Consider the univariate logistic model without intercept for the sample

(x1, y1),...,(xn, yn) with xi ∈ R; that is, ????′

x = ????x. Let p(x, ????) = e????x

∕(1 + e????x

) = P(y = 1).

(a) Show that An(????) = ∑n i=1

(yi − p(xi, ????))xi is decreasing in ????.

(b) Call ????̂

n the ML estimator. Assume ????̂

n > 0. Add one outlier (K, 0) where K > 0; call ????̂

n+1(K) the MLE computed with the enlarged sample. Show that limK→∞????̂

n+1(K) = 0. State a similar result when ????̂

n < 0.