proving that log likelihood of logit is globally concave

[abhijit]

Hi! I'm new in this group.
Can anyone help me proving that likelihood fn of logit is globally
cocave? I have tried it. But the calculation has become too messy. I
need the answer urgently.

[vadimcher]

logL = sum{ y*x'b - log(1 + exp(x'b)) }

The first part is linear.

The second one.
d-log(1+exp(x'b))/db = - exp(x'b)/(1+exp(x'b))*x = -
[1-1/(1+exp(x'b))]*x
dd-log(1+exp(x'b))/dbdb' = - exp(x'b)/(1+exp(x'b))^2*xx'

ddlogL/dbdb' = - sum{ exp(x'b)/(1+exp(x'b))^2 * xx' }
i.e. a sum of negative-semidefinite matrices.

Now, if matrix X=(x1, x2, ..., xn) has full rank and n>k, where k:
b=(b1, ..., bk), or if n>k and X is random and has full rank with
probability 1, then for every a there exists xi such that axi<>0,
hence, axixi'a'>0.

Finally, it's globally strictly concave iff X has full rank.

[abhijit]

Thanks for the help.
Abhijit