# hw1_001 - 3. Adaptation of problems 2.8-2.12 in MN....

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HW 1 for Stat 7249 - Spring 2006 Due January 25 Reading in text for this assignment Chapter 2 Datasets none for this assignment 1. 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. 2. Gamma distribution: (a) Express the gamma distribution in the form of an exponential dispersion family, identi- fying the relevent components. (b) What is the canonical link? (c) Derive the mean and variance using the cumulant generating function.
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Unformatted text preview: 3. Adaptation of problems 2.8-2.12 in MN. Consider the following density, f x ( x ; , ) = (1-x 2 ) -1 / 2 (1-2 x + 2 ) B ( + 1 / 2 , 1 / 2) ,-1 x 1 . (1) for &gt;-1 / 2 and-1 1 (note: B ( , ) is the beta function.) (a) Show that for xed , the density given above is in the exponential dispersion family. Identify the relevant components and identify the cumulant generating function. (b) Suppose X 1 , . . . , X n are iid with density given above. Derive the maximum likelihood estimate for for xed . Show that the mle is independent of by showing that the Fisher information matrix for ( , ) is diagonal. [Hint: feel free to use results from problems 2.8-2.11 in MN]...
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