Exponential family of distributions

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Exponential family of distributions Examples Norm
al Exponential Gamma Chi-square Beta Dirichlet Ber
noulli Binomial Multinomial Poisson Negative
binomial Geometric Weibull
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Bernoulli
3
The function G is called the link function. The
inverse of the function q is the canonical link
(makes the math easier).
(logit link)
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Normal
(identity link)
5
Poisson
(log link)
6
Negative-Binomial
(log link)
7
For over-dispersed count data
Minimum dispersion Maximum dispersion
In WinBUGS
Yi dnegbin(p,a) p lt- b/(b1) a Some
positive integer valued density b Some
non-negative continuous density
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The assignment given to 3 students was to count
the number of cars passing a particular
intersection on 4 different occasions. Student
1 2 3 12 21 28 11 17 34 18 26 39 7 14 19
5min 10min 15min
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model for(i in 112) yi
dnegbin(p,ri) p dbeta(1,1) pred1
lt- negbin(p,5) pred2 lt- negbin(p,10)
pred3 lt- negbin(p,15)
model for(i in 112) yi
dpois(lambdai) log(lambdai) lt-
b1b2ri for(i in 12) bi
dnorm(0.0,1.0E-6) log(mean1) lt-
b1b25 log(mean1) lt- b1b210
log(mean1) lt- b1b215 for(i in 13)
predi dpois(meani)
list(bc(0,0))
list(rc(5, 5, 5, 5, 10, 10, 10, 10, 15, 15, 15,
15), y c(12, 11, 18, 7, 21, 17, 26, 14, 28, 34,
39, 19))