pnbeta {DPQ}

Noncentral Beta Probabilities

Description

pnbetaAppr2() and its inital version pnbetaAppr2v1() provide the “approximation 2” of Chattamvelli and Shanmugam(1997) to the noncentral Beta probability distribution.

pnbetaAS310() is an R level interface to a C translation (and “Rification”) of the AS 310 Fortran implementation.

Usage

pnbetaAppr2(x, a, b, ncp = 0, lower.tail = TRUE, log.p = FALSE)

pnbetaAS310(x, a, b, ncp = 0, lower.tail = TRUE, log.p = FALSE,
            useAS226 = (ncp < 54.),
            errmax = 1e-6, itrmax = 100)

Arguments

x

numeric vector (of quantiles), typically from inside [0,1].

a, b

the shape parameters of Beta, aka as shape1 and shape2.

ncp

non-centrality parameter.

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].

useAS226

logical specifying if AS 226 (with R84 and R95 amendments) should be used which is said to be sufficient for small ncp. The default ncp < 54 had been hardwired in AS 310.

errmax

non-negative number determining convergence for AS 310.

itrmax

positive integer number, only if(useAS226) is passed to AS 226.

Value

a numeric vector of (log) probabilities of the same length as x.

Note

The authors in the reference compare AS 310 with Lam(1995), Frick(1990) and Lenth(1987) and state to be better than them. R's current (2019) noncentral beta implementation builds on these, too, with some amendments though; still, pnbetaAS310() may potentially be better, at least in certain corners of the 4-dimensional input space.

Author(s)

Martin Maechler; pnbetaAppr2() in Oct 2007.

References

Chattamvelli, R., and Shanmugam, R. (1997) Algorithm AS 310: Computing the Non-Central Beta Distribution Function. Journal of the Royal Statistical Society. Series C (Applied Statistics) 46(1), 146–156, for “approximation 2” notably p.154; doi:10.1111/1467-9876.00055.

Lenth, R. V. (1987) Algorithm AS 226, ..., Frick, H. (1990)'s AS R84, ..., and Lam, M.L. (1995)'s AS R95 : See ‘References’ in R's pbeta page.

See Also

R's own pbeta.

Examples

## Same arguments as for Table 1 (p.151) of the reference
a <- 5*rep(1:3, each=3)
aargs <- cbind(a = a, b = a,
               ncp = rep(c(54, 140, 170), 3),
               x = 1e-4*c(8640, 9000, 9560, 8686, 9000, 9000, 8787, 9000, 9220))
aargs
pnbA2 <- apply(aargs, 1, function(aa) do.call(pnbetaAppr2, as.list(aa)))
pnA310<- apply(aargs, 1, function(aa) do.call(pnbetaAS310, as.list(aa)))
aar2 <- aargs; dimnames(aar2)[[2]] <- c(paste0("shape", 1:2), "ncp", "q")
pnbR  <- apply(aar2,  1, function(aa) do.call(pbeta, as.list(aa)))
range(relD2   <- 1 - pnbA2 /pnbR)
range(relD310 <- 1 - pnA310/pnbR)
cbind(aargs, pnbA2, pnA310, pnbR,
      relD2 = signif(relD2, 3), relD310 = signif(relD310, 3)) # <------> Table 1
stopifnot(abs(relD2)   < 0.009) # max is 0.006286
stopifnot(abs(relD310) < 1e-5 ) # max is 6.3732e-6

## Arguments as for Table 2 (p.152) of the reference :
aarg2 <- cbind(a = c( 10, 10, 15, 20, 20, 20, 30, 30),
               b = c( 20, 10,  5, 10, 30, 50, 20, 40),
               ncp=c(150,120, 80,110, 65,130, 80,130),
               x = c(868,900,880,850,660,720,720,800)/1000)
pnbA2 <- apply(aarg2, 1, function(aa) do.call(pnbetaAppr2, as.list(aa)))
pnA310<- apply(aarg2, 1, function(aa) do.call(pnbetaAS310, as.list(aa)))
aar2 <- aarg2; dimnames(aar2)[[2]] <- c(paste0("shape", 1:2), "ncp", "q")
pnbR  <- apply(aar2,  1, function(aa) do.call(pbeta, as.list(aa)))
range(relD2   <- 1 - pnbA2 /pnbR)
range(relD310 <- 1 - pnA310/pnbR)
cbind(aarg2, pnbA2, pnA310, pnbR,
      relD2 = signif(relD2, 3), relD310 = signif(relD310, 3)) # <------> Table 2
stopifnot(abs(relD2  ) < 0.006) # max is 0.00412
stopifnot(abs(relD310) < 1e-5 ) # max is 5.5953e-6

## Arguments as for Table 3 (p.152) of the reference :
aarg3 <- cbind(a = c( 10, 10, 10, 15, 10, 12, 30, 35),
               b = c(  5, 10, 30, 20,  5, 17, 30, 30),
               ncp=c( 20, 54, 80,120, 55, 64,140, 20),
               x = c(644,700,780,760,795,560,800,670)/1000)
pnbA3 <- apply(aarg3, 1, function(aa) do.call(pnbetaAppr2, as.list(aa)))
pnA310<- apply(aarg3, 1, function(aa) do.call(pnbetaAS310, as.list(aa)))
aar3 <- aarg3; dimnames(aar3)[[2]] <- c(paste0("shape", 1:2), "ncp", "q")
pnbR  <- apply(aar3,  1, function(aa) do.call(pbeta, as.list(aa)))
range(relD2   <- 1 - pnbA3 /pnbR)
range(relD310 <- 1 - pnA310/pnbR)
cbind(aarg3, pnbA3, pnA310, pnbR,
      relD2 = signif(relD2, 3), relD310 = signif(relD310, 3)) # <------> Table 3
stopifnot(abs(relD2  ) < 0.09) # max is 0.06337
stopifnot(abs(relD310) < 1e-4) # max is 3.898e-5