.rm = FALSE
)
# S4 method for Beta,numeric
me(distr, x, na.rm = FALSE)
# S4 method for Beta,numeric
same(distr, x, na.rm = FALSE)
vbeta(shape1, shape2, type = "mle")
# S4 method for Beta
avar_mle(distr)
# S4 method for Beta
avar_me(distr)
# S4 method for Beta
avar_same(distr)

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 estimators 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.

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).

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.

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")