# theta.mle: MLE of Theta for Negative Binomial In npreg: Nonparametric Regression via Smoothing Splines

## Description

Computes the maximum likelihood estimate of the size (theta) parameter for the Negative Binomial distribution via a Newton-Raphson algorithm.

## Usage

 ```1 2 3``` ```theta.mle(y, mu, theta, wt = 1, maxit = 100, maxth = .Machine\$double.xmax, tol = .Machine\$double.eps^0.5) ```

## Arguments

 `y` response vector `mu` mean vector `theta` initial theta (optional) `wt` weight vector `maxit` max number of iterations `maxth` max possible value of `theta` `tol` convergence tolerance

## Details

Based on the `glm.nb` function in the MASS package. If `theta` is missing, the initial estimate of theta is given by

`theta <- 1 / mean(wt * (y / mu - 1)^2)`

which is motivated by the method of moments estimator for the dispersion parameter in a quasi-Poisson model.

## Value

Returns estimated theta with attributes

 `SE` standard error estimate `iter` number of iterations

## Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

## References

Venables, W. N. and Ripley, B. D. (1999) Modern Applied Statistics with S-PLUS. Third Edition. Springer.

https://www.rdocumentation.org/packages/MASS/versions/7.3-51.6/topics/negative.binomial

https://www.rdocumentation.org/packages/MASS/versions/7.3-51.6/topics/glm.nb

`NegBin` for details on the Negative Binomial distribution
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```# generate data n <- 1000 x <- seq(0, 1, length.out = n) fx <- 3 * x + sin(2 * pi * x) - 1.5 mu <- exp(fx) # simulate negative binomial data set.seed(1) y <- rnbinom(n = n, size = 1/2, mu = mu) # estimate theta theta.mle(y, mu) ```