Assume that a continuous random variable U has the following probability density function:
f(u|alpha ,zeta )=(zeta ^((1)/(alpha ))e^((u)/(alpha )))/(alpha )
where alpha >0,zeta >0 and U_(1),dots,U_(n)f(u|alpha ,zeta )vec(x)_(1),dots,vec(x)_(n)vec(x)_(i)=(x_(i0),dots,x_(ip))zeta U_(i)pdff(u|alpha ,eta )vec(x)_(i)vec(eta )eta _(j)mu _(i)neta _(0)eta _(0)zeta alpha -infty .
Now assume that we observe independent response variables U_(1),dots,U_(n)f(u|alpha ,zeta ) and associated covariate vectors
vec(x)_(1),dots,vec(x)_(n), where vec(x)_(i)=(x_(i0),dots,x_(ip)). Here assume that zeta is a known constant.
(a) If we want to assume that the response U_(i) has pdff(u|alpha ,eta ), define the response and link functions for the linear
predictor vec(x)_(i)vec(eta ) under the standard generalized linear model with canonical link function. Hint: If you are not able to
obtain the canonical link, just make one up and carry through with the rest of the question as best as possible.
(b) Explain how one would interpret the parameter eta _(j) in terms of mu _(i) for the canonical link function.
(c) Assume that the only covariate in your model is an intercept term, i.e., a single vector containing the value 1 for all n
elements and there is one parameter, eta _(0). Derive the maximum likelihood estimator for eta _(0) assuming that zeta is known
and use this then find the maximum likelihood estimator for alpha .
Assume that a continuous random variable U has the following probability density function:
C1/aeu/a f(ua) a
where a > 0, > 0 and -o< u< log
Now assume that we observe independent response variables U.Un f(u,) and associated covariate vectors 1,...,n, where x, = (o,...,Xip). Here assume that ( is a known constant.
a If we want to assume that the response U; has pdf f(ua,), define the response and link functions for the linear predictor ; under the standard generalized linear model with canonical link function. Hint: If you are not able to obtain the canonical link, just make one up and carry through with the rest of the question as best as possible.
(b) Explain how one would interpret the parameter ; in terms of ;for the canonical link function
(c) Assume that the only covariate in your model is an intercept term, i.e., a single vector containing the value 1 for all n elements and there is one parameter, So. Derive the maximum likelihood estimator for Po assuming that ( is known and use this then find the maximum likelihood estimator for .
