BiasedUrn-Univariate {BiasedUrn} R Documentation

## Biased urn models: Univariate distributions

### Description

Statistical models of biased sampling in the form of noncentral hypergeometric distributions, including Wallenius' noncentral hypergeometric distribution and Fisher's noncentral hypergeometric distribution (also called extended hypergeometric distribution).

These are distributions that you can get when taking colored balls from an urn without replacement, with bias. The univariate distributions are used when there are two colors of balls. The multivariate distributions are used when there are more than two colors of balls.

Please see vignette("UrnTheory") for a definition of these distributions and how to decide which distribution to use in a specific case.

### Usage

dWNCHypergeo(x, m1, m2, n, odds, precision=1E-7)
dFNCHypergeo(x, m1, m2, n, odds, precision=1E-7)
pWNCHypergeo(x, m1, m2, n, odds, precision=1E-7, lower.tail=TRUE)
pFNCHypergeo(x, m1, m2, n, odds, precision=1E-7, lower.tail=TRUE)
qWNCHypergeo(p, m1, m2, n, odds, precision=1E-7, lower.tail=TRUE)
qFNCHypergeo(p, m1, m2, n, odds, precision=1E-7, lower.tail=TRUE)
rWNCHypergeo(nran, m1, m2, n, odds, precision=1E-7)
rFNCHypergeo(nran, m1, m2, n, odds, precision=1E-7)
meanWNCHypergeo(m1, m2, n, odds, precision=1E-7)
meanFNCHypergeo(m1, m2, n, odds, precision=1E-7)
varWNCHypergeo(m1, m2, n, odds, precision=1E-7)
varFNCHypergeo(m1, m2, n, odds, precision=1E-7)
modeWNCHypergeo(m1, m2, n, odds, precision=1E-7)
modeFNCHypergeo(m1, m2, n, odds, precision=0)
oddsWNCHypergeo(mu, m1, m2, n, precision=0.1)
oddsFNCHypergeo(mu, m1, m2, n, precision=0.1)
numWNCHypergeo(mu, n, N, odds, precision=0.1)
numFNCHypergeo(mu, n, N, odds, precision=0.1)
minHypergeo(m1, m2, n)
maxHypergeo(m1, m2, n)


### Arguments

 x Number of red balls sampled. m1 Initial number of red balls in the urn. m2 Initial number of white balls in the urn. n Total number of balls sampled. N Total number of balls in urn before sampling. odds Probability ratio of red over white balls. p Cumulative probability. nran Number of random variates to generate. mu Mean x. precision Desired precision of calculation. lower.tail if TRUE (default), probabilities are P(X \le x), otherwise, P(X > x).

### Details

Allowed parameter values
All parameters must be non-negative. n cannot exceed N = m1 + m2. The code has been tested with odds in the range 10^{-9} \ldots 10^9 and zero. The code may work with odds outside this range, but errors or NAN can occur for extreme values of odds. A ball with odds = 0 is equivalent to no ball. mu must be within the possible range of x.

Calculation time
The calculation time depends on the specified precision.

### Value

dWNCHypergeo and dFNCHypergeo return the probability mass function for Wallenius' and Fisher's noncentral hypergeometric distribution, respectively. A single value is returned if x is a scalar. Multiple values are returned if x is a vector.

pWNCHypergeo and pFNCHypergeo return the cumulative probability function for Wallenius' and Fisher's noncentral hypergeometric distribution, respectively. A single value is returned if x is a scalar. Multiple values are returned if x is a vector.

qWNCHypergeo and qFNCHypergeo return the quantile function for Wallenius' and Fisher's noncentral hypergeometric distribution, respectively. A single value is returned if p is a scalar. Multiple values are returned if p is a vector.

rWNCHypergeo and rFNCHypergeo return random variates with Wallenius' and Fisher's noncentral hypergeometric distribution, respectively.

meanWNCHypergeo and meanFNCHypergeo calculate the mean of Wallenius' and Fisher's noncentral hypergeometric distribution, respectively. A simple and fast approximation is used when precision \geq 0.1.

varWNCHypergeo and varFNCHypergeo calculate the variance of Wallenius' and Fisher's noncentral hypergeometric distribution, respectively. A simple and fast approximation is used when precision \geq 0.1.

modeWNCHypergeo and modeFNCHypergeo calculate the mode of Wallenius' and Fisher's noncentral hypergeometric distribution, respectively.

oddsWNCHypergeo and oddsFNCHypergeo estimate the odds of Wallenius' and Fisher's noncentral hypergeometric distribution from a measured mean. A single value is returned if mu is a scalar. Multiple values are returned if mu is a vector. A simple and fast approximation is used regardless of the specified precision. Exact calculation is not supported. See demo(OddsPrecision).

numWNCHypergeo and numFNCHypergeo estimate the number of balls of each color in the urn before sampling from an experimental mean and a known odds ratio for Wallenius' and Fisher's noncentral hypergeometric distributions. The returned numbers m1 and m2 are not integers. A vector of m1 and m2 is returned if mu is a scalar. A matrix is returned if mu is a vector. A simple approximation is used regardless of the specified precision. Exact calculation is not supported. The precision of calculation is indicated by demo(OddsPrecision).

minHypergeo and maxHypergeo calculate the minimum and maximum value of x. The value is valid for Wallenius' and Fisher's noncentral hypergeometric distribution as well as for the (central) hypergeometric distribution.

### References

Fog, A. 2008a. Calculation methods for Wallenius’ noncentral hypergeometric distribution. Communications in Statistics—Simulation and Computation 37, 2 doi:10.1080/03610910701790269

Fog, A. 2008b. Sampling methods for Wallenius’ and Fisher’s noncentral hypergeometric distributions. Communications in Statistics—Simulation and Computation 37, 2 doi:10.1080/03610910701790236

vignette("UrnTheory")
BiasedUrn-Multivariate.
BiasedUrn.
fisher.test
# get probability