convexity of log of moment generating function Why is log of a moment generating function of random

Matilda Webb

Matilda Webb

Answered question

2022-05-13

convexity of log of moment generating function
Why is log of a moment generating function of random variable Z is convex? that is
log E [ exp ( λ . Z ) ]
My logic says since expectation is linear so it is in particular convex and exponential is convex therefore E [ exp ( λ . Z ) ] is convex but how to know if apllying log doesnt affect convexity?

Answer & Explanation

Aibling6n2re

Aibling6n2re

Beginner2022-05-14Added 16 answers

Holder's inequality says:
E ( U V ) ( E | U | p ) 1 / p ( E | V | q ) 1 / q
for any 1 < p , q < with 1 p + 1 q = 1
Put U = exp ( ( 1 θ ) λ 0 Z ), V = exp ( θ λ 1 Z ), p = 1 1 θ , q = 1 θ for any 0 < θ < 1, take logs of both sides, and you get
log E ( exp ( ( ( 1 θ ) λ 0 + θ λ 1 ) Z ) ) ( 1 θ ) log E ( exp ( λ 0 Z ) ) + θ log E ( exp ( λ 1 Z ) ) .
Jamir Melendez

Jamir Melendez

Beginner2022-05-15Added 2 answers

You can also argue the following:
Boyd Convex Optimization p. 106: generally, if f ( x , y ) is log-convex in x for each y C then
g ( x ) = c f ( x , y ) d y
is log-convex.
For the MGF:
M ( t ) = p ( z ) e t T x d z
clearly f ( t , z ) = p ( z ) e t T x is log-convex (actually log-affine) in t for each z.

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