theta.md: Estimate theta of the Negative Binomial

Description

Given the estimated mean vector, estimate theta of the Negative Binomial Distribution.

Usage

theta.md(y, mu, dfr, weights, limit = 20, eps = .Machine$double.eps^0.25)
theta.ml(y, mu, n, weights, limit = 10, eps = .Machine$double.eps^0.25,
         trace = FALSE)
theta.mm(y, mu, dfr, weights, limit = 10, eps = .Machine$double.eps^0.25)

Value

The required estimate of theta, as a scalar. For theta.ml, the standard error is given as attribute "SE".

Arguments

y: Vector of observed values from the Negative Binomial.
mu: Estimated mean vector.
n: Number of data points (defaults to the sum of weights)
dfr: Residual degrees of freedom (assuming theta known). For a weighted fit this is the sum of the weights minus the number of fitted parameters.
weights: Case weights. If missing, taken as 1.
limit: Limit on the number of iterations.
eps: Tolerance to determine convergence.
trace: logical: should iteration progress be printed?

Details

theta.md estimates by equating the deviance to the residual degrees of freedom, an analogue of a moment estimator.

theta.ml uses maximum likelihood.

theta.mm calculates the moment estimator of theta by equating the Pearson chi-square \(\sum (y-\mu)^2/(\mu+\mu^2/\theta)\) to the residual degrees of freedom.

Examples

Run this code

quine.nb <- glm.nb(Days ~ .^2, data = quine)
theta.md(quine$Days, fitted(quine.nb), dfr = df.residual(quine.nb))
theta.ml(quine$Days, fitted(quine.nb))
theta.mm(quine$Days, fitted(quine.nb), dfr = df.residual(quine.nb))

## weighted example
yeast <- data.frame(cbind(numbers = 0:5, fr = c(213, 128, 37, 18, 3, 1)))
fit <- glm.nb(numbers ~ 1, weights = fr, data = yeast)
summary(fit)
mu <- fitted(fit)
theta.md(yeast$numbers, mu, dfr = 399, weights = yeast$fr)
theta.ml(yeast$numbers, mu, limit = 15, weights = yeast$fr)
theta.mm(yeast$numbers, mu, dfr = 399, weights = yeast$fr)