DispersionEstimatorFunction
is an option for generalized linear model fitting functions that specifies the estimator for the dispersion parameter.
Details
- DispersionEstimatorFunction is an option for GeneralizedLinearModelFit, LogitModelFit, and ProbitModelFit.
- With DispersionEstimatorFunction->"PearsonChiSquare", the estimator is
where 
is the number of data points, 
is the number of parameters, and 
is the variance function for the distribution. - With DispersionEstimatorFunction->Automatic, the following estimates are used:
-
"Binomial" 1 "Gamma" 
"Gaussian" 
"InverseGaussian" 
"Poisson" 1 "QuasiLikelihood" 
- Non‐default values can be used to model overdispersion in "Binomial" and "Poisson" models.
- With the setting DispersionEstimatorFunction->f, the common dispersion is estimated by f[y,
,w] where y={y1,y2,…} is the list of observations, 
={
,
,…} is the list of predicted values, and w={w1,w2,…} is the list of weights for the measurements yi.
Examples
open all close allBasic Examples (2)
Fit a Poisson model using Pearson's 
to estimate the dispersion:
m = GeneralizedLinearModelFit[{...}, x, x, ExponentialFamily -> "Poisson", DispersionEstimatorFunction -> "PearsonChiSquare"]Compute the parameter covariance matrix using the default dispersion estimate:
m["CovarianceMatrix"]//MatrixFormSet the dispersion estimator when querying the covariance after fitting:
m = GeneralizedLinearModelFit[{...}, x, x, ExponentialFamily -> "Poisson"]Estimate the dispersion by the mean squared error:
m["CovarianceMatrix", DispersionEstimatorFunction -> (Mean[(#1 - #2) ^ 2]&)]//MatrixFormScope (2)
Define the estimate within the FittedModel:
data = {{1, 1}, {2, 1}, {3, 4}, {5, 6}};m = GeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Poisson"]m["CovarianceMatrix"]m["CovarianceMatrix", DispersionEstimatorFunction -> (2&)]m["CovarianceMatrix", DispersionEstimatorFunction -> "PearsonChiSquare"]m["CovarianceMatrix", DispersionEstimatorFunction -> (Mean[(#1 - #2) ^ 2]&)]data = {{1, 0}, {1, 1}, {1, 1}, {2, 0}, {2, 0}, {2, 1}, {2, 1}, {2, 1}, {2, 1}};logit = LogitModelFit[data, x, x]Estimate the dispersion by the sum of squared errors:
logit["EstimatedDispersion", DispersionEstimatorFunction -> (Total[(#1 - #2) ^ 2]&)]probit = ProbitModelFit[data, x, x]Estimate the dispersion by the mean squared error:
probit["EstimatedDispersion", DispersionEstimatorFunction -> (Mean[(#1 - #2) ^ 2]&)]Text
Wolfram Research (2008), DispersionEstimatorFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/DispersionEstimatorFunction.html.
CMS
Wolfram Language. 2008. "DispersionEstimatorFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DispersionEstimatorFunction.html.
APA
Wolfram Language. (2008). DispersionEstimatorFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DispersionEstimatorFunction.html