db: The db ("discretised Beta") distribution.
In dbd: Discretised Beta Distribution

Description Usage Arguments Details Value Note Author(s) See Also Examples

Density, distribution function, quantile function and random generation for the db distribution with parameters alpha, beta and ntop.

ddb(x, alpha, beta, ntop, zeta=FALSE, log=FALSE)
pdb(x, alpha, beta, ntop, zeta=FALSE)
qdb(p, alpha, beta, ntop, zeta=FALSE)
rdb(n, alpha, beta, ntop, zeta=FALSE)

`x`	Numeric vector of values at which the “density” (probability mass function) `ddb()` and the cumulative distribution function `pdb()` are evaluated. Normally these would be integer values between `nbot` and `ntop`, but they need not be. Note that `nbot` is 0 if `zeta` is `TRUE`, and is 1 if `zeta` is `FALSE`. A result of 0 is returned by `ddb()` for values of `x` that do not satisfy the foregoing criterion. A warning is issued by `ddb()` if any of the values in `x` are non-integer. See section Note for a little more information. Missing values (`NA`) are allowed; the corresponding results are `NA`.
`alpha`	Positive scalar. The first “shape” parameter of the db distribution.
`beta`	Positive scalar. The second “shape” parameter of the db distribution.
`ntop`	Integer scalar, strictly greater than 1. The maximum possible value of the db distribution.
`zeta`	Logical scalar. Should zero origin indexing be used? I.e. should the range of values of the distribution be taken to be `{0,1,2,...,ntop}` rather than `{1,2,...,ntop}`? Setting `zeta=TRUE` may be useful for example when the values of the distribution are to be interpreted as counts.
`log`	Logical scalar. Should logs of the probabilities calculated by `ddb()` be returned, rather than the actual probabilities?
`p`	Vector of probablilities (i.e. values between 0 and 1). The corresponding quantiles of the db distribution are calculated by `qdb()`. Missing values (`NA`) are allowed.
`n`	Integer scalar. An independent sample of size `n` from the db distribution is generated by `rdb()`.

In the predecessor of this package (hse versions 0.1-15 and earlier), the probability function of the distribution was calculated as dbeta(x/(ntop+1),alpha,beta)/ sum(dbeta((nbot:ntop)/(ntop+k),alpha,beta)) where nbot and k were set to 1 if zeta was FALSE, and nbot was set to 0 and k to 2 if zeta was TRUE.

However the probability function is calculated in a more “direct” manner, using an exponential family representation of this function. The Beta distribution is no longer called upon (although it still of course conceptually underlies the distribution).

The function ddb() is a probability mass function for an ad hoc finite discrete distribution of ordered values, with a “reasonably flexible” shape.

The pth quantile of a random variable X is defined to be the infimum over the range of X of those values of x such that F(x) >= p where F(x) is the cumulative distribution function for X. Note that if we did not impose the “over the range of X” restriction, then the 0th quantile of e.g. an exponential distribution would be \code{-Inf} (since F(x) >= 0 for all x) whereas we actually want this quantile to be 0.

Consequently qdb(p,alpha,beta,ntop) is equal to the least value of i such that pdb(i,alpha,beta,ntop) >= p. The set of values of i to be considered is {1,2,...,ntop} if zeta is FALSE and is {0,1,2,...,ntop} if zeta is TRUE.

For ddb() and pdb() vectors of probabilities.
For qdb() a vector of quantiles.
For rdb() a vector of length n, of integers between nbot and ntop, independently sampled from the db distribution, where nbot is 1 if zeta is FALSE and is 0 if zeta is TRUE.

In the predecessor of this package (hse, versions 0.1-14 and earlier) the density/probability function threw an error if any values of argument i were not in the set of integers nbot:ntop. In accordance with a suggestion from Duncan Murdoch this behaviour was changed so that the density/probability function returns 0 for such values. It also issues a warning if any of the values are non-integer. The criterion used for “non-integer” is that abs(i-round(i)) > sqrt(.Machine$double.eps). The new behaviour is analogous to that of other probability functions used in R, dbinom() in particular.

Rolf Turner r.turner@auckland.ac.nz

meDb() mleDb()

parz <- list(c(0.5,0.5),c(5,1),c(1,3),c(2,2),c(2,5))
for(i in 1:5) {
    p1 <- ddb(1:15,parz[[i]][1],parz[[i]][2],15)
    names(p1) <- 1:15
    eckslab <- paste0("alpha=",parz[[i]][1]," beta=",parz[[i]][2])
    barplot(p1,xlab=eckslab,main="db probabilities",
            space=1.5,col="black")
    abline(h=0)
    if(i < 5) readline("Go? ")
}
x <- c(-1.5,-1,-0.5,0,0.5,1,1.5)
ddb(x,2.5,1,5,TRUE) # Produces 0 for all but the 4th and 6th
                     # entries of x, and issues a warning.