By Annette J. Dobson
Advent historical past Scope Notation Distributions concerning the traditional Distribution Quadratic types Estimation version becoming creation Examples a few rules of Statistical Modeling Notation and Coding for Explanatory Variables Exponential family members and Generalized Linear types creation Exponential relatives of Distributions homes of Distributions within the Exponential relations Generalized Linear Models Examples Estimation creation instance: Failure occasions for strain Vessels greatest probability Estimation Poisson Regression instance Inference Introduct. Read more...
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Extra info for An Introduction to Generalized Linear Models, Third Edition
6). If H0 is correct, then 1 (S0 − S1 ) ∼ χ2 (J − 1). σ2 If H0 is not correct, then (S0 − S1 )/σ 2 has a non-central chi-squared distribution. However, as σ 2 is unknown, we cannot compare (S0 − S1 )/σ 2 directly with the χ2 (J − 1) distribution. Instead we eliminate σ 2 by using the ratio of (S0 − S1 )/σ 2 and the random variable S1 /σ 2 with a central chi-squared distribution, each divided by the relevant degrees of freedom, F = (S0 − S1 )/σ 2 (J − 1) S1 /σ 2 (S0 − S1 )/(J − 1) = . 4, F has the central distribution F (J−1, JK− 32 MODEL FITTING 2J).
3) where s(y) = exp d(y) and t(θ) = exp c(θ). If a(y) = y, the distribution is said to be in canonical (that is, standard) form and b(θ) is sometimes called the natural parameter of the distribution. If there are other parameters, in addition to the parameter of interest θ, they are regarded as nuisance parameters forming parts of the functions a, b, c and d, and they are treated as though they are known. Many well-known distributions belong to the exponential family. 1. 1 Poisson, Normal and Binomial distributions as members of the exponential family.
This too can be checked graphically (see below). Additionally, the sum of squared residuals (yi − µi )2 provides an overall statistic for assessing the adequacy of the model; in fact, it is the component of the log-likelihood function or least squares expression which is optimized in the estimation process. Secondly, consider residuals from a Poisson model. Recall the model for chronic medical conditions E(Yi ) = θi ; Yi ∼ Po(θi ). In this case approximate standardized residuals are of the form ri = y i − θi .
An Introduction to Generalized Linear Models, Third Edition by Annette J. Dobson