Convex Optimization · Problem 7.6

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · estimation-mean-variance

Problem

Estimation of mean and variance. Consider a random variable

x \in \mathbf{R}

with density

p

, which is normalized, i.e., has zero mean and unit variance. Consider a random variable

y = (x+b)/a

obtained by an affine transformation of

x

, where

a > 0

. The random variable

y

has mean

b/a

and variance

1/a^2

. As

a

and

b

vary over

\mathbf{R}_+

and

\mathbf{R}

, respectively, we generate a family of densities obtained from

p

by scaling and shifting, uniquely parametrized by mean and variance. Show that if

p

is log-concave, then finding the ML estimate of

a

and

b

, given samples

y_1, \ldots, y_n

y

, is a convex problem. As an example, work out an analytical solution for the ML estimates of

a

and

b

, assuming

p

is a normalized Laplacian density,

p(x) = e^{-2|x|}