8.7 Suppose Y = ( y1,…, yn)T are independent gamma random variableswith density f ( yi ∣ k, ????i) = xk−1exp(−yi∕????i)∕

(

????k i Γ(k)

)

, with unknown shape k, scale θi, andmean kθi = g(Σj????jxij), where g is a known function, ???? = (????1,…, ????J) are unknown regression coefficients, and xi1,…, xiJ are the values of covariates X1,…, XJ for unit i. For what choice of g does Y belong to the regular J + 1 parameter exponential family, and what are the natural parameters and complete-data sufficient statistics?