VGAM (version 0.9-1)

hyperg: Hypergeometric Family Function

Description

Family function for a hypergeometric distribution where either the number of white balls or the total number of white and black balls are unknown.

Usage

hyperg(N = NULL, D = NULL, lprob = "logit", iprob = NULL)

Arguments

N

Total number of white and black balls in the urn. Must be a vector with positive values, and is recycled, if necessary, to the same length as the response. One of N and D must be specified.

D

Number of white balls in the urn. Must be a vector with positive values, and is recycled, if necessary, to the same length as the response. One of N and D must be specified.

lprob

Link function for the probabilities. See Links for more choices.

iprob

Optional initial value for the probabilities. The default is to choose initial values internally.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, vgam, rrvglm, cqo, and cao.

Warning

No checking is done to ensure that certain values are within range, e.g., $k \leq N$.

Details

Consider the scenario from dhyper where there are $N=m+n$ balls in an urn, where $m$ are white and $n$ are black. A simple random sample (i.e., without replacement) of $k$ balls is taken. The response here is the sample proportion of white balls. In this document, N is $N=m+n$, D is $m$ (for the number of ``defectives'', in quality control terminology, or equivalently, the number of marked individuals). The parameter to be estimated is the population proportion of white balls, viz. $prob = m/(m+n)$.

Depending on which one of N and D is inputted, the estimate of the other parameter can be obtained from the equation $prob = m/(m+n)$, or equivalently, prob = D/N. However, the log-factorials are computed using lgamma and both $m$ and $n$ are not restricted to being integer. Thus if an integer $N$ is to be estimated, it will be necessary to evaluate the likelihood function at integer values about the estimate, i.e., at trunc(Nhat) and ceiling(Nhat) where Nhat is the (real) estimate of $N$.

References

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

See Also

dhyper, binomialff.

Examples

nn <- 100
m <- 5   # number of white balls in the population
k <- rep(4, len = nn)   # sample sizes
n <- 4   # number of black balls in the population
y  <- rhyper(nn = nn, m = m, n = n, k = k)
yprop <- y / k  # sample proportions

# N is unknown, D is known. Both models are equivalent:
fit <- vglm(cbind(y,k-y) ~ 1, hyperg(D = m), trace = TRUE, crit = "c")
fit <- vglm(yprop ~ 1, hyperg(D=m), weight = k, trace = TRUE, crit = "c")

# N is known, D is unknown. Both models are equivalent:
fit <- vglm(cbind(y,k-y) ~ 1, hyperg(N = m+n), trace = TRUE, crit = "l")
fit <- vglm(yprop ~ 1, hyperg(N = m+n), weight = k, trace = TRUE, crit = "l")

coef(fit, matrix = TRUE)
Coef(fit)  # Should be equal to the true population proportion
unique(m / (m+n))  # The true population proportion
fit@extra
head(fitted(fit))
summary(fit)