Let yi, i = 1,,N, denote N independent binary random variables. a. Derive the log-likelihood for the probit model 1[????(xi )] = j ????jxij.

Let yi, i = 1,…,N, denote N independent binary random variables.

a. Derive the log-likelihood for the probit model Φ−1[????(xi

)] = ∑

j ????jxij.

b. Show that the likelihood equations for the logistic and probit regression models are

∑

N i=1

(yi − ̂????i)zixij = 0, j = 1,…, p, where zi = 1 for the logistic case and zi = ????(

∑

j ????̂

jxij)∕ ̂????i(1 − ̂????i) for the probit case.