Beta: The Beta Distribution

Description

Density, distribution function, quantile function and random generation for the Beta distribution with parameters shape1 and shape2 (and optional non-centrality parameter ncp).

Usage

dbeta(x, shape1, shape2, ncp = 0, log = FALSE)
pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
rbeta(n, shape1, shape2, ncp = 0)

Arguments

x, q

vector of quantiles.

vector of probabilities.

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

shape1, shape2

non-negative parameters of the Beta distribution.

ncp

non-centrality parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

Value

dbeta gives the density, pbeta the distribution function, qbeta the quantile function, and rbeta generates random deviates.Invalid arguments will result in return value NaN, with a warning.The length of the result is determined by n for rbeta, and is the maximum of the lengths of the numerical arguments for the other functions.The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Source

The central dbeta is based on a binomial probability, using code contributed by Catherine Loader (see dbinom) if either shape parameter is larger than one, otherwise directly from the definition. The non-central case is based on the derivation as a Poisson mixture of betas (Johnson et al, 1995, pp.\ifelse{latex}{\out{~}}{ } 502--3). The central pbeta uses a C translation (and enhancement for log_p = TRUE) of Didonato, A. and Morris, A., Jr, (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios, ACM Transactions on Mathematical Software, 18, 360--373. (See also Brown, B. and Lawrence Levy, L. (1994) Certification of algorithm 708: Significant digit computation of the incomplete beta, ACM Transactions on Mathematical Software, 20, 393--397.) The non-central pbeta uses a C translation of Lenth, R. V. (1987) Algorithm AS226: Computing noncentral beta probabilities. Appl. Statist, 36, 241--244, incorporating Frick, H. (1990)'s AS R84, Appl. Statist, 39, 311--2, and Lam, M.L. (1995)'s AS R95, Appl. Statist, 44, 551--2. This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant. The central case of qbeta is based on a C translation of Cran, G. W., K. J. Martin and G. E. Thomas (1977). Remark AS R19 and Algorithm AS 109, Applied Statistics, 26, 111--114, and subsequent remarks (AS83 and correction). The central case of rbeta is based on a C translation of R. C. H. Cheng (1978). Generating beta variates with nonintegral shape parameters. Communications of the ACM, 21, 317--322.

Details

The Beta distribution with parameters shape1 $= a$ and shape2 $= b$ has density

f (x) = \frac{Γ (a + b)}{Γ (a) Γ (b)} x^{a - 1} {(1 - x)}^{b - 1}

for $a > 0$, $b > 0$ and $0 \le x \le 1$ where the boundary values at $x=0$ or $x=1$ are defined as by continuity (as limits). The mean is $a/(a+b)$ and the variance is $ab/((a+b)^2 (a+b+1))$. These moments and all distributional properties can be defined as limits (leading to point masses at 0, 1/2, or 1) when $a$ or $b$ are zero or infinite, and the corresponding [dpqr]beta() functions are defined correspondingly.

pbeta is closely related to the incomplete beta function. As defined by Abramowitz and Stegun 6.6.1 $B_{x} (a, b) = \int_{0}^{x} t^{a - 1} (1 - t)^{b - 1} d t,$ and 6.6.2 $I_x(a,b) = B_x(a,b) / B(a,b)$ where $B(a,b) = B_1(a,b)$ is the Beta function (beta).

$I_x(a,b)$ is pbeta(x, a, b).

The noncentral Beta distribution (with ncp $ = \lambda$) is defined (Johnson et al, 1995, pp.\ifelse{latex}{\out{~}}{ } 502) as the distribution of $X/(X+Y)$ where $X ~ chi^2_2a(\lambda)$ and $Y ~ chi^2_2b$.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, especially chapter 25. Wiley, New York.

Examples

Run this code

x <- seq(0, 1, length = 21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)

## Visualization, including limit cases:
pl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) {
  if(isLim <- a == 0 || b == 0 || a == Inf || b == Inf) {
    eps <- 1e-10
    x <- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1)
  } else {
    x <- seq(0, 1, length = 1025)
  }
  fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b))
  f <- fx; f[fx == Inf] <- 1e100
  matplot(x, f, ylab="", type="l", ylim=ylim, asp=asp,
          main = sprintf("[dpq]beta(x, a=%g, b=%g)", a,b))
  abline(0,1,     col="gray", lty=3)
  abline(h = 0:1, col="gray", lty=3)
  legend("top", paste0(c("d","p","q"), "beta(x, a,b)"),
         col=1:3, lty=1:3, bty = "n")
  invisible(cbind(x, fx))
}
pl.beta(3,1)

pl.beta(2, 4)
pl.beta(3, 7)
pl.beta(3, 7, asp=1)

pl.beta(0, 0)   ## point masses at  {0, 1}

pl.beta(0, 2)   ## point mass at 0 ; the same as
pl.beta(1, Inf)

pl.beta(Inf, 2) ## point mass at 1 ; the same as
pl.beta(3, 0)

pl.beta(Inf, Inf)# point mass at 1/2

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