# hw1 - fying the relevent components(b What is the canonical...

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HW 1 for Stat 7249 - Spring 2009 Due January 19 Reading in text for this assignment Chapters 3-5 Datasets none for this assignment 1. Problem 3.8 in DB 2. Cumulants and Cumulant generating function. Let M y ( t ) be the moment generating funtion for a random variable Y (we will assume it is ﬁnite for t in a neighborhood of 0). The cumulant generating function is deﬁned as K y ( t ) = logM y ( t ). Expressing K y ( t ) as a series expansion, K y ( t ) = j =1 K j t j /j !, the coeﬃcients K j are the cumulants. (a) Derive the cumulant generating function for the exponential dispersion family. (b) Connect the ﬁrst three cumulants to the ﬁrst three moments. 3. Gamma distribution: (a) Express the gamma distribution in the form of an exponential dispersion family, identi-
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Unformatted text preview: fying the relevent components. (b) What is the canonical link? (c) Derive the mean and variance using the cumulant generating function. 4. Consider Y i ∼ Bin ( m i ,π i ) ,i = 1 ,...,n . Assume the logit link is speciﬁed. Derive the likeli-hood equations, the asymptotic covariance matrix for ˆ β , and the deviance. 5. Show that if we specify the link function to be the canonical link, then the Newton-Raphson and Fisher Scoring algorithms are identical ( Hint : just need to show that expected and observed information matrices are the same for this case.)...
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