asgn1-2010sp - STA 7249 Generalized Linear Models...

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STA 7249: Generalized Linear Models Assignment 1 1. This problem concerns the Inverse Gaussian distribution. Let Φ denote the standard normal CDF and consider the function F ( y ) = 0 , y 0 , Φ q λ y - 1 + y μ + e 2 λ/μ Φ - q λ y 1 + y μ , y > 0 . (a) Show that F has density f given by f ( y ) = 0 , y 0 , λ 2 πy 3 1 / 2 exp n - λ ( y - μ ) 2 2 μ 2 y o , y > 0 . (b) Show that the density can be written in exponential dispersion form. Identify: the canonical parameter θ and the dispersion parameter φ (in terms of λ and μ ); the cumulant function, b ( θ ); the canonical link; and the variance function for this model. 2. The idea behind this exercise is to walk you through a way to construct an exponential dispersion family and to derive some basic properties of such a family. The properties that you will derive (and particularly the results of parts 2e through 2g) can be derived directly from the form of the exponential dispersion distribution given in class, so the construction given here is arguably just a curiosity, though it is at least a fairly interesting curiosity. We begin with a review of some basic terminology and results. Let Q represent a probability distribution (i.e., a probability measure) on ( R , R ), where R represents the Borel subsets of R , and let Y be a random variable with distribution Q , i.e., P ( Y A ) = Q ( A ), A R . (Don’t worry too much if you don’t know about the Borel sets: suffice it to say that not all subsets A R are Borel sets, but you’re unlikely to come up with one that isn’t without working pretty hard at it.) The moment generating function (mgf) of Q (or equivalently of Y ) is defined to be M ( t ) = E ( e tY ) = R e ty Q ( dy ). Because e ty > 0, this integral is well-defined for all t , but it may be infinite. Let Θ = { t : M ( t ) < ∞} . Of course M (0) = 1 and thus 0 Θ is assured. (a) Show that Θ is a convex subset of R . The only convex, nonempty subsets of R are intervals (here we count a singleton { t } as an interval through the representation { t } = [ t, t ]), and hence Θ is an interval. It can be shown (see Billingsley, Probability and Measure , 3rd edn) that M ( t ) is dif- ferentiable at all t Θ (the interior of θ ) and that its derivatives can be computed by differentiating across the expectation, i.e., M ( j ) ( t ) = E ( Y j e tY ) for all t Θ . If 1
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0 Θ , then Y has finite moments of all orders given by E ( X j ) = M ( j ) (0) and its dis- tribution Q is determined (i.e., uniquely identified) by either of its moment generating function or its sequence of moments. Henceforth we will assume that 0 Θ .
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