A show that negative binomial distribution belongs to

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(a) Show that negative binomial distribution belongs to the exponential family if r is known and define the canonical parameter, the functions a ( · ), b ( · ) and c ( · ; · ). (b) Obtain expressions for E ( Y ), Var( Y ), the variance function V ( · ), and the canonical link. (c) Suppose Y 1 , . . . , Y n are random samples from Negative Binomial distributions with pos- sibly different success probability π i but a common known r , i = 1 , . . . , n . Let x i = (1 , x i 1 , . . . , x ip ) 0 indicate a vector of covariates associated with each Y i , η i = β 0 + β 1 x i 1 + · · · + β p x i,p denote the linear predictor, and β = ( β 0 , β 1 , . . . , β p ) 0 be the vector of regres- sion coefficients. Instead of the canonical link function, the log link function is commonly used for the negative binomial distribution to link the mean μ i = E ( Y i ) to the linear predictor η i . Find the specific form of the score vector and information matrix for β under the log link function, and explain how you would obtain the maximum likelihood estimates of β . 2
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3. Consider a setting where Y i 1 , . . . , Y im i are m i independently distributed Poisson random vari- ables with Y ij Poisson( μ ij ), j = 1 , . . . , m i , i = 1 , . . . , n . Moreover, assume that the Poisson counts are generated by a time homogeneous Poisson process with μ ij = λ i t ij , where λ i is an underlying rate assumed to be common for Y i 1 , . . . , Y im i and t ij is the duration of observation leading to the count y ij , j = 1 , . . . , m
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