Let y_i, i = 1,...,n, denote n independent binary random variables.

a. Derive the log likelihood for the probit model Φ^€“1 [Ï€(x_i)] = ˆ‘_j β_j x_ij.

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

where z_i = 1 for the logistic case and z_i = Ï•(ˆ‘_j β̂_j x_ij)/π̂_i(1 €“ π̂_i) for the probit case.

Transcribed Image Text:

E(y – â,) z,xij = 0, j = 0,..., p,