~ T Suppose you want to regress n observations Y ind G(i, Vi), i = 1,...,n,...

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~ T Suppose you want to regress n observations Y ind G(i, Vi), i = 1,...,n, where the means i = E[Y] are related to a set of regressors X by g(i) =x B, and the precisions vi are related to another set of regressors Z (more on that below). The gamma log-likelihood is given by (up to a constant that does not depend on or v): l(, V; y) = Vilog C(v;y) n (1 i=1 where () is the gamma function. - Yi Yi (a) Defining the i-th deviance component as -2/log +vi log Vlog (vi), Yi Yi - Hi d = -2 log - di show that, given di, the log-likelihood above can be written as a function of di and depends only on Vi. (b) Now, show that the distribution of d; belongs to the exponential family and so we can regress d; on Z with h(E[d;]) = zy. Identify the canonical parameter and the cumulant function b(0) of this distribution. = E[d] as a function of the (c) Using the cumulant function, find the mean Ti precision v. Using the fact that log vi > (vi) since vi > 0, show that Ti > 0, as expected. Here (x) = d(log(x))/dx is the digamma function. (d) Next, obtain the variance of d; as a function of the precision v. Now, using the fact that (vi) > v since v > 0, deduce that the variance of d is positive, as expected. The trigamma function is (x) = d(x)/dx. (e) Describe a numerical procedure to obtain maximum likelihood estimates of and y. In particular, how are you exploiting the fact that computing d; requires only yi and but not v? ~ T Suppose you want to regress n observations Y ind G(i, Vi), i = 1,...,n, where the means i = E[Y] are related to a set of regressors X by g(i) =x B, and the precisions vi are related to another set of regressors Z (more on that below). The gamma log-likelihood is given by (up to a constant that does not depend on or v): l(, V; y) = Vilog C(v;y) n (1 i=1 where () is the gamma function. - Yi Yi (a) Defining the i-th deviance component as -2/log +vi log Vlog (vi), Yi Yi - Hi d = -2 log - di show that, given di, the log-likelihood above can be written as a function of di and depends only on Vi. (b) Now, show that the distribution of d; belongs to the exponential family and so we can regress d; on Z with h(E[d;]) = zy. Identify the canonical parameter and the cumulant function b(0) of this distribution. = E[d] as a function of the (c) Using the cumulant function, find the mean Ti precision v. Using the fact that log vi > (vi) since vi > 0, show that Ti > 0, as expected. Here (x) = d(log(x))/dx is the digamma function. (d) Next, obtain the variance of d; as a function of the precision v. Now, using the fact that (vi) > v since v > 0, deduce that the variance of d is positive, as expected. The trigamma function is (x) = d(x)/dx. (e) Describe a numerical procedure to obtain maximum likelihood estimates of and y. In particular, how are you exploiting the fact that computing d; requires only yi and but not v?