How can one prove that $\log{\Phi(x)}$ is a concave function in x?
I tried taking second derivative, but so far it isn't helpful. I read a hint on my textbook that says it is easy to show its first derivative $\frac{\phi(x)}{\Phi(x)}$ is decreasing, but I have no clue on how I should work that out either.
Can anybody give me some hint? Thanks very much!
EDIT: Sorry if I didn't make it clear. $\Phi(.)$ refers to the cumulative distribution function of the standard normal distribution, while $\phi(.)$ is its derivative - the density function.