Home /
Expert Answers /
Statistics and Probability /
problem-2-poisson-regression-you-observe-nonnegative-integer-valued-data-y-1-dots-y-n-and-an-pa273

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.