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# (Solved): Problem 2. Poisson regression. You observe nonnegative integer-valued data Y_(1),dots,Y_(n) and an ...

Problem 2. Poisson regression. You observe nonnegative integer-valued data

Y_(1),dots,Y_(n)

and an

n\times k

data matrix

x

where the

i

th row,

x_(i)^(TT)

, is a

k

-dimensional vector of covariates for individual

i

. Assume

Y_(i)?Pois(\lambda _(i))

independently and

\lambda _(i)=exp(x_(i)^(TT)\beta )

, with

\beta

an unknown

k

-vector of regression coefficients. If you don't like working with vectors or matrices, you may choose to solve this problem for

k=1

, in which case

\lambda _(i)=exp(x_(i)\beta )

with

x_(i)

a scalar covariate and

\beta

a scalar parameter. (a) Write down the likelihood function for

\beta

. Identify a minimal sufficient statistic. (b) Show that the likelihood has a unique maximum, and provide the equation that defines the MLE

hat(\beta )

. (c) Specify the approximate distribution of the MLE for

n

large.

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