# theta.md

##### Estimate theta of the Negative Binomial

Given the estimated mean vector, estimate `theta`

of the
Negative Binomial Distribution.

- Keywords
- models

##### 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)

##### 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.

##### Value

The required estimate of `theta`

, as a scalar.
For `theta.ml`

, the standard error is given as attribute `"SE"`

.

##### See Also

##### Examples

```
# NOT RUN {
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)
# }
```

*Documentation reproduced from package MASS, version 7.3-51.6, License: GPL-2 | GPL-3*